 Okay, so let me record you where we stand last time. So what we discussed is this genequilibrium theory, which is a mathematical theory to describe economies as a whole. And what we showed is that under some circumstances, which are complexity assumptions, etc., and complete market and perfect market, perfect competition, then one can ensure that the allocation that is reached by markets would be Pareto efficient. And so that any Pareto efficient allocation can be reached by transfers. Okay, so these are the welfare theories. So in this sense, markets solve this problem of the individual optimization problem of each of the individuals and to make it essentially a global option. Okay, and so this is what we said, but was realized really long ago by Adam Smith. Okay, so now let's start discussing markets. So what are financial markets good for? Okay, so the idea of markets, financial markets, has to do with uncertainty and risk. Okay, so the problem is that, say, you may want to reallocate your wealth across time in order to deal with uncertainty and risk. If you think about it, pension schemes and pension funds do this job and also insurance contracts are a way to deal with risk. Okay, so we pay an insurance contract today to cover us from risk that we could incur in the future. Okay, and so now let me discuss a very simple problem just to give you an idea of what is the main conceptual issue. Okay, so imagine that we are in a very simple world and in this world there are just two days, today and tomorrow. And tomorrow there are two different situations that can arise. Either it is rainy or it is sunny. Okay, so if it is rainy, I will need an umbrella. If it is sunny, I will need sunglasses. Okay, so the question is that if I have a certain amount of money now, then I don't know what is the best way to do. Should I buy an umbrella or should I buy sunglasses? Okay, whatever I do, I may end up in the wrong state tomorrow. Okay, so now you could think that one particular solution to this problem is to have what are called contingent commodity markets. Okay, what is a contingent commodity? So a contingent commodity is a commodity, a good, that is conditional on a certain condition. Okay, so for example, sunglasses if rain is a contingent commodity that is equal to sunglasses if it rains tomorrow, and it is equal to nothing if it will not rain tomorrow. Okay, and so you can have essentially a contingent commodity market for each possible contingent commodity. And if these markets, so you should have a price for each of these contingent objects. And so the idea is that you can solve the problem if these markets are there, then you can solve your problem by going and shopping and buying a sunglasses sun by this good here sunglasses if sun and umbrella if rain. And then this means that if tomorrow it will be sunny, you will have sunglasses. If tomorrow it is rainy, you will have an umbrella. Okay, so this is the idea of say general equilibrium theory. And so with this idea it is enough to have enough markets for all these contingent commodities and you can solve the problem of risk and uncertainty of dealing with uncertainty. Okay, so the problem is that in reality these markets are very difficult. I mean, in reality they do not exist and they are very difficult to implement. Okay, so for example, no one will buy umbrella, I mean, if sun, so this contingent commodity will not be very much, I mean, so if you think at a shop that sells umbrella if sun, this shop will not have many customers. Okay, so then this is why essentially financial markets can be useful. Okay, so let me give you an idea of how financial markets can solve this problem without introducing contingent commodity markets. Okay, imagine that there is a financial market and in this financial market there are two stocks. Okay, one is what is called a bond and one is what is called a stock. So a bond is a price of one today and it will give a return of one tomorrow. So I give to the bank one dollar today and the bank will give me one dollar tomorrow. Okay, instead there is another stock, the risky stock S that today cost one and tomorrow it will pay back one plus you if it is sunny and one minus D if it is rainy. So you see this stock is a random variable. Okay, it depends on the state that will be realized in the future. Okay, now imagine that I want to have a certain amount C rain if it rains and this is how much I need to buy an umbrella tomorrow and a certain amount C sun of euros tomorrow which is what I need to buy sunglasses. Okay, so the idea is that I can go into the market and buy a portfolio of ZB units of bond and ZS units of the asset in such a way that my return tomorrow which is ZB times one plus one plus you times ZS if it is sunny. Okay, and these are the expression if it is raining so that the value of my portfolio tomorrow will be exactly equal to what I will need. Okay, now you see this is just two equations for two unknown and so they can be solved and I can find exactly what are the values of ZB and ZS what is my portfolio by inverting this equation and so I can find out how much this portfolio will cost and how much I have to spend today in order to have tomorrow what I need to buy either sunglasses or umbrella. Okay, and this is just the value of the portfolio today which is the sum of these two things. Now the interesting thing that one finds is that this price of the portfolio can be expressed in this simple case in this way which is essentially if you think these are two numbers that add up to one okay so this can be interpreted as probabilities so it can be interpreted as an expected value over some probability distribution which in this case is D divided by U plus D or U divided by U plus D of what is the value the risky value of my contingent claim to motor okay and this is actually you can generalize this example and this is essentially the basis of what is called asset pricing theory and you can generalize this example and develop a theory where essentially every contingent claim every say value any random value of the sum of money tomorrow can be expressed as the expected value over the same probability distribution of this random value and this is really interesting for several reasons one because essentially what you see in all this game what you manage to do is to get rid of risk altogether and the other thing is you can measure how much this contract will cost you this portfolio will cost you just by taking an average over a probability distribution which is called risk neutral probability distribution the interesting thing is that this probability distribution has nothing to do with the real probability distribution with the real probability that tomorrow it will rain or tomorrow it will be sunny okay it is a probability distribution that is only derived from the prices as in this example of the different stocks okay now this is very very nice and looks like magic so it is important to realize what are the assumptions that we took well first of all we assume that this is perfect competition so that when you buy in the financial market the prices are not affected by how much you buy okay and this is not exactly true then also we assume that we have full information that we know what are the returns or we know what is the probability distribution of returns of the different assets in the financial market tomorrow and this is also somewhat of a strong assumption also well another assumption is that these works because you and the are positive you see that if you and the are not positive then this probability distribution can be negative and in this case this game does not work so this is a condition that is called no arbitrage that has to be satisfied in other words the prices should be such that there is no portfolio that will make a positive gain for sure without running any risk and this is the no arbitrage okay and then the most important thing is that you see we solve this problem because there is a 2 by 2 so an equation for so there are two equations for two unknowns so there are essentially two equations for the two states and for two variables which are the two stocks okay so what if now we have three states okay what if tomorrow it can be sunny, rainy or cloudy and if it is cloudy you don't need either sunglasses or rain okay now it is clear that if there are three states tomorrow then you cannot no longer solve this problem which means you can no longer eliminate completely the risk so this assumption that the number of states is equal to the number of stocks is what is called the complete market assumption and when the number of states is larger than the number of stocks which is typically the case then we say that the markets are incomplete okay and in this situation you will not get away with the risk okay so this is one way in which financial markets are designed to solve this problem of dealing with risk and uncertainty and indeed if you think about this is what financial markets have been doing in the last so trying to enforce these two conditions perfect competition and market completeness is what has driven the evolution of financial markets at least in the last 30 years okay because you see the reason why financial markets are not perfectly competitive is because the market participants are not many market participants so that some of the participants have monopoly power they can fix the price but if you allow many participants to access the market then markets will become more competitive and this is what has happened so for example this is the growth of hedge funds in the last in the last decades and the other thing is that if you want financial markets to be complete then you should allow the creation of new stocks which have nothing to do with standard stocks like I don't know cotton or say gold or oil okay but they have to do with financial securities or derivatives like options or swaps etc or what has been happening in the last say 30 years is the development of these credit derivatives which are essentially assets which are built on other assets okay and these other assets are based on credit instruments like mortgages or say credit to the false swaps of firms etc etc okay so the idea behind the proliferation of all these financial instruments is that essentially the market will become more and more complete so it will approach these limits where these we have a complete market and then we can get away with our uncertainty and risk about the future with asset pricing tier so we have questions I know this is a lot of stuff but I hope you got the main idea I have a question yes please why the increasing number of assets leads to the completeness of market yes so because essentially if you have many assets so you can think of these many assets like in the example that I showed you as vectors okay as random variables in a space of in this space where say every component of this vector is the realization of the return of this financial instrument for a particular state of the world okay for a particular contingency okay so if you have many of these vectors which are linearly independent then you can use them as a basis to essentially reproduce any other vector of completeness okay and if instead you have your spaces the number of states is very high and you have few vectors then you cannot reproduce all possible say contingent claims so all possible vectors okay thank you I couldn't understand what do you mean by no arbitrage no arbitrage means that given the assets that you can trade in a market there is no portfolio that you can form in such a way that this portfolio for sure will give you a positive return and non-negative return a return which is say non-negative which means that you will not lose anything but that at least in one state will give you a positive strictly positive return okay so this condition is a condition if you think about this picture of assets as vectors it means that the space of vectors that you have must be convex in the sense that there is no way in which you can combine vectors in order to have a vector which has only non-negative components okay so does this helps Benjamin we will see this in another example okay so no arbitrage is also called no free lunch hypothesis okay which means that there is no way in which you can get a lunch and you without paying okay so you if you well of course you can get it from your mom but sorry I have another question this no arbitrage is a means outcome of a complete market why to again assume no arbitrage this complete market will have number of assets equals to number of stocks and there is no risk anyway no so these two are two different things okay no arbitrage hypothesis is you can think it as the second law of thermodynamics of financial markets okay it tells you that you cannot get anything for free you cannot get something for free okay the completeness instead tells you that you can always represent whatever contingent claim whatever say random random return in the future is a linear combination of the returns of the stocks that you can buy in the market okay so these are two different concepts okay so if you want to say understand more about this I suggest you to look at the first chapter of this book by Plishka that I put in the references because this explains very easily all these concepts on the basis of essentially linear algebra okay okay okay so well this was very brief introduction to finance what I want to do in the next half an hour is to address the question when we have learned all these nice principles all these nice theories about economies and financial markets but what do these tell us about real economies real financial markets okay so of course we have seen for example that we can discuss an economy with two goods and two consumers but what we would like to do is actually discuss understand what are the properties of an economy where there are millions of firms and consumers and millions of goods okay and likewise we would like to have to see how these ideas or complete markets etc work in the case where markets become very very complex okay with thousands and millions of different stocks okay so the idea to do this is essentially the same idea of statistical mechanics that is to pick some theory at the micro scale and try to derive from this consequences for the behavior at the macro scale okay and this type of idea that you could use this set of statistical mechanics to study also human society actually dates back to Ludwig Wolfsburg okay or IEMSENCH you should be able to do statistical mechanics of economies and financial markets okay so statistical mechanics as you guess know is what allows you set of tools that allows you to go from what is the theory of the micro state to a theory of the microscopic state of collective behavior and the properties of this that essentially when you go from the microscopic behavior to the macroscopic behavior what you find is that the collective behavior is largely is very robust is very robust in the sense that even if you take very different gases which are very different composition the type of laws that they obey are essentially they are very similar and actually you can put them one on top of the other if you scale the variables properly so this type of this is the idea that essentially does not really matter microscopic details does not do not really matter so an idea is that if this is so then essentially you can study very simple model and then think that essentially you can compare the behavior of simple model with the real world data so this is the general idea and so what we will try to do is I will try to explain a few attempts to do these things for economies okay so let's start with economies okay so as I told you general equilibrium theory tells us that economies are these three main actors consumers, firms and markets and consumers buy goods and maximize utility firms transforms inputs into outputs and maximize profits and markets say clear prices so fixes prices so that demand matches okay so now you can think so mathematically you have a space of commodity everything is taking place in a space of this x is a vector of quantities of different goods each component is a quantity of a different good so you have this big C capital C commodities then one side you have these consumers and these consumers have initial endowments and then they have a utility function and what they do is to maximize the utility function subject to some budget constraints okay and the budget constraints is essentially whatever they consume they should be able to afford whatever they consume okay then there are firms firms essentially transforms input transforms inputs into outputs and they have a specific function which is essentially their technology by which they do so and essentially they have a profit and what they do is to maximize their profit okay to optimize the inputs how much inputs they should buy in order to maximize the profit okay and then the markets markets essentially adjust these prices in such a way that essentially the demand which is what is on this side which is the demand for goods from the consumers from the consumers and from the inputs of the firms should equal the supply which is essentially the endowments plus the outputs okay now essentially what they want to do is to take a model of this type where essentially both the number of firms and number of goods go to infinity but with a fixed ratio and divided by C okay and this ratio you can think of it as a measure of industrial development so how many technologies you have in your economies in your economy as a with respect to the goods to the different goods okay now the goods themselves you can think they are divided into different classes so there are primary goods these are primary goods these are things that you say dig out of the ground or that grow on the trees so these are things that cannot that are already there in the initial endowments okay then there are final goods final goods are what consumers care about so the consumers eat only a certain number of goods a certain type of goods they don't eat they don't consume all the goods so we don't eat anything that say we find that we can dig out of the of the ground then there is waste waste are so but say notice that some of the final goods can also be primary goods such as apples for example but some of the final goods like for example the iPhone does not grow on the trees it has to be produced so then there are waste are goods that are that are not final and that are just they can be primary or not but that are in surplus in our economy and then there are intermediate goods intermediate goods are goods that are not primary they are produced but then they are used to produce a final goods which is what we care about for example you can think at a part of your car is an intermediate good or say a component a computer component like a chip is an intermediate good because in itself it is not something that we use for consumption but it is used to assemble things that we use for consumption okay okay so then these are the consumers the consumer are separable utility function which makes life easier so and only depends on the final goods okay and also this thing tells you that a priori at time zero goods are different because some of them are primary some of them are not primary okay but each of the final goods is as good as any other goods and finally firms have these linear production functions which means that essentially the the amount of inputs the amount of output that is produced depends linearly with the amount of inputs okay so each firm the technology which firm is essentially a vector in this space of commodities okay now do you have questions on this I know this is a little bit complicated but it is essentially putting mathematics into what we have been saying okay so one issue here is that what I am going to assume is that this you know I have an infinite number of number n of firms that is going to infinity and I am going to assume that each production function is drawn at random so each of these vectors is drawn at random for each firm okay so this means that we are going to have we are going to study the general equilibrium of a large random economy okay okay so now the nice thing is that this can be done analytically with those of statistical mechanics I don't want to get into any detail just scare you with math but so in the end you can figure out what are the order parameters and reduce this problem in a way which is tractable and also there is a lot of insight that you can get from these subtle point equations let me just discuss the results okay now the result is that essentially if you look at phase diagram like this one where here you put this n which is essentially an index of industrial development what is the fraction of technologies in your economy to the number of goods and here is the fraction of primary goods then you find two phases so one phase which is essentially a phase where your economy does not take off there is no production in this all the firms are not producing anything in this region here and then you have a sharp transition to a regime where essentially instead the firms start producing so you can think of these as essentially an industrial revolution as the number of goods increases as the number of technologies as you invent more and more technology at a certain point your economy will undergo phase transition where mass production will start and you see this the scale of production of firms jumps to a positive value and the fraction of technologies that are adopted are zero up to this point and then they jump to a finite one okay essentially like industrial revolution and you see that this industrial revolution occurs before where is expected to occur before in places which have a large fraction of primary goods okay an economy where there are very few primary goods will not have will not experience this industrial revolution as early as one that has a lot of access to primary goods so this is interesting because you can think that essentially industrial revolution occur mainly in countries that had access the big colonial empire so wide access to primary goods but is essentially countries that base their economy on just one single very abundant results which is like oil or gold minerals etc etc typically have their industrial development has been slowed down a lot okay okay so this transition is very interesting so also if you study this how do you have a question yeah please in the previous slide I couldn't get means the small n is the ratio of the number of forms to the number of commodities right this n yes so even a small n is too which may mean for forms producing say two commodities then without any primary good you have almost industrial revolution sorry you have no you can have an industrial revolution yes so yes left graph fraction of primary goods as a function of small n yes yes if you have few primary goods then you need more technologies to get into this regime where you have mass production because otherwise inside of the phase diagram you have all these technologies but they are not used okay the firms so the scale of production how much the firms produce is equal to zero okay so if you have access to large number of primary goods then this transition occurs needs less technologies okay thus this model macroeconomics from microbehaviour consider the interrelation between different final products and primary goods so the there are no substitution effects if this is what because essentially utility function is separable but other than that you have all the general equilibrium effects between primary and final goods and I am going to discuss a few of these if in the next slide do this number of intermediate goods okay so the number I am going to get to this in the next slide so let me first discuss what are the recipes for economic growth that you will get out of this model so essentially you can compute what is the analog of a GDP which is essentially total value of the goods produced is the total amount of money that flows in the economy and you can see that essentially as you increase the industrial development this GDP increases and then it saturates so you have these two regimes and but also the important thing is to ask what should you do if you are in a particular situation what should you do in order to increase GDP okay and you find that when the industrial development is below a certain critical value let's say which is essentially two in this model then in order to increase GDP you should introduce a new technology increase and move in this direction but when you are in this other regime which is this regime where GDP is saturated what you should do to increase GDP is introduce new goods like the iPhone or the iPod something that a new good that didn't exist before and it's invented but the technology is to produce that goods were already there okay okay so this is interesting because it tells you that more or less we went through this so let me get to the issue of the different goods so if you compare these economies in different stages of development so what you will find is that say in this when you are at lower stage of development so there is there is a distinction between so this is a PDF probably the distribution of how many goods are consumed by the consumers okay and you see that there are a lot of abundant primary goods and few scarce manufacturing goods but as you increase the industrial development then essentially the amount of produce goods increases and more and more the primary goods are used for production okay and also you have that initially in the initial stage of development the economy is very efficient produces a lot of waste but as you enter deeper into this phase the waste decreases let me get to intermediate goods okay so imagine that now you are in this situation here up here what you could do is to introduce new goods so this increase the number of goods seen with the same number of technologies okay so this would correspond to moving along this line here okay what is interesting is that essentially as I told you the GDP will increase up to a certain point and this is what you see here but then after a certain point it will start decreasing and at certain point your economy will collapse okay so this is very strange so by introducing more and more goods into an economy you may also drive it to collapse because and when you can study what is the mathematical reason for this I will not get into the details so but just to mention that the number of intermediate goods in this economy is not set a priori it is the equilibrium of the economy that decides which firm operates which firm does not operate which good is an intermediate good and which good is waste I hope I reply to your question okay so after all this one is interesting because if you read accounts of the last four centuries of industrial dynamics then what you see is that more or less our economies have been first moving in this direction increasing their technological the technological repertoire expanding the technological repertoire and then essentially moving in the other direction so which means expanding the number of intermediate goods so for example how a machine a car was built in the early 20th century so there was a firm that was just taking raw material like wood, metal say leather etc all in the same place and then there was production process that were just producing a car okay nowadays essentially if you take a car the firm does not even buys different pieces from different other firms so they may buy the steering wheel from a particular place the motor may come from different firms etc etc this is like there is more and more of this outsourcing there has been more and more of this outsourcing okay so you have questions on this I couldn't understand the collapse which you described with more number of produced goods why should the GDP of a country collapse sorry I didn't understand and no I went to point out that you mentioned that the GDP of an economy will collapse if the number of produced goods becomes larger and larger I couldn't get the reason behind that okay so okay so one reason is that yes so the intermediate goods are mathematical special goods because essentially they impose homogeneous constraints on production so where is a primary good so if you go back to the definition of an economy so this one so you see that essentially you have a market clearing condition for each good okay so each commodity each good imposes a constraint okay so if you have an intermediate good then the why here is not there because it is not a primary good and the W here sorry and the X here is not there because it is not consumed okay so it is a linear it is a linear constraint on just production okay it's a homogeneous constraint on production where is that if you have a primary good these does not impose a linear an homogeneous constraint so if you have too many homogeneous constraints then you may have that the set of feasible production plants may shrink just to zero okay and this is why you have this phase transition essentially I mean this is why here you have no production this is why you have this industrial revolution so you can understand this from a mathematical point of view okay okay so now I think we just have let me spend means I hope you don't mind if I take a little bit longer but I will last like just to tell you that the same type of ideas can be applied to deal with a complex financial market okay so you remember this financial market stocks and we have these portfolios now you can essentially think of generalizing this picture to a case where you have many states big omega states and many assets say large and assets okay you think in a system where you have an industry bank that essentially trade these stocks these offer these financial instruments to investors and these investors form a portfolio and the way they do these is to find a Z which maximizes their utility okay so you have a utility so this mathematical is not so much so different setting than before and then you can think about financial innovation again it's like inventing a new asset okay and again you can study the situation where the number of assets goes to infinity and also the number of states goes to infinity okay and and you can draw these states at random so you can look at say a large random financial markets okay and there are two main variables here one is what is called financial complexity the ratio between the number of assets and the number of states and the other one is essentially the risk premium so how much the banks charge for say for the assets assets they are trading okay then you do a lot of what and you use all these tricks for statistical mechanics and in the end you end up with a phase diagram which is very similar to the one we discussed before okay you have a situation where your financial market is stable here where essentially the optimization problem of the consumer is well defined and a situation here where financial market is unstable because essentially you have a line of solutions okay and the phase transition which is this line here you have the situation where the solution exist but is degenerate there are say flat directions okay and actually the interesting thing is that complete markets correspond to exactly these points where essentially you have these flat directions these flat directions are important because at these points the susceptibility diverges which is essentially how much the portfolio of investors changes if you change the price of what they want to buy okay so in this line essentially you get what is essentially instability so let me cut a very long story short so you can also understand why is it that a financial market would tend towards this line when you increase the number of assets and these happens because essentially when the financial industry introduces a new asset what they will do is to hedge these assets with other assets that they buy again in the financial market in order to reduce their risk okay so this is essentially what is called hedging which means essentially in order to reduce your risk you try to replicate the product that you are selling with other products that can match the losses of this portfolio of this product now if you do this then what you obtain is that the more you convert to the situation where you have a complete market the more the risk premium should be zero and at the same point the number of the amount of trading in the intermarked market when you get to this limit of complete market should diverge this is essentially in line with what we saw in financial markets so essentially there is a reason why as you increase n in this direction you also expect that epsilon should decrease so you should land on this line of complete markets so essentially what you understand is that essentially if you put these things together then the more your markets get complete the more prices become unstable the more the volume of interbank trading increases and this is essentially more or less what has been going on say for example in Iceland in 2007-2008 essentially you can draw out of this model a face diamond say what is a stable economy an economy where the amount size of the financial sector is not so large and the instability and the susceptibility also is not so large and you can predict that essentially if you have financial markets for example it is subject to foreign exchange fluctuations like the one of Iceland then this will move in this direction and if it increases in size it will move in this direction and it will become constant so I think we are done let me just give you a take-home message what you learn from apply statistical mechanics to these economic systems essentially most of material economics has been developed with the idea of efficiency in mind of optimality but what you learn from statistical mechanics is that there is a trade-off between stability and efficiency so and if you push your system to be as efficient as possible then at some point the system may become unstable and this is a situation that we that was seen also for other system like ecosystems in a very famous paper by Robert May but in ecosystems you find that the ecosystems with a very large diversity of species in spite that live on very few resources but this is because these are systems that have evolved under this condition for a long time whereas in the financial markets the rules are changing very fast and the system is a system that has no that is adapting at the same time scale which it is changing so there is a very very different system there is also a very interesting system okay so yes so can this model tell anything about the economic crisis like 2008 yes it has a lot of as I was discussing the case of Iceland it tells you that say there is a limit to the size of the financial system that beyond which your system becomes unstable okay and this is essentially what has been happening in finance with proliferation of all these assets and yes there are other lessons that you can learn on say how for example in 2000's august 2007 all of a sudden all the banks stopped lending each other money so there was a say collapse of trust in finance and this you can understand as a game okay so what is happening is say is an agreement a social norm that at a certain point stopped to be compatible with incentives of the people and so I mean there are a lot of interesting things that one can learn from starting this models but I think we should stop here and I'm sorry this was and so I think we should stop here and I so encourage you to attend the colloquium that we have now and so otherwise we will meet tomorrow for head gas lectures and apologies again for today's change of problem