Hello and welcome to this lecture. My name is equal to and I will be leading the second half of this module in microeconomic analysis.
I have to say that I normally teach parts of the game theory module instead of this one.
But one of our colleagues who used to be teaching this module is currently on medical leave, so I'm covering for him.
Stepping in to teach this module, I've prepared the contents for the first time, so if you have any suggestions, comments, feedback along the way, please just let me know. I will be very happy to see them and act upon them. To improve this module.
MSA economics is also very special for me because not long ago I was the director of studies for this program.
Before Ian took over in September.
That means that the most likely I've seen your applications when you apply to the program.
And you might have guessed that I.
Recommended an acceptance to application and that's why we're meeting here.
Right, so in normal times I think I would be welcoming you in a lecture Theatre. We would have the chance to chat a bit face to face and introduce to each other so.
While that would be nice, but given the.
In the current circumstances, that video lectures are the way we are doing now, so let's try to make the best out of it.
And I thought it might be a good idea to introduce myself to you a bit and.
So that's you know, my academic background, my research interests and other roles that I am on within the University.
Right, as I mentioned that my name is equal to.
And I'm a reader an economics at some management school.
I am also representing the University of Liverpool at the NWS HTTP partnership.
If you're interested in applying for a PhD, you might have heard of this collaboration and I'm.
I'm going to spend a bit more time there so that you know if you are interested in applying for a PhD from this partnership.
And then you know where to find the information etc.
I would start a little bit of introduction about myself. My background, as you might have guessed, that I am originally from China and this is.
A picture of the high school that I attended.
And it's a small city in China. It's got about 1 million people, so it's small compared to other Chinese cities.
And after my high school I went to Shanghai.
Undertones University for my undergrad studies.
I started engineering and business studies as well.
So after that I went to Germany. This is a very nice place, very beautiful place. Chordify book in bicycle. It is close to the Black Forest and also very close to Basel. the Sioux city.
And I started to finance their. Had enjoyed my time very much there. Afterwards I became interested in.
In a career in academia, so went on to pursue a PhD and I went to a document.
Many of you heard of document might because of the football team. They have brochure document.
And this is the city quite similar to Liverpool as well. Used to be very important. Dustrial city.
And they both got a very nice, very famous football team as well. So if you are a fan of football.
Altamont and Liverpool are very nice places to do a study or to be working.
Your career.
So right so after five book, I went to endorsement for my PhD. I specialized in game theory.
So here's a picture you might have seen in your previous text books. This is a game of income from incomplete formation, a signaling game. I said nice to be nice to include here, so we get the feeling that's what game theory look like.
And of course, I'm.
I didn't my picture to economics, so apply game theory to economic problems in particular in understanding how marketing works and also think about policy problems. If market is not working efficiently, what kind of policy we can think of to improve it a bit more on this topic.
So actually my main research interest is in the field called in Dutch organization. I'm not sure if you have had this subjects in your undergrad cities.
Because I, as I mentioned, it's not long ago I was the director of studies for the MSA cannot economics program?
And I know that we are so not all of you are coming from an economics background. You might have studied this something religious objects in finance or in.
Implanted studies.
And I also think that some of you may have a technical background in physics or engineering, so you may not have heard of the industrial conomiques or industrial organization this subject.
So in a nutshell. So we are looking at a market.
Normally a market has two sides. That one is the demand side and the other is supply side. On the demand side. Here we are looking at a whole consumers search for their product. What information is available to consumers?
Can they compare prices, prices for instance and how? How do they make decisions etc? So that's the demand side of the market.
And on the supply side of market, we look at the number of firms there, the size and their sizes, whether they are more or less equal or it's characterized by a few dominant firms, for instance. So this turns out to be very important to the efficiency of the market. So industrial connotation is basically looking at a particular particular markets and understanding.
Whether the market is functioning efficiently or not, and if it is not efficiently.
Not working efficiently.
What kind of public policy we can think of to correct it? For instance, if the market is dominated by a few very large firms, then we might be thinking about to investigate whether they have been abusing first to establish whether they have market power.
Not always that in a highly concentrated market that firms automatically have market power with. It's an empirical crushing to what extend the behavior of the firm is not disciplined by the market forces.
So that roughly in the idea of market power.
So first it's an empirical question to ask whether firms have market power.
And went whether they abuse it or not.
And the weather.
If if they abuse it toward what kind of damage they caused to the consumers?
If all these questions but understood clearly, then we can think about public policies, either to regulate them, to limit their ability in exercising their market power, or other policies. For instance, one might think of to split a big firm into smaller firms, for instance, so to restore competition in the markets.
So that's the central questions in Industrial Organization and basically what I do is to apply game theory to the behavior of the firms and derive equilibrium of the markets and a compared to the benchmark. Is that so we are interested efficiency back marks.
Right so.
Of course, the field has developed rapidly over the last few decades. We now understand quite a lot.
Of the market.
It loads about many markets, in particular those markets satisfied satisfied the standard assumptions that we make in economic theory. In particular rational consumers, for instance. So consumers are able to compare prices and always pick the one that is best for them, so they will create the disciplinary forces.
For for the firms, because the firms have to compete for them.
We understand quite a bit about such markets. Example, for example the beer market in standard. Now with standard markets.
Nothing too complicated and there have been quite a lot of empirical studies about it.
Another example for firm would be cinnamon roll breakfast cereal markets. For instance, there's a lot of study about that as well, so these are standard markets are functioning well, so we understand quite a lot.
What is less well understood is that.
When we have non standard consumer behavior for instance.
And one particular area that I'm particularly interested in order order be a strategic obfuscation.
This is an area starting from the observation that consumers do not make rational choices. Imagine that you are taking up a credit card for the first time. You go to the bank. You will be presented.
Quite lengthy documents with all the terms conditions in small prints. Even if you have all the time, it might be difficult and challenging for you to understand everything they say in this documents. So basically you have limited understanding of their offers and which.
Would hinder your ability to compare different offers. A credit card offering from Bank a compared to the offering from Bank B without understanding what exactly they're offering. It's difficult for you to compete down to two.
To compare.
So.
And without the ability without the ability for the consumers to compare products, firms do not compete that aggressively with each other. For instance, if we lower their prices or making more favorable conditions as it does not necessarily translate into.
Higher high demand, so in that sense that the competition is impact.
So this is a new area, quite new area so.
I'm particularly interested interested in and the main questions that I ask is that when consumers.
Exempting such biases or limited ability in making choices, how firms would react to that.
And to what extent the firms are able to reduce competition in the market and what kind of policy the government agencies can think of to restore efficiency to the markets etc etc. So it's quite extent exciting examples going to be as a financial products that I just mentioned can also think about the telecommunications contracts when you take up mobile phone. Often the contracts are quite.
Complicated to understand it as well and even even simple stuff like apples and orange. If if Apple is quoted in supermarket A for a price per unit like 1 Apple, how much you have to pay another supermarkets. They quote the price per.
Paquillo, for instance, how much you pay for 1K of apples.
And then consumers often find it difficult to compare them.
The products so that could also be studied within the framework that I just explained.
And here is a very nice cartoon that I would like to use to show to my audience or when I presented this topic.
So this I think most many of us can relate to that, in particular when we go to the bank taking up tipping out financial product, we are asked to sign sign on the dotted line without understanding what exactly we are signing for, right?
So.
I'm I've, I've done quite a few papers on this topic and look at the mechanisms that firms can do can implement it too too.
And make consumers even more confused.
I know them to study what are their incentives?
To do that?
And also look at Apollo public policies, various public policies and whether to educate the consumers so that consumer if you give for example, if you give a consumer consumer Calculator. Basically we have our.
Phone mobile phones now with us every time. So if you if we give consumers calculate every time they will make them eat will make it easier for them to compare prices quoted in different units, for instance. Or we also.
Can run educational program telling people what exactly many financial terms mean and so they can understand what they're signing up for when they're taking up a financial products. For instance, right? That's the educating the consumer side. We can also regulate the firms. Or we can.
You can outlaw certain practice that firms and the firm can use. For instance, we can limit the number of.
Small printer Britain points in the terms account terms and conditions, and we can require them to state everything in plain English. For example there so.
What, what? What are public policy can do?
So in a set of papers.
We compare this time different different policies and try to find the conditions under which which type of part, which type, which type works better than the other.
So I'm I've been approaching this problem both theoretically using game theory and also empirically using a lab experiments. So basically I invite subjects into the lab and performing the interactions that we would have.
Expect that the firms are playing and derive empirical.
Empirical evidence is.
So in a nutshell that I do both theory and experiments to study how markets functioning in particular when consumers facing limited cognitive ability or suffer from various biases. For example, if you're interested in finding out more about this line of research of mine, you can.
I look at my official school website and there you can see many of the papers that I published there. I have a few.
Working paper on this topic as well. It's not published, and the.
But I'm happy to to make it available if you're interested to learn. If you're interested in learning more about this topic, so please get in touch.
Also more recently.
I became interesting understanding big data, so consumer in particular consumer data, so not more often than ever. We are moving we are doing.
Most of all of our activities online, in particular with this pandemic we.
We buy things online for ecommerce was on the rise and I think the pandemic just give. Just give it another significant push so assessor assessor really rated the progress.
And so when we do a lot of things online, we leave digital footprints. We manage. Firms have lots of information about consumers.
Come, they they know our purchase history, for instance, who buys things from Amazon where they will know what products we buy, how many we often do, we buy it and.
Put our preferences are and etc etc said the lots and lots of data available to firms.
This data can be very useful and they can. They can be used to improve their services, improve the quality because they know better what consumers are expecting and what consumers.
Value, so they might use this data to improve their services. That's good. They also can use data to improve their design and research for new products. New services as well. But equally likely they can also use the data to do something less straight, less clearcut. For instance, like they may use the data to make a personalized offers.
Since they know what you like and what sort of income, probably your your consumption behavior, your your purchase history might give some hints about your your income, social status, etc etc. They might give you a personalized offer, so an offer that.
An offer for product you would not normally buy, for instance, for at the uniform at it surprised that available to everybody. But if they give you a special discount voucher, you might find it can.
Beneficial to by then, so this type of pricing behavior so and then it begs the question, how does it affect competition among firms? And whether whether consumers are harmed or the social welfare thing.
Is impact positively or negatively? And if it is negatively affected, what kind of policy we can think of, etc etc so?
It's new, it's a new phenomenon and the studies are not many yet. So there's a huge demand for rigorous and academic analysis on this topic. So if you are interested in pursuing a PhD, this might be a good topic to get into, because right now it's very hot, and so that means that whatever you do on this topic there will, you will get a head start.
Come on people to be interested in your research.
That's always a nice thing to have.
Right so.
Here and a few.
Interesting and interesting pictures, and there's a lot of discussion about data, how valuable they are. But before we can analyze it, we first have to understand what is data, how it is different from the traditional resources. For instance, people compare data with oil, water. What are the commonalities between oil and data, and what are the differences? Alright, so we have to understand those those things.
Before we can understand how they are going to affect the functioning of the market and then and then we can understand how we can best approach suggest the policy.
Policy changes.
Writes serve in a in a in a more recent paper I look at the practice of the firms using data to make a personalized offers to consumers.
And look at how they're going, how this type of practice can change the behavior of the firms, and how that's going to affect consumer Sir plus and social welfare etc etc so.
There's again so nice pictures explaining how much data firms are collecting and besides like firms like an Amazon, they can collect their data through their engagement towards their own customers. But as other firms are, for instance Unilever, for example, on not taking any legal responsibilities but just to add example, an ordinary firm Unilever.
Or Jaguar to local firms that are in this area.
Li Li can buy data from a 33rd party supplier, often a data broker.
And they can use the data to various insert either to improve their product design or to make personalized the office, for instance. So this data are widely available and eyes are collectors new, your own business or buy it from a third party.
But the idea is that in particular, if a firm is engaged in online business, this data can be used very efficiently.
Data from various sources can be combined and they have sophisticated algorithm to make a better offer for the consumer through this process, for instance.
And.
Right so.
In a series of papers, I have two papers on this topic understanding investigating the, the competition implications of firms accessing big data. One paper is particularly about this. The timing of the pricing of the firms because the traditional literature would assume that the firm will not assume, or indeed they were. Traditional literature will derive that firms normally would like to be a price.
Follower instead of price leader. But in reality we see lots of price leadership as well.
And then we in one paper we explained that the access to Big Data can explain this phenomenon and we invested competition policy implications of this type of behavior as well.
Another paper we look at this specifically in the the data broker industry. Those are the specialized firms are selling data to other firms to two firms are producing the end product or end of service.
Right, so this time, since if, as I said, these are the very, very interesting and Hot Knew topic, and there's lots to be done. And if you're interested in this area, I would think that this is a very good area for your PhD as well so.
Right, so that's that's broadly I do. Also other things that I'm basically I'm applying game theory to many, many different.
Areas, different applications. I'm also interesting contest, for instance of papers on that. I'm also interesting. Price, tenacity, electricity in home. This elastic demand affects the behavior of various standard models, for instance. So in a nutshell, that's anything that I can apply game theory to. I would be interested in so if you have something along that line.
And please also feel free to get in touch and if I'm not the best person to be discussing the research with you, I'm going to. I will help you to identify the right contact colleague with the Department.
Right, so that was my research.
Right now I'm talking a little bit about zero. I took over after my terms as director of studies for MSA Economics.
From this September I became his institutional lead.
For NWS HTTP, this is an SRC. The economics, economic and Social Research Council funded.
Training collaboration.
We have four partner institutions at the NWS SDP.
Q Lancaster, Liverpool and Manchester.
So the idea is that yes, I say well found.
A large number of students chips for PHD's and there will be allocated across this four partner institutions.
And.
The NWS DTP which stands for Northwest Social Science.
Doctor training platform.
It covers a range of subjects, a economics and social Sciences. Of course, if I'm interested in applying a PhD economics, you are very welcome to go to their website.
Which is.
Which you can find here NWST p.ac.uk and to find out some procedure.
Procedures here.
So that's a Liverpool if you apply a PhD.
At a Liverpool, and to be considered for the NWS SNDP scholarships then.
Normally you apply first to the management school for four PHD's as usual and in the following sources funding sources question. Then you put in N.W.A.
SSDP or other funding scholarships, for instance, like graduated teaching fellowship that is available also from school. So you make it clear there where your score is. Your sources of funding then you.
Colleagues will.
A well puts you together with other applications that apply for this scholarship.
Please be reminded that the school, the University of our management school has declined slightly earlier than the one advertised at NWS SNDP website. The idea is that we first do a screening of all applications and the strongest ones will be recommended for competition at the NWS HTTP.
Because this is a quite a prestigious surgeon student ship.
So very competitive and what is the funding?
The level of support is also very generous, so and also it's a very prestigious student ship, so if you are on this student ship, there's definitely something for your CV when you go on to look for a job, etc.
It signals the very high quality of your.
A research potential.
So if you are interested, I would definitely recommend you to go to website.
And find out more about the procedure and of course that's feel free to drop me a line and also represents other disciplines as well. But I also know economics Biggers before the.
The academic literal I was possibly wrapped for economics for Liverpool.
Right?
And this is also related that.
As I just I briefly mentioned before that, besides the NWS DTP student ship, the management school also have the Graduate Teaching Fellowships. The lab. It's also quite generous support. The only difference, not the only difference is that the main difference I can think of between those two studentships is at the Graduate Teaching Fellowship comes with certain number of hours of teaching.
This again can be rewarding as well and good training for your future job as well. So if you're interested, please go to the website. It's there for the.
For two 2021 entry it's already open so.
Here is a.
He and the web address that you can. You can go on to find out more.
And for economics, both the NWS HTTP and GTF share the same deadline. As I explained that the we we are going to screen for the strongest for the interprets SSDP student ship and.
That's why the boat both studentships have the same deadline and there is an ace of January. So if you're interested in applying for a PhD or with us, I think as a time to start before you start to prepare your application, would it be now?
And I would strongly recommend you too.
Pair up with a member of staff in the area that you're interested in. If you're interested macroeconomics, we have a set of excellent macroeconomics.
If you're interested in chromatics, we also have colleagues there. If you're interested in game theory or industry organization, I'm going to be more than happy to talk about it with you and also help you to understand more about the process and maybe also work together on the research project as well.
Right, so I mean research proposal that you need for your application. So if you're ever interested.
Please.
This is the right time to start preparation, so if you leave it to Christmas it will be too late for the January then like.
Alright, now was a little bit of background information about myself.
And my main research interests and as a service rose that I'm taking off at the moment.
By explaining the roles that I play.
For the NWS SDPI, also remind you that there are multiple.
Opportunities for applying for a PhD student ship. If you ever have such an interest, please just.
Feel free to get in contact and maybe I can help you a bit on that as well.
Alright, let's start with today's lecture.
Um?
First, a bit of housekeeping about this half of the module.
The plan is that I'm going to post record lectures of our canvas each week in week seven and two week 11.
From the next week week eight, you will also have a tutorial session which will be led by yet.
Um?
On the topics as is related to the lectures.
In Week 12 you are going to have a 24 hour assessment.
The amount of time you required to complete assessments will be around two to two slightly more than two hours, but you will be given 24 hours to complete it and submit it online.
And this format was decided.
In the summer when I was not aware of taking up this module, so there's not much I can do regarding the formats or changes or whatever.
So I just take it is as given I I operate under the instructions that have been given that it's.
The assessment will be 24 hours to complete in Week 12.
I will explain more about the formats and what you can expect from the assessments later.
In this module as we progress.
Regarding consultation.
My policy would be please feel free to email me for an appointment whenever you have a question or suggestion or feedback regarding the module. I also tried to run a weekly office hours and I will post the details on Canvas so this is a typo.
Because the two systems are running simultaneously, something scary confusing. Which platform we are using for for this module and it's going to be canvas, so I'm going to post office all details on Canvas.
Then, depending on the demand that if if demand is not that high, then I think the PT appointment system works better so it's more efficient if there are lots of demand for questions and officers obviously is more efficient. So same question. I can explain just once instead of 10 times.
So it will be depending on the demand for office hours.
And yet, and who is teaching? The tutorials will also hold office hours I think, and he will also be available through email. You can email him so the email addresses for him is this one and my email address is that one so.
Please do drop drop me a line if you if you have any questions.
And the book a letter we are going to use for this module is various microeconomic analysis. I think this is this is almost the perfect book for this module, which intend to give you a advanced yet accessible treatment to various topics on microeconomics. It seems quite outdated, quite old, but I think the classical results in Michael Conomiques.
Has not changed so much since the 90s, and even if you get to pick up a more modern textbook on this topic, you will find him more or less the same set of contents. So this still remains a very well written treatment of the subjects and variant. As you know, probably no, he isn't.
Well, well known he's a professor at UC Berkeley and also chief economist for Google is known for designing the auction mechanism for Google to who?
Who are able to extract so much revenue from advertisers by selling?
Selling slot.
On their search results so.
Right, so this is a very good book. Iaccessible public is proper treatment of the topic, do not.
Involved that much.
Unnecessary technical details. So all around or on the very good book.
My lecture slides are more or less based on this. Various chapters of this of this book, so I would recommend recommend you to find a copy which is available sooner library, so I think even during this time library is to open so you can get a copy of the book. So from the library.
And if you like another.
Ternative book for the for the for the module.
You might consider the sort of the classic.
Let's work on this topic in muscular Winston Green microeconomic theory. It's it's. It's a involved. It's more rigorous than variant.
But at the same time it may.
Not be that accessible in particular if you're not coming from a strong mass background or have studied extensively economics before.
From my experiments as previous director studies, I understand that some of you do not necessarily have taken very advanced economics module before, so this might be a little bit tough to get home. On the other hand, the variance book is quite successful even if you only had a very basic training economics.
And there's another alternative for the textbook, which if by June Rainy it's called advanced microeconomic theory, it's also very good book as well. So if we cannot get to any of the other two box and privacy, you can check if you can get ahold of this this book as well. So in each of the Champ lecture I will listed relevant to chapters in this three books. My lecture notes based on variant.
But you can also read relevant chapters in the other two books as well. Gives you more or less.
That covers more or less the.
The same contents, but with a different style. Perhaps also notation wise.
You have also need. You need to invest a little bit time to get used to the notation as well, so if you can get a variant book if you cannot, maybe you can find it alternatives as well so.
And.
It is my plan to cover several chapters in the variance book and picking up a monopoly in our company as well. So this topic actually I mean or relates to my research in that organization.
Here are two.
Specialized books on the function of the of the markets.
And it's again supplements the our treatments.
Of treatment of monopoly and oligopoly. Because our treatments are being a microeconomic analysis module only at we treated at a very general level. If you are interested in this topic in particular, how markets work at all firms behave how they compete with each other. What are the completion policy czar?
These two boxes are very good. The first one is.
I have covers also auctions and mechanism design. The second book by Martin Pipes and.
And he scores.
It's more on the oil topics, so in particular, if you're interested in finding out the prices or discrimination. For instance, in which we're going to cover and the chapter of a normal monopoly, but only at very general level if you want to find out more about price discrimination, for instance, you can find any of the two is toolbox to help yourself understand it better.
And the level of master mathematics. It depends on the topic.
But but I think by and large, if you are rather comfortable with this material provided by Martin Osborne for free on his website, then I think you are you OK with you? OK with this the rest of the module in terms of mathematics in there, and I think it's a good idea that you have taken the mass krammer training at the beginning of the semester. So you you should be.
Up to speed are already.
Right, so now we get onto the first proper lecture. I thought it might be a good idea to provide some continuity from Christians topic. I understand he talked about choices, utility, etc.
Galatea, but I understand that he was talking about the choices Ahmad Terministic sets of outcomes. Basically you have voices comparing Apple and Orange. You know whatever gets and the choices or outcomes of their choices are deterministic.
But in real life, we often face situations were the choices, the outcome, the choices is uncertain.
Wait, for instance, if we have some type of, have some money to to invest in the stock market.
If you invest certain months with a particular stock, how the price were going to develop or in the future is an uncertain event, right? If you know if you knew that.
How the stock price is going to behave then you if you knew for certain, then there's a easy way for you to get instantaneously. Very very rich. But the markets do not behave that way. Many of the assets are fundamentally uncertain. The return of these assets are fundamentally uncertain here. For example, I have a stock.
Stock price and project. I think in the in the in the span of a year and you can see that.
Here is a slight slightly dip.
Here they correspond to.
The news of the Pandemic COVID-19 breakout breaking out, and then since then the stock price has increased.
Overtime and also substantially so.
And you can guess which stock it is.
And if it, of course, I mean you can find the answer on the top left corner of the.
Of the of the picture.
But you can get. You can have a guess what kind of stalker has increased so much since the pandemic. Since the beginning of the pandemic.
Rent some.
Right so.
This is our motivation to continue.
The.
The choice the utility treatment that treat utility utility function, extra, etc. To extend this treatment to uncertain outcomes. So because often we face in real life will feature face decisions that are the outcomes are certain.
So.
To build up the theory of.
Of of how to choose among uncertain outcomes? We knew we needed the concept of lotteries.
Um?
Imagine that the choices facing the consumer takes the form of a Rotary. Rotary is denoted by this expression, which basically means that you have this price of X with probability P and a price of Y ways. If you take and if you accept this, if you choose this route ring, then you have a probability P getting the price of X and probability 1 -- P, getting the price of.
Of why?
And these prices can be anything. Can be money, can be bundle of goods consumption or even further lotteries. So in this way you can come if this why is another Rotary, for instance queue times a the price of a.
Compounded by 1 -- Q, The price of B. Then if if Y is.
Equal to that then?
The original lottery contains more price.
Since then, the simply these two rotary's, so this is rather flexible way of modeling. So our approach is not limited to two prices. It's a can accept any finite number of prices. So this is the lottery. Simple but powerful concept.
To conceptualize the.
Idea, we need some assumptions about the lotteries.
The first assumption is rather.
Simple, straightforward. It says that if.
Apart, another price has a probability of zero of happening, then it is as if.
The same as a distant deterministic outcome of X.
So as you can see that.
Here the probability of Y happening is 0. So which means that even if it's written as a lottery, but it's the same as a deterministic outcome of X.
And the second assumption.
Tells you that you're a consumer. Do not care which price is presented first.
And which prices presented second. For instance, if you presented X first or present X later than these two lottery should be equivalent for consumers.
Our people might argue that, um, it's rather straightforward, but people might argue that psychologically this may may make a difference.
But if in most circumstances this seems to be a non controversial assumption.
The second one.
Is that the the third one is called a reduction of compounded lotteries. It's me, it seems that it says that, as explained before, so if you have a lottery of Q probability cube having this lottery.
And probability Mormon school having having wine then?
It should be equivalent to these other actions that we accept it and without proof, so we should. We should consider it should be treated as the same as with probability P&Q having the outcome of X and probably 1 minus P 1 -- P * Q of having problems having doubts comma why so?
This is somewhat controversial since there's some evidence to suggest that the consumers treat compounded lotteries differently than one shots motorists, so this is substantial or something.
Once we have these three assumptions, then we understand that we can define the space of lottery.
They noted there was elf available to consumers that contains all the possible lotteries and that is facing the consumer.
Consumer is assumed to have preferences over the lottery space.
Meaning, given any two lotteries ** *** can choose between them and we assume the preferences are complete.
Reflexive and transitive transitive. So that's the standard concept that you have.
You are familiar with from the first half of the letter.
The lecture, First off the lecture the module, so we're just extending this to the space of lotteries instead of the only considering deterministic outcomes.
Well.
So, as I briefly mentioned before, the fact that's the Rotary has only two outcomes is not restrictive with the carbon reduction of compound lotteries.
So because the prices can be allowed to take further lotteries.
This is an example. I think I also explained it. For example, if we want to detect that there's a lottery with three prices, each of them having.
1 / 3 of the possibility of happening. Then we can write the in this way.
2/3 of the first lottery and one set of the one set of the Outcome Z, and with the first lottery being 1/2 of X and 1/2 of wine. Then in the end they should be.
Equivalent to.
With one over 3.
Of eggs and 1 / 3 or wine and 1 /. 3 of the refraining from using this sign. Because this is only only defined for combining 2 two prices, it has not been extended to.
To represent three prices, for instance, so it this lottery should be current to the original idea that we want to have an ordering with three three prices, each with equal probability of happening.
Right so.
Once we have this set of lotteries, and consumers are endowed with complete preferences, it's it's reflexive and transitive. Then it's not difficult to come up with a utility function words. Then we I believe that's a Christian has explained it. So once that preferences satisfies these type of set of conditions, then there exists a utility function such that if consumers prefer.
I ternative A to B if and only if for his utility of a is larger.
Then his utility of be, as that's clear so.
It's.
The fact is that we are considering lotteries in the space instead of determines outcomes does not violate the theorem. Say that that we have.
There exists a utility function.
And and and what? What is what is important, though? What is?
Hum.
More important to understanding is whether there exists a particular type of utility function such that it has to convenient property property of having this.
A relationship exists. The utility of a particular.
Lottery is equal to.
The probability of the first price. The utility of the probability times they are clear of the first price and probably times utility of the second price.
So this is not obvious that we can do this right? Because previously it's easy to find a utility function function too.
On the space on the space of.
Of lotteries, but it's not clear that we can do do this too, so to speak to we can put the probabilities out of the utility function and and solve it in this way, right? It's not clear that we can do. We can do this, and in what follows we're trying to find out what are the conditions that we have to satisfy for this for the existence of expected utility function.
And I think that's your previous studies. For instance, in a module of game theory, that's where you talk about a mixed strategy. For instance, right, so you allow a player to randomize over the center of pure pure strategies, and then you, when you do the usual Nash analysis lexical analysis, you can try to come up with the best response. You first need to understand how a player to maximize their utility.
We we.
We usually we usually assume that.
The players are endowed with the expected function, meaning that.
Although he is facing an uncertain choice environment and the one of the possible, there are multiple possible outcomes for one action he chooses, but we can summarize his preference using an expected utility, meaning that when you can do.
Do calculations in this way so.
We almost take it for granted that we can do it, but never sort of.
Question the underlying assumptions for us to be able to do it in this way, and here we are going to find out.
On top of the three axioms we we introduced before.
The three actions are.
With the 1st one second, the only sort of controversial one is that the reduction of compound lotteries keep that in mind.
And what are we required on top of the.
The list of assumptions that we make here.
So for the for the for us to find the expectantly ceron.
We need first.
For any given 3.
Um?
Lotteries from the lottery space.
Then this set.
So any?
Given three authories.
We can always find that we can combine the first one, the X&Y XYZ. Combine the first 2 and the third One Z, and then we ask.
Would P hear what probability here from zero to 1?
With this condition holds so X&Y. Combining excellent why and which the consumer would prefer to to Z.
We can do for any for any three given alternatives.
Three given lotteries.
You can always come up with this set right so?
Likewise.
We can find the Euler probabilities such that Z.
Is preferred to this combination right?
These could preferred to to this combination and the important assumption about these two sets are that they are closed.
And then you can see that this is this is a continuity property. So if you have a sequence of P that converges to a limit.
And why when you have this relationship holds for all the pieces in the sequence. Then this relationship should hold in the limits as well. So this is a continuity property.
So this turns out to be quite important.
For the existence of this utility function, so it puts on some structure on the utility on the.
On the preferences for the of the consumers.
The second.
Is that if you if we have extra Y equivalent, then it doesn't matter if we attach another component to it if X&Y.
I couldn't then, if in the same day they happen with probability P, for instance, only happen with the same property, then then.
If we attach another price together is also probably 1 -- P, then these two lotteries should be equivalent as well. But again this this points to something like the independence of irrelevant choices. Something like this so.
It's not straightforward, will have, will have to sync sync very hot, some quite hard to justify this assumptions, but this is what we we are the standard treatment then people understanding.
People wonder people do in standard treatment for having an expected utility function, one can.
Really think about whether this is a good example. You're not good, not so good, except this assumption indeed, by the end of lecture we're going to review some cases were they expecting tilt steering.
Sound does not hold right? So they exist.
Consumer choices that will violate Cizek special utility predictions so, and indeed it can be traced back to violations of one of the extras here.
As something three and three is just for convenience, it's not so bad, it's just.
To say that so.
Who is in the space of lotteries? There exists something called the best Rotary, and there's just something called the worst Rotary.
And.
Or other lotteries in the space. The consumers would have put it in between of the best and the worst. This is not controversial. Requires exist as long as can maximize over the preference, then you can do do this type of risk. Something is.
Is that is fine.
So.
The last one.
Is.
Now we make use of this best and worst lottery. We see that if these two combined two lotteries and the first lottery is preferred to the signatory, if and only if P is larger than Q. So as you can see this gradually towards.
Gradually telling us how we can find a.
Utility function that has the expected utility property.
This is.
This is one of the.
Well, that one of the assumptions that make it quite convenient, although one can notice that this is something can be derived from the other six actions.
So what is the expected utility theory and the expected utility theory and sassetta if the space of lottery and the preference over?
Over the of the space of lotteries satisfies assumptions. At 1203, the one that we first encountered and there's some things you want to you for. Then there exists a utility function U defined on the space of lotteries that says that it satisfies the expected utility property, meaning that there exists at least the existing actually function.
When we applied to the lottery, then the utility function gives us that this condition. So we can put it. We can put the properties out of the utility function.
Went so.
And do it in this way so.
I think you have my.
For instance, if you have lottery, have 1/2 of probability have probability of X and 1/2 of probability having the price of avoid, then this is the utility and this utility function tells you what's.
What's the level of utility you are going to having when you have this?
Lottery and it says that the Internet is the same as 1/2 of the utility.
When you have the deterministic price of X + 1/2 of the utility having the deterministic.
Um?
Utility of why so, right? So it's not obvious letter without proving it's not obvious there exists such a utility function, but when we prove after we offer proof, we can be confident that there exists such utility function then that provides us the justification when we talk about the mixed strategy equilibrium. For instance where we make use extensively expected this hearing.
And in many other areas, the phrase is a risk analysis. We sometimes also use expected utility of consumers to describe their choice behavior, and this is the fundamental.
Assumptions that we need for us to be able to carry out such an analysis.
The key to the proof is really the observation of how to construct the utility for a particular lottery. So for convenience, define the utility.
Of the best outcome as well, and the utility of the worst outcome of zero, you can always normalize it in this way, right? That's nothing substantial, so.
Now the construction of the utility function is crucial. Notice how the utility is.
Is is constructed saying now you give when you give him a lottery and arbitrary Rotary in the in space.
Your task is trying to find.
An equivalent lottery that is constructive.
Bye.
The best outcome and the worst outcome.
So any given lottery in the space you try to reconstruct there, trying to find an equivalent Robert Lottery that is.
That is.
Constructed by the two of the best outcome and the worst outcome and this probability that makes this equivalence work, is your.
Utility.
Right this so, this definition of the utility function and it's important to recognize that by this construct.
The utility is defined as the probability you attach to the best outcome.
Now the question is, does.
Can we always defined such a such a such a probability that you attach to the best outcome?
And the answer is yes by the continuity, so any.
So, forgiven Z and.
Remember, we have three given lotteries. Now you have we have Z&P&W, so you put Z BMW in the place over in the place of X&Y here next. And why here?
Think about X being the lottery or BY being the lottery of W. The worst case then.
Any number from zero to one must belong to one of the two sets, right? So? And because the sets are closed?
And because the interval of 0 to one is connected.
That means that we can always find.
Find a PC that makes this equivalence work that requires the continuity. If if we had not having the continuity. If we have the open sets here then there might not exist no PC such that's making the equivalence work. So first time first.
First question, where does it exist? Yes, check is it unique.
Re buy something for here.
Look, if we have two different PC that makes the equivalence work.
Then because they are.
Different so at least then one is larger than the other one. That meaning that these two lotteries will be one is preferred to the other than violets.
Assumption that suppose I equivalent to Z, right? So then it is unique, so it exists as unique.
So then what is left to check is? Does this utility function have the expected utility property?
And it is you build a utility function at all.
Wait to check.
Whether you have the expected utility property, we can think of.
This one.
So.
Given two lotteries X&Y, we.
I'm all two prizes, X&Y or two 2 lotteries. It doesn't matter whether we need to show, is that if I apply the utility function to this lottery, do I get to this second right right hand side, meaning that I can put in the P the probability is out of the utility function? Whether I can do it or not?
No, no to notice the following transformation. So first we replace X with this one. So we basically find the equivalent.
Uh.
Lottery that is constructed by the best and worst outcome we can do that we have seen before. While we can also.
Do it for why? Because it exists, right? We can exist the PX it exist the PY and it is unique so we can always do that. So that's the first step. The second step is.
By the assumption before where the players are, the consumers care about is the probability of certain prices coming. Suffer certain prices of emerging so you can put all the property together and which is going to be equivalent to this one.
To this one.
And then.
Because by definition PX is UX. Remember how we construct the utility function. The utility is defined as the probability that makes makes this.
Equivalent so PX is just the UX utility of X. Likewise pys utility or of UY.
And likewise for the four W.
It's here Y minus bu.
X -- 1 -- B * U Y.
So now we established this lottery is the same as is equivalent to this lottery. Then we apply the utility function to both sides.
Then we can have.
Utility of this one. The left hand side is equal to utility of their one.
Rights and utility of that one can you can notice that.
Because this is the construction of the best outcome and worst outcome.
Right, and by definition of the utility function, that number in front of the.
The best outcome is your utility, remember.
The number in front of the number in front of the best outcome. Best price is your utility.
So this is by definition utility of this.
Lottery is just the number in front of the best outcome, which is this number right? So in this way we have just assumed that the there this construction of utility function satisfies the expected utility property.
Right so.
This is rather clever.
Bye.
Constructing the traveling his way. Then you can shoot the demonstrates that it satisfies.
The property easily.
Now last question to check it that is it a utility function? Meaning that if I prefer A and prefer X / Y if and only if my utility of X is larger than my utility of, why?
So that if X is PX such that you can make this equivalence work.
And why is the PY? That makes this sequence work?
And.
By the by the 4th or something by U4, then X is preferred to Y, meaning that this.
This lottery is preferred to the second one, or if only if.
PX larger than PY, meaning if only if UXU of X large then Y of X, so check it is the utility function.
Right?
To summarize, to summarize.
With this
assumptions.
Excuse me
with this, something is given as before. We can always find utility function that satisfies the expected utility property.
We can we we? I think we should review the important assumptions, in particular, the importance of the continuity here guarantees the existence of such construction.
And.
And this the uniqueness of this construction, so these are quite important assumptions, in particular, right? The continuity is is is important.
So these are the.
So we have demonstrated that so we can find the expected utility.
That has this convenient property that that was substantially simplify our analysis in real applications for it right. For instance, if you look at the utility of particular assets or stock that you hope you're holding.
It's quite difficult actually conceptualized to what is utility of holding this.
And this talk.
And in particular, when you want to combine with other analysis. But if you can do the expected utility, then you.
You can you only need to know what is utility for deterministic outcome. Then you average out.
And gets your expected utility.
So this is substantially.
Simplifies alot of analysis data in real applications as we are doing so important.
Will you you have seen that the utility functions are normally not unique and when we talk about it's administered cause any monotonic transformation of utility function will be an utility function meaning satisfies that I prefer a.
Overby if and only for my utility of a is larger than my identity of of B.
However.
Not all such transformation.
Has the expected utility property.
Right so previously.
What is required is only preserve the ranking among alternatives. In the deterministic case.
But the actual utility has the meaning in the expected to ceron.
So.
Not all monotonic transformation will be well preserved. Expect utility property.
Actually there is a theorem says that an expected utility function is unique up to an affine transformation, meaning that if you apply apply an affine utility affine transformation to an expected utility function.
And to any utility function at satisfies expected.
Utility property.
Then your GNU utility function will also have the expected utility property.
In
Conversely, if a transformation if a monotonic transformation.
Preserves the expected utility.
Property.
Then this monotonic transformation has to be affine, so that's the two.
There are two sides of the result.
And the proof is actually rather easy. So first, for instance, given an affine transformation.
With a larger than zero to preserve the direction.
Have a project direction, then we need to show that this knew.
Utility function V will have the expected utility property if you.
Right, let me start again. The GNU utility function of V has the expected utility property. If you has the expected city property. OK, how do we? How do we do that?
The utility of the.
The V function of this lottery.
And this step is just.
The definition.
Of the following transformation, the definition because V function is equal to 8 times you proceed. This is applying this definition and becausw you has the expected.
Utility property we can.
Do this step right. The utility of this Monterrey equal to P times U X + 1 -- P * U Y so we can do. We also can.
Can decompose because here is C * 1 decompose this one into P and 1 -- P so that I can move I can identify one appear and be here to group them together and if we do that then we have tee times a * U + C right? So that's VX.
So basically what we're doing is P times.
Hey of UX.
*****.
Plus 1 -- P.
8 * U of y + y, so this is equal to.
This part.
Just by moving the the C or old then we can have this this property, but this is equal to this part.
OK, the direction is slightly different.
So.
These two parts equal to each other. Then we have we have demonstrated if you apply this V function to a lottery then you have the same property. The fan experience ability properly. So if you do an affine transformation of utility function that has the expected utility property, then you knew function will also have the expected utility for protein.
No.
The converse direction.
Let's FBR Monotonic transform transform of U.
That has the expected utility property. Now we need to show that F have to be affine.
So the observation is that the first we can.
Use that.
The first line we use that to you has the expected utility property that that's why we can write in this way we can put P out of the function of U, then constructing. This way, the second one is that we using the starting position that F preserves the expected utility function, preserve the expected utility property.
Then we can say that we can move P out of the F function because it has the excitedly function property. Then we can have this this equation because there's always hold then.
These parts.
Is identical to that to that part, so the left at the right hand side of the first function is always always hold to the right hand side of the second function, right? So this always hold.
No.
This always hold only when F is fine.
What we can do to show it is actually quite quite simple. You take it derivative of both sides with respect of P.
And taking the first time and then take it again, then you quickly I can quickly find that the F double line should be equal to 0 so that second load derivative of F should always be equal to 20. If you take the 1st order of discarding P to the on the right hand side and you get to F&FFUX and minus FQ, why you take it again? You get to 0, so the right hand side it's obvious searchable.
And on the on the left hand side. Then if you take it once and twice then you have you are left with.
Who is an expansion?
But this should.
Always equal to 0, but only when F is a fine. Do you have this condition right and then we have demonstrated that if Monica transformation preserves the expected utility property, it has to be a fine, right?
OK, So what is important to note is that with the traditional utility function with result uncertainty and all that anymore than the transformation will be an utility function. Here you need to be a fine to preserve the expected utility property.
Right then.
And this is a building block coming only, although only with two outcomes.
But it can be extended to any finite number of outcomes. For instance, if you have an.
And different prices then the same structure can be extended to that rather straightforwardly, because you can combine the component component component coupon. Then you have this.
This results meaning that.
When you're facing a lottery with N different prices, each with its own probability, then they expect is the expected utility.
The utility of this lottery will be the expected utility of the deterministic outcomes of access so.
So the utility of this lottery becomes the expected plenty of different outcomes. So and for any number of.
Of outcomes.
This can also be extended to a continuous variable, for instance.
Variable district and random variable X distributed on the continuous interval and with some technical details. One can also extend the hour the same essence that we have developed can be applied to the continuous case and likewise.
Your utility of this random variable.
Is equal to the expected utility.
The expected utility of different outcomes, right so?
In this with this notation.
Integration is just another way of saying the average so.
Right?
And the expectation of this random variable.
The expected utility.
Is the expected utility associated with this lottery? So basically it means that we can extend from 2 outcomes to M outcomes.
And even to an Infinity case, the continuous variable.
Now with this expected utility construction, it's easier for us to capture consumers.
Choice behavior in face of uncertain outcomes run at different territories.
One of the areas that we can meaningfully discuss. What does it mean by what does it mean?
How when we see a person an individual is a risk averse?
Oh, it's a risk loving, for instance, what does that mean? So we can give it give some structure to this type of discussions so.
So consider a lottery. Space consists only of gambles with money prices. If the consumers choice behavior satisfies the various required axioms, we are guaranteed with representation of utility that has the expected utility property.
This means that we can describe describe the consumers behavior. Overall, monetary gambles if we only know this particular utility function.
For example, to compute the consumers expect utility of a gamble with probability P having price X and 1 -- P price of why then we only look at?
This is not without loss of generality only. Look at this.
This is expected utility right?
If this expected utility is less than the utility of the expected outcome.
Remember, now we are looking at the excellent wise our money monetary payoff.
So we can also calculate the expected value.
Of this of X&Y, right so?
And what I ordered the seeds.
It seems that the expected utility of this lottery.
If the experiment if if the utility.
Because the utility of this surgery can be represented by the expected utility of this form. If it is less then.
The utility of the deterministic outcome at the level of the expected value of this lottery.
Then we see that the player or the consumer is risk averse.
If this is larger than 0.
Then we say it's a risk loving so intuitively.
This is average over there.
Utility and this is the right hand side is average of the money. So if this is smaller, averaging over the utility's, you get the less utility compared to less suited by Harrington Lottery compared to just the straightforward and give you the expected value of this utility deterministically, right?
So if you're having less value when you're faced with the lottery than you are risk averse. We do not like the.
The different probability of having different outcomes. You would rather have the expected value of the.
Of the lottery, right?
For instance, if you have 1/2 of a property getting 10 and it will have a preventive Cantin 20 if there are utility of this.
Lottery is less than the utility of.
Getting
1/2 of.
10 + 1/2 of 20.
Which is.
Well, half of 15 right? So know that.
The utility of the utility or 15 so.
That means that you would rather prefer the terministic of the expected value of the lottery, so you naturally you're risk averse. On the other hand, if it is.
You would prefer the lottery is that of the average or expected value of the lottery. Then your risk arriving you would rather take the gamble.
So with the utility function we can easily.
Represented this case of risk aversion or risk loving by the utility funding opportunity, and.
The monetary payoff, for instance if it's concave, meaning that if you have.
2.
X1X22 outcomes, for instance, if it is P.
1/2 of chance this happens and 1/2 of chance. This happens now. The expected utility.
Is this one?
Because this is you X2 and this is US1 and this is 1/2 of UX1 plus or half of you X2, so that's the expected utility.
And now if you.
Do an experiment value of the lottery is here.
So this is.
This point is you of.
1/2 of X1 plus.
1/2 of
X2. So this means that if.
If this one is larger than 1/2 of U, X1 plus 1/2 of UX two so this is risk averse, you would rather prefer the terministic outcome of the average value instead of this two.
The the average of this.
Utility rates, so this is and you can see this is the curvature. This concave concavity of this utility function measures gives you the idea. The difference between these two words so.
So, so to speak. If we, if we have another function.
His more.
Something like this. It's in this range. It becomes more risk averse, right? So because the difference between this these two is even larger. On the other hand, if you have.
A convex.
If you have a convex utility function, then the deterministic utility of the deterministic.
Outcome is lower than the average of the of the.
Two utilities and then. This is a risk arriving right because you would rather prefer a gamble instead of something deterministic, so.
With the same expected value.
So this quite nicely helps us to conceptualize risk aversion.
So I think we have done that.
So as I already alluded to that with ideally we need some.
Way to compare across individuals to be able to say which individual is more risk averse than the author. So it looks like the concavity of the of the utility function, which is measured by the 2nd.
2nd.
Order derivative of the utility function seems captures how concave utility functions and you can. We can use it to measure the.
The risk aversion of the of consumers and allow comparison cross across the different individuals.
However, think about a U.
And toys review. For instance, if you.
If you only take the 2nd order ative then you have.
You
I would do it step by step. The first alternative would be.
This one, the 2nd order two would be.
This also.
You know this is enough affine transform transformation when it's for interesting, we buy, we scale you by the factor 2 is still preserves exact utility properties, so it is also a good utility function. So if we only look at the 2nd order derivative.
It's.
It is not in very.
It's 2.
To the.
To the to the flying transformation. So good idea people. Clever people have thought about why don't we look at the values something like? Normalize it by the 1st order achievement because then the fine transformation we should be invariant to affine transformation in this way.
And this is indeed what is known as Arrow Pratt's measure of risk aversion, because normally risk averse, risk averse consumers have concave utility function.
You double prime will be negative, so we put a negative sign in front of it to make it a positive measure so we can compare if it's larger than the consumer is more risk averse. If it's smaller, it's Alaskan risk averse. Now we can see that the this error products virtual measure.
In this case.
Is defined by math.
With an affine transformation.
It's defined in this way because it's non normalized by the 1st order, first order Condition, 1st order it if and in which he is.
So because then I find transmission should capture the same behavior of the individual. So a good measure of risk aversion aversion should not discriminate too much bye bye.
By this affine transformation, right? Because it should in principle.
Should capture the same behavior of the consumers so.
It was a good idea to normalize the 2nd order condition 2nd order it if with the 1st order it if.
So.
And this is the definition of Arrow Pratt's measure of risk aversion.
What does it measure? It has something to do with the 2nd order.
Dirty rights, apart from being normalized so to understand awarded to measures perhaps.
One can think of think of a gamble represented by a pair of numbers XY and X2.
Will consumer get the extra 1 if?
Event E Happens Annex two if non E occurs, so only two outcomes. Then we can define the consumers acceptance set to be the set of all gambles. Their consumer would accept it as an initial level of wealth for his level was level of W for instance. This is initial level of wealth.
And W + 0 W plus zero. So let's deterministic, it's it's given.
Now we ask.
When it is.
It is equivalent to have this this one right?
The X1 and X2 can be either negative or positive. If XY is negative positive.
Assuming that both are desirable outcomes, XY next to a desirable outcome gives you increased utility.
Then if you think what XY being positive you you get more of X1. This means that how much X2 you are willing to.
To give up.
So that you are indifferent between these two.
Two outcomes, right? So basically we can draw the indifference curve like that.
So for instance here.
Here you get here.
X1 is.
X One for his disappointed here.
X1 is less than 0 minutes. You get more of X one, but the X2.
Is less than zero mean that you have to give up some X2 to be just to be indifferent right? Of course anything to that direction. You were strictly prefer to write. You would accept such a gamble.
And anything to the other side of the indifference curve. You were not accepted, and this what we call acceptance set.
So now if we consider if we want to.
I want to compare risk aversion across individuals.
It might be a good idea to compare, for instance, if another consumer.
Also, starting at the worst level of.
Also, starting the wealth level of of W and also have doing such a thought experiment can to find out the indifference curve so this indifference curve for for another consumer.
Gives you a smaller set of acceptance gambles, acceptable gambles right so the acceptance that of this blue consumer is strictly contained in the acceptance of the red consumer, meaning that.
The blue consumer might be needs more to accept. The fewer gambles and more risk covers right? So it is in this sense that we can compare risk aversion between consumers. So the reader consumer is more tolerant towards the risk. For instance, for the same amount of X, one part of amount of X1.
The blue consumer can blue consumer only willing to give up that amount. The red consumer is willing to give up more.
Of Of of the of X2 right so in this sense that the blue consumer is more risk averse, it does not like to.
Does not have the same level of risk tolerance as radican super. So now what is what is important that the curvature?
In the local area of this part, the curvature will be different, right?
So in the sense that in a sense, that.
It is more.
Giving a smaller set for the blue consumers through the curvature at this point should be should have a different as more risk averse it has. Then we can see that it will have a larger.
Larger and hold on. How can we do it? So first we need to identify that has something to do with indifference curve and has something to do with the curvature of the indifference curve. Now we can see whether.
The blue consumer should have a higher.
Oh, this is an error product measure because according to this measure, the more risk averse you are, the more.
The higher are you will be, and in this case a smaller set of acceptance leads to a higher risk aversion so.
OK.
So the way we can do it.
So this part is just explaining what.
What X2 of X1 means? That is an amount that you can you giving up or receiving, depending whether it's positive or negative.
Just to be indifferent. Alright, so this is your indifference curve.
With UX.
The.
Right so.
Now what we can do is to take the 1st order.
Condition so we can do.
With respect to X1.
Take the 1st order condition with respect to X1 and evaluated at 00. The original West level of W. Then we have this.
Condition.
And the slope of the acceptance sets is minus P / 1 minus pink when it's the 1st order.
It has something to do with the curvature, so we will. We would like to have a have a second order.
Directive as well, so we keep on, we do it one more time. We take the 1st order derivative respect to X one again to the left side hand side, then we are. We will end up with this condition right hand side there will be 0 anyway and we substitute the 1st.
Order Directive evaluated 0 into the.
Into the into the into this condition. Then we can see that this is.
At the indifference curve, the curve is second order.
Condition is no second order durative evaluated of the indifference curve evaluated at.
The origin, the initial wealth level is equal to this one. This is proportional to.
To the risk aversion. Imaginary practice measure might so a higher.
A higher X2 double prime.
Corresponds to a more curvature.
More curved in.
Office indifference curve, on the other hand, the red one is more flat movement than the blue one. If it is more flat.
Then this value will be smaller.
So because this is proportional to the arrow Pratt anyway, because it's just a normalized measure, so it's proportional to error at risk measure.
So the air pressure.
Images The flatness of the curvature of the indifference curve error at the local level of 0 and 0.
It is more flat as lower air pressure measure if it's more curved than it should have a higher air pressure measure and it's perfect. Corresponds perfectly to our interpretation of.
If it is have higher risk aversion then the acceptance set should be smaller and this indeed blue one is strictly contained in the right one, so it's all very nice so the story.
We can close the story or the concepts can be linked together. Acceptance being contained a curvature and the 2nd on the 1st order derivative and all that has a very nice concept.
So.
OK, so I think I will take a slight break here for five minutes just to drink some water. You can.
Well, for you you can pause anytime you want.
So yeah, I'm going to pause a bit.
And start again.
Alright, let's let's continue.
You notice that the previous discussion surrounded the local area of the initial value wealth level of W, so order acceptance, array acceptance, sets discussion and requires a local level of wealth. And often we can compare this not programmatically. We can. We can compare two individuals at the given values level and compare which individual is more risk averse than the other.
But sometimes the four applications, for instance macroeconomics, utility functions, we we sometimes it, would be nice and convenient to assume one consumer is always more risk averse and another consumer for all possible levels of wealth so.
Remember, this is a stronger assumption, so local risk aversion. We can always compare.
A consume A, maybe more risk averse than be at this level, but a maybe less risk averse compared to be at another level, so there's less component Matic so, but sometimes it would be nice to have global risk aversion, so compare the two individuals for all levels of wealth, so this is not.
So we can compare this, but.
But it's not always the case that one individual is always more risk averse than the other at all possible level of of worse levels, so.
This is sometimes there has to be an assumption to be made that.
Well, we see global risk aversion.
So.
They are there seems to be 3 different ways too.
To measure risk aversion, first is the error practice risk measure instead of only required only compared to at the local level of wealth of W. For instance, if we require this condition, this measure holds for all words levels. If if individual a.
This measure is always larger than individual beats a value for all possible level of W. Then it's natural to say.
A is more risk averse than B.
Alright, so another.
Idea of measuring risk averse. Last time we say that.
It's about the curvature of the concave utility function, so there not sensible way of saying that Agent A is more risk averse.
Then then then be.
Is saying that A's utility function is a monotonic and concave trend concave transformation of visual function we need, which means that the ace utility function is as if this is B and this could be.
More concave
more concave transformation of of B.
This is a for instance, so if this relation holds for all possible values of W, we are considering then.
They should be more risk averse than be right. Says it another way of interpreting this.
And a certain way of interpreting this is that we can ask the question that if if I gave you if a given individual, a small gamble with expected.
The expected value of this gamble being 0.
And.
How much so? How much damage? How much damage it makes for consumer to take on this small gamble, right? And remember, this is a random variable of expecting 0, so in four wells level, if you are risk neutral, you shouldn't be should be indifferent. You shouldn't be worried about the difference of a risk neutral.
Expected value.
Is as good as the expected utility so.
However, I'm for risk averse consumers.
Consumer A is more risk averse and consumer be. It's a means that the more damage it will do to this consumer compared to the other, and that's where their risk of premium idea comes from. So in other words that you would rather pay this amount of money to award the.
To award that avoid having the small gamble so.
And this is called a risk premium.
And if we have this ace risk, premium is always large beast. Risk variant mean that a always require more compensation to take on the small gamble.
Or means that it creates more damage for A to take on the game.
Take on the gamble, then beat us, then it does to be. Then a is more risk averse than B and if this holds for all possible values of wealth, then a is globally more risk averse than be right so.
So we have three ways of representing global risk aversion.
And the press. Ceron says that any of this measure is as good as the other. So indeed, these three measures are equivalent from first marriage. You can get the second image is the current a second major secondary is equivalent to a certain measure, and certain measure is equivalent to the first measure. So all three equally good.
So that's the main theorem there.
It's it's rather straightforward to carry out the proof, and I don't think that it had too much value that we do to hear. If you're interested, please read the relevant chapters in the book and you can follow goes through the roof and verify whether indeed you are convinced that all three measures are the same, so.
Right?
So with this representation that we can use this our new tool to analyze.
And airlines practical problems. The first one is the portfolio.
A problem they consider us.
Two period portfolio for problems involving two assets.
Along with a risky return, the one with the safe return. Or you can think about one.
Being fixed, income, instruments and other would be like stock market stock assets.
Since the rate of return on a risk acid is uncertain, we denoted by denoted by a random variable of theater R.
Let WB initial wells and a beta Dollar Mountain invested in the risky assets. The budget constraint implies that the W -- A is the amount invested in the show assets. For convenience, we assume the suas shore assets has a zero rate of return. Again, this is the normalization or is different in that.
Well, it's just normalization.
What is important is the relative. What is the return rate of return relative to the risky assets? And that's what it measures so.
In the case of second period, words can be written as.
If you invest A and it has the return of a tutor, then you have a * 1 plus are due to. Plus what do you have in the sure assets which gives you a zero turn, so you keep the value so which means that a + 8 times are tutor plus W -- 8 is equal to a times are tutor plus, so that's your.
Whilst level, remember this is the random variable, right? You're worse, could it be fresh, frustrating fluctuating depending on the realization of our.
So in previous terms, this is.
Also lottery right? You can then you apply the utility of.
Here is V.
Utility of this one.
You
so you are. You are having a utility.
You look at the utility of this.
Reading this.
Lottery that as the outcome is, is uncertain.
And because we know that they exist.
Utility function that has the expected tree property. Then we can see that this is the same as the expected utility.
Of this one.
Oh no.
They expected today from investing A.
Is equal to this one right? By changing a you are changing the distribution of the future wells. Next period wells and.
We can represent the utility of this kind of this random variable.
Why it's expected utility?
So the question is that how much you should invest.
The 1st Order condition is very simple, will take the 1st order condition with respect to a you have.
If you have W a + R theater, then it's just you prime.
And our tutor lets you take the first Directive directive.
Straight on the random variable, right?
And the expected turn carries over because it's this time just means that they're taking average across order.
Other realizations and conditional and taking into account the proper distribution of this random variable.
This is a first order condition ideally.
If it is equal to 0, then gives you the optimal choice of investment right? And the 2nd order derivative is obviously.
Right?
Negative.
Why? Because this is R-squared. Even if it's a, it's a random variable. There elevation can be different, but away if you squared, then even if the renovation is negative then square is positive, so it's always positive. The 2nd order conditioning.
And because of concavity of the tree function is less than 0, right?
Because it's concave.
So.
You have everything negative, then you take it.
The expectation.
Then of course this is negative, so the 2nd order condition is satisfied. So this is what the 1st order condition says.
And it gives you the the.
The optimal investment level.
No. First, we also be mindful of the corner solution. Is it possible that the consumer were invested? Nothing. What happens if equal to 0 invest nothing, then the derivative first alternative is your prime which is larger than 0.
And if the extractor return is higher than the true assets.
Then the investment the investor, even if being risk averse, he would still invest some amount into the risky assets, right? Because simply the marginal utility is higher at equal to 0.
If it is negative, for instance, if there is the really bad asset, not only it's risky, it also has a negative expected return.
Obviously a risk averse.
Consumer will put nothing into it, but it has no as it's positive, expected return. Then there isn't an end. Consumer will invest in a positive amount.
So let's assume this is much as zero then the 1st order condition gives us the interior solution.
Now we can ask the question, given the optimal choice, we can ask the comparative statics question. What happens if someone becomes richer?
Meaning higher W.
Does he invest more or less so? Then we take the 1st order condition as identity because in an equilibrium nor any optimal choice this has to hold. Then we differentiating with respect to W so that we can get.
Get DADW right so the effects of wells on a so taking.
The 1st order condition with stress W we will have this one. This is rather straightforward.
Doesn't involve too much for it, So what is important is that now we have this part identified. This is where we are interesting and by rearranging the quality.
Regent arranged the identity we can have.
The.
The direction of wealth on risky assets is given by this expression.
Note that the denominator is the 2nd order condition, right? So it's negative.
We have a minus sign in front of it, so how wealth is impacting optimum investment?
Has the same sign as this spot.
Which is the 2nd order condition times the random variable itself. The 2nd Order Directive of the function times the random variable itself.
So now we claim.
This part is positive, meaning that you increase your investment as your.
Wealth increases.
If RW is decreasing, mean it when you.
When you will see increases, you become the less.
Risk averse. Then you invest more. So invest more. If you become more.
Risk averse.
If I is increasing then you decrease your optimal investment. It's rather intuitive, right?
The proof is rather simple.
We we only take a One Direction. Let's see. Risk aversion is decreasing when you have more wealth than you become, the less absolute risk averse, right? It's also intuitive, like if a billionaire for the same gamble of $10, for instance gaining or losing $10 a billionaire might be more willing to take it because.
It doesn't make too much difference to his wealth level, but compared to someone who?
With the more wealth constraints.
Gaining $10 or losing $10 can be quite can have quite a substantial consequences, right? So I did so intuitively. If your wealth increases, you becomes less risk averse or more willing to take the risk.
We shoot for this direction because the same steps can be applied to show the other direction as well. It's not impossible for someone to to be more risk averse when it's when his wealth increases, right. Theoretically, it's still.
Possible rent, I mean, as long as our function is.
Is increasing then you have this case?
So first suppose R is our prime is less than zeros of decreasing risk aversion. Then we need to show this part is positive, the 2nd order.
Condition which is a negative for any second order. Conditioning negative 2nd order director is negative times a random variable and the product of this.
Only expectations should be like senzu.
Actually, it's a stronger than that. First consider.
Positive realizations of the random variable. Because R is decreasing.
Then when you have a positive reputation, then you're.
Risk aversion?
This part is.
Smaller than.
Your original level of risk aversion, meaning that.
You have the 2nd order second order derivative, the large then this level.
Meaning that it is less negative.
Compared to this one, right?
Because remember 2nd order condition, 2nd order is negative.
Because I is larger than though we can time both sides with our so the left hand side is less negative compared to the right hand side.
Likewise, if we consider negative realizations because it's the negative realizations stand or is increasing, be fearful, smaller values of worlds, the NYS increasing is larger than we have. This again by writing.
This letter inside out. Then we have this one, which means that.
Is.
Which means this is less than this one, which means that the left hand side is more negative.
Then the right hand side.
Now, because our tutor is less than 0, so we time both sides by a negative number. Then the sign of the incredulous which right? So switch.
So this is positive, so this is more positive than this one.
Because.
Both for positive relation of our in negative realizations so far.
We always have the same air quality.
Which, which means that if we take the expectation this.
Still hold it holds for negative radishes. It's hold for.
Hope positive relations, then of course, on average also holds right? So this holds.
But on the other hand.
This spot.
If we take it there.
First, we first we see that these parts.
Holds for both part of relations and or negative relations of our tutor.
Then what we do is that we take the expectation of both sides.
We don't have to know the extra distribution of this R random variable.
But if it holds when it's positive it holds when it's negative, then it of course hurts when you take the expectation with.
Now this part is because this is a constant basically minus RW. This is a constant and this is.
And this is the first order condition mean that if the consumer is optimally investing, this has to be 0 or expect terms. So if this is zero then this part.
The.
Right hand side is 0, so this is larger than 0.
This shows that.
If oil is decreasing, if oil is decreasing.
This part is positive. This part is positive then, meaning the more wealth you have, the more you invest so.
Very nicely in the concave utility function.
And all these other constructions allow us to do a meaningful analysis of asking the question how an individual adjust its investment level as his wealth or wealth increases.
Right, so that's the listening example.
Sometimes it's also helpful to think about the relative risk aversion.
Because as I said that if you have constant.
Absolute, risk averse. It doesn't seems to be too.
To reasonable because when you, when your wealth grow and you might.
Move might be more willing to take small gambles.
As before, small the same small absolute value gambles so.
So it makes sense to think about the relative risk aversion as well. And in many applications in finance macroeconomics this it is important to be able to to compare to have a concept of relative risk aversion.
So consider consumer with wealth W.
I suppose that is she is offered a gamble of the form of probability. Sure received X percent of our current wells and 1 -- P receive Y percent of her current wells.
Right, so now accent wise are in percentage.
In percentage, so basically means that if the consumer evaluates lotteries using expected utility and utility of this Rotary is P * U of X * X percent of W + 1 -- P utility of Y percent of W.
And.
So this are relative gambles and we can ask when at one consumer will accept more small relative gambles than another at a given wealth level. We can repeat the analysis that before and as in the absolute gamble case.
And in the end, we find that.
If we times the absolute measure of risk aversion by.
The wealth level. Then we have the relative.
Measure of risk aversion. This is also called an arrow Pratt.
Measure of relative risk aversion.
Well serve.
We would argue that that if a constant.
Attack.
Constant absolute risk aversion is not such a good idea, but.
But maybe it's good, but if you think about it.
Relative risk aversion is not so.
So bad to think about them.
A constant relative risk aversion and this will help.
Quite a lot of analysis of with the insurance applications to know that, so it's quite convenient. It often allows to allows us to derive closed form solutions with constant relative risk aversion, so it's it's quite convenient and it's not that important, implausible, right? So a gamble of 5% of your wealth you might be able to take it to it when you are rather poor and.
You are also going to sort of a St have same.
Eriska attitude when you're facing.
The same 5% gamble of your wealth when you are relatively rich. So in percentage terms is not so absurd to think that this risk attitude stays constant.
End for analytic analytical news reasons.
This would be very convenient in many applications.
Right so.
Sometimes even.
I mean, we often build a very complicated models with very variables and.
So in that case, in that type of situation is trackability. Tractability becomes very important, sometimes only have implicit solutions are not good enough. I sometimes we only have the existing tools and we know it. Their solution exists, but we cannot find it away because we are not able to characterize it.
So.
Sometimes we have to make a trade off, so we make it a less general so but more tractable.
For instance, one type of simplification we can make with utility function is that we ask ourselves when we can focus on.
Instead of the entire distribution of the random variable, we can focus on a few summary statistics of the random variable.
For instance, the 1st and the second or second moment of.
The random variable is insufficient. Well, such possibilities on focusing on on the mean and variance of random variable. It's gotta mean variance utility.
One way of of of trying to summarize the random variable with only the 1st and 2nd order set up first and second moments, meaning only by mean and variance would be to think about. Consumers might have the quadratic utility function W -- B square, right?
Right, so with this quadratic utility function then you can see the expected utility.
Equal to.
Special value of the random variable and B times. They expect very of the quadratic of the random variable. It is indeed only depending on the.
The mean and variance of this random variable. So the first time, the second moment completely summarizes the distribution.
In regards to an expected utility.
However, if you look at this utility function W -- B, W squared is equal to W.
1 minus PW, so it has.
O as root and 1 / B.
He said 1 / B has 1 / B as won't be there very close to 0 degrees to overby's another route so it it looks like this.
So this is a static utility function, so with W over here and utility over here. So in this area when W increases, the utility goes up nicely, but however.
When?
W is.
Larger than one.
Rising one over to be then when wealth increases, utility decreases, so it's not so nice, so it has this drawback.
Also it has increasing absolute absolute risk aversion so.
It's not so much desirable.
So if it's not about the utility function, can we think about that?
The.
Characteristics of the of the random variable which allow us to focus on the 1st and the 2nd moment.
One of such distributions would be the normal distribution, as as we can see here.
The normal distribution and the utility function takes the form of E -- E to the part of WN here.
Are nicely is a risk aversion.
Is the absolute risk aversion up?
And the expected utility of this.
Extend utility of this normally distributed wells level is equal to.
Um?
The probability density function of the normal distribution, and if you and this is actually using the moment generation function, it's very easy to drive.
To be.
To be this form and it's your eyes are you eyes are substituted a probability density function into this equation and try to integrate it out or using their results.
Of the moment moment generation function of their normal distribution, then it's straightforward to get this results and then you can see that.
This part.
Completely summarizes the.
Normal distribution and it depends only on the 1st moment and the second moment of the distribution.
And.
And this utility function.
OK, it's already here so.
Completely characterize, it would be the same as.
So we can only we can focus only on the 1st and 2nd moments of this distribution, and the characterized consumer behavior in this way, since rather convenient. And indeed this utility function.
Is widely used in many applications. Consumption decisions in the macro models and the world.
Where you would like to.
Do away with the complications of more than two moments of the of the district of the random variable, so quite useful, often years, quite tractable models as well.
Right, that's the expected utility function. So we when we when we know that.
Consumers decision facing the.
A lottery satisfies the set of axioms, then we know they exist and exist.
Utility function that has the expected utility property.
However.
There are.
Several problems with this type of characterization.
Without going into details here, I just would like to highlight two Paradox little violates.
That will give the contradiction to the expected utility theory and the first is quoted Ali paradox.
It goes as follows. So you are asked to choose between the following two gambles, A or B, 100% chance of receiving 1,000,000 to this deterministic at you for sure get to 1,000,000 be 10% chance to get 5 million, 89% chance to get 1 million and a 1% chance to get nothing before going any further. Please take a moment and and write down your preference.
Of these two.
Choices right?
I might prefer.
A.
To be.
Why, because a for sure I get a 1,000,000 be. I have 89% chance to get a million.
A 10% chance get 5 minutes nice but also have 1% chance of getting nothing. So I maybe do not like that. So I prefer AO over B.
And then we can compare another two lotteries. See you have 11% chance of 1,000,000 and 89% chance of nothing.
And DA 10% chance of 5,000,000 and 90% chance of nothing, right?
So it looks like a deep.
You only have sacrifice.
You only have 1% chance more to get nothing.
But with the rest of the 10% chance you get 5,000,000 instead of 1,000,000, right? So it looks like I might prefer D. / C, right?
I'm I don't know how would you choose among these two alternatives, but I think a majority people would have to choose a / B and D / C.
Know if these are.
If our decision can be characterized, it expected to see eternity.
So and then what are we basically saying is that we should prefer the first one if we prefer a over?
B then the utility of getting 100% of 1,000,000 is larger than 10% of 5 million, 89% of 1,000,000 and 1% of nothing to prefer. The first one, 8 / B.
If we rearrange in this expression, then we have zero 11% of getting 1 million larger than.
10% of getting 5 million and Joe and 1% are getting nothing.
Now.
This is a crucial step. We add both sides, the 80 percent 89% chance of getting nothing. Then we have.
11% of any 1,000,000 + 89% of nothing should be larger than 10% or 5,000,000 and 90% of nothing right? However, this this would say that you would have we would prefer C / D.
And actually, we just argue that most people would prefer D. / C, So.
The logic consequences of a preferred to be and together with expected true servant and then the logical consequences would be you people would predict would prefer C / D. However, in actuality many people prefer the oversea, so it's a violation.
So.
If we look at the.
The set of actions that we just talked about.
And.
The point is that whether we can add both sides.
Um?
89% of chance of getting nothing to an existing 2.
It.
If the axioms are satisfied, then we can do that, but.
In reality, one would question whether we can add.
In this way, as if we are adding some real numbers or.
Right, so it's it's not that straightforward, so one can go back into to argue that what happened in this paradox and which aspect. Which axioms are violated right so?
Let's one paradox called Ali Ali paradox.
Another paradox has something to do with whether we can represent.
Subjectively, the utility, the proper distribution so.
It's called a Ellsberg paradox.
It has nothing to do with ambiguity and.
This is a.
Ever since this paradox being proposed, people have been trying to modify the utility ceron return representation servant. Try to accommodate this progress has been made by the one cannot see that this issue has been fully addressed.
The Ellsberg paradox goes as follows there.
You are told to. You are told that an urn contains 300 balls. 100 of the boys are red and 200 of the birds are. Eyes are blue or green.
Let's hit you. Had you do not have any further information beyond that? These two 100 balls can be either blue or green.
You do not know. I don't know how many of which it even possible.
That 199.
Of the boys are blue and one is green, or vice versa, or indeed 100 balls. Blue and another 100 would be green so but I do not have this. Same to information you only know that the $200 our eyes are blue or green.
The gambles are you receive.
1000 if the ball is red and you receive 1000 if the ball is blue.
Which gamble do you prefer?
Many people would prefer a over beef, right?
Because you know that 1 / 3 of the chance you get 1000.
But blue it's difficult for you to come up with a number with it, right? It could be anything from.
From almost
and.
1 / 3 Soccer 302.
199 / 300 is at all possible situation. Suppose that there are green blue balls indeed.
It can be anything ranging from this probability. On the other hand, if it's red, you know that this is going to be 100 / 300, so you know the number is there, so there's no ambiguity, so to speak. Here the proper distribution is unknown, so.
Great and many people do not like Rambo.
Richie and they would prefer a / B. OK, that's fine.
But are considered now falling into gamblers and you receive 1000 if if the ball is not red, so it's either blue or green. If it's not red.
And see if it is not blue.
Right?
And many people would prefer C / D because again you have you are comfortable come up with proper distribution because you know that you have two.
Over 3.
The chance getting this 1000 with Blue again the property can be all over the place tonight so.
Oh, it's not blue eyes, green or red. That could be either all over the place. So many people would prefer C 2D, so a / B and C of C2D right now. Let's look at what are the logical consequences would believe that we can represent.
These events using probability.
Subjective probability, right?
So.
So let's all be the events that the boys read the.
Negative are to be the events at the ball is not a red and a B and the negative be not be.
And for blue as well so.
This I think we more or less agree.
That
PR should be longwise Pete.
Of not all right? So this is the is not very contentious to say that it's either red or not red, and the probability of these two events should start up to one and normalize.
Zero payoff to 0 for convenience than a preferred to be mean. Half the probability of our Red Times 1000. The utility of heavy 1000 should be larger than.
Probability of having brew of U1000 right?
So which means that probability are.
Has to be large then probability or be.
If we believe that we can use the expected utility theorem.
Of course this is 1 minus. You have 1 -- P.
Our.
And.
Almost PR.
Of zero, of course. Here you have 1 -- P, B of 0, but we normalized the other event to utility of the other invented to 0. So this what do we have?
So you subjectively you have to have.
Think about it, the probability of our red has to be larger than the probability of B happening blue happening to justify your preferences of your preference of a / B.
On the other hand, the C / D means that the probability of not R.
Of you, 1000 has to be large than property, not be times you 1000. If you see, think that you can still use the subject to behave subjective.
Expected utility.
No what?
Of course, I mean there's another event of getting nothing. We have limited omitted that as well.
So this is a logical consequences, which means that P, not R, has to be larger than P not be right?
And this 2 cannot hold simultaneously, alright?
So again, if we combine we, if we assume that people can write down probabilities, probability distributions for the events, and we.
And we can apply the expert utility ceron and this cannot hold at the same time.
So this is called a Ellsberg paradox.
When the properties are not specified so early paradox.
Here everything is well specified that you know what is the probability of each event happening and what the payoff comes with this event. So this is not about whether people are able to write down the probability or not, it's more about whether you can add or subtract independent.
The environments of independent alternatives, so to speak. This is more of that that problem, and with Ellsberg paradox. It's more about the weather people are able to write down the.
The proper distribution so.
And when there is ambiguity when the proper distribution is unknown, so these two paradoxes highlights the problems with the standard treatment.
But not with standing of this issues. The experiment this hearing is building block for many modern economic theory, and the application is wild.
I mean, there's still the stand that we're using if we.
Make do.
Without expecting to hear and then lots of assurance.
The large body of economic theory has to be revised and whether we still be able to get a meaningful predictions and the results is still doubtful.
But there is a trend. Many people, theoretical people proposing new message. Try to address these issues and some people were not like do not like it expected this year and they will do a lot of.
Other nurses, for instance, come up with other utility functions that is more robust towards ambiguity, for instance, and also.
Other ways of like, for instance maximum preferences for instance so that you do not rely on the expected utility ceron. Wait alright, that will be today's lecture and here I highlighted the relevant sections in the three box, that's your.
That are relevant for our module. I would strongly recommend it to look at the variant book because this is this notes is based on this book.
And for other treatments you can look at other toolbox. Alright, so any questions, suggestions, feedback, please do let me know and I wish you.
I could a week I think I'm already over using my time and I try to compensate it in the next lectures, maybe taking less time in the next week, trying to make up for it, so otherwise I wish you a good week and see you next.