 and quasi hyperbolic discounting. So what's all about? The key term here is discounting. Discounting the future. So let me first give you some examples. All of us have to make some kind of intertemporal choice, being it smoke now or we want to spend now or save for retirement. It doesn't now or be slimmer later. And finally, we have $100 now or $120 in a year. So but how do people compare costs and benefits that occur in different periods of time? Well, most people are impatient and they always prefer to have $1 today rather than $1 tomorrow, which means that each future dollar has a smaller equivalent present value, hence the term discounting. And today I'm going to talk about different rules which lead to a particular form of discounting. And let's take a look at the definition first. So suppose that Alice is indifferent between $1 at time t and dft dollars today. Then we say that dft is her discount function. Right? So then the discounted utility for Alice of the consumption stream is the sum product of the discounted function multiplied by utility function. And we assume that the set of outcomes in each period is the same and we also may have infinite time horizon. And there are several types of discounting, actually a number which have been proposed in recent studies, but there are two main types of discounting which are the most popular. Well, the first one is exponential discounting. And it's predominantly used in business world. I'm sure most of you are familiar with it because it's used in cost-benefit analysis in investment appraisal. And its key feature is that the marginal rate of substitution between two consecutive periods is always the same. It's a constant number. And the second type of discounting I'm going to discuss today is quasi-hype-bullet discounting. There is experimental evidence that quasi-hype-bullet discounting actually fits the data better than exponential discounting. And it's key feature is that the marginal rate of substitution between the second and the first period is smaller than the marginal rate of substitution between the third and the second period and other consecutive periods on. All right, and so we have this, suppose that we have these consumption streams. And we say that discounted utility represents our preferences on the set of these consumption streams. If one consumption stream is preferred to another if and don't leave, its discounted utility has a higher value. And the question is what type of rules, which type of axioms allows to say that the preferences of a decision maker can be represented by discounted utility model with the discounting in a particular form in the form of exponential or quasi-hype-bullet discounting. And of course I'm not the first person who's interested in this particular type of question. And I'd like to have a look at the existing axiomatization systems first without going into much detail now. And well, the most famous result here is Kupman's axiomatization system. And recently it has been generalized and simplified which gave way to axiomatize quasi-hype-bullet discounting as well. And here we consider these preferences over consumption streams which are non-stochastic. So we work in deterministic environment. And this fundamental mathematical results which allows to obtain additive representation is DeBrow's additive utility representation theory. And mathematical instruments used here is mainly topology. Another possible approach is this very setting when we allow our preferences to be over wateries. And these wateries have the whole consumption stream as an outcome. Working in that particular environment, Epstein and Hayashi axiomatize exponential and quasi-hype-bullet discounting. And again, here in order to obtain this additive representation, they used very famous theory by von Neumann and Morgenstein. So based on that, it's possible to formulate exponential and quasi-hype-bullet discounting. Well, what's my suggestion here? It just that we can, now we can consider again preferences and there's also some element of randomness, so to say. So we consider preferences over set of wateries. So there's a lottery in each period of time. And working in this environment, it allows us to use some known very fundamental result by Anscom and Neumann from subjective expected utility. So it was mentioned in Matthew's yesterday talk. And the advantage here when we work in this very setting is that when we apply this mixture axioms, it allows us to give the clear and shorter axiomatization in comparison with Hayashi, for example, where he needs to use four assumptions in order to obtain the same result. So let's take a look into this with more details. First, I need to introduce some definitions. The key definition here is that each set of outcomes is a mixture set. So X of T is called a mixture set. It's for every X and Y. In X, T, there is such an element which satisfies these three equations here. And we say that the utility function is linear if the following equation is satisfied. And now we are ready to define the mixture of two consumption streams using this very notation. And in order to formulate Anscom and Neumann result, it's necessary to, he uses five assumptions on the preferences. But I'd like to draw your attention that his results was formulated originally for the states of the world, whereas I changed the interpretation and look at the states of the world at time periods rather than states of the world. So it's just a change of interpretation itself. And so here is some notation as well. So A is single outcomes. A single outcome A is better than B if the whole consumption stream of A is greater than the consumption stream of B. So the preferences keeps whole consumption streams in the very sense. So the first axiom is pretty standard. It's weak order. So the binary relation is complete and transitive. The second axiom is the essentiality. It says that there are always at least two such consumption streams that one is strictly preferred to another one. So this assumption allows us to avoid such a situation when our discounted utility function is just a constant. So we avoid triviality. Axiom three is mixture independence and it's the most powerful axiom here which gives us that very nice separable structure. Axiom four is continuity and it's more of technical nature I would say. And axiom five is axiom of monotonicity which states that if we prefer one consumption stream to another in each period of times then overall one consumption stream is preferred to another one. So with all these five ingredients in place it's possible to formulate this Anscom and Alman result in 1963 which states that the preferences on X satisfy axioms one to five even only if there exists a linear function which is unique up to positive linear transformations and weights for all of t such that for every x and y and x for every consumption stream the following condition is satisfied, right? And now, when we have this very result it's possible to add some theorems and to build on it and to obtain exponential discounting. So we have, we start with the same five axioms here and we add what we need to do is to change essentiality to essentiality of periods one and two. Then it's necessary to also do it in patients and axiom of in patients well intuitively it's quite clear what it means so if you prefer a to b then you will also prefer such a consumption stream where a goes first before b rather than the other way around. And finally axiom number seven is axiom of stationarity which is the strongest axiom here. It says that if you have this particular preferences and there's a common consumption in the first period then if you shift both consumption streams by the same amount of time so your preference will remain the same. And with this seven axioms we can formulate, we can easily obtain this exponential discounting form. All right, and speaking about quasi-hypiballic discounting the transition to quasi-hypiballic discounting is not that complicated in this particular case either. So we begin again with the same set of axioms and what we need to do is we need to generalize in patients and to relax stationarity to quasi-stationarity which is effectively stationarity but from the second period on. And the important addition here is present bi-axiom which states that if we have this indifference in our preferences and so the first consumption stream is impatient whereas the second consumption stream is patient because a is greater than c whereas b is smaller than d. Then if we shift both consumption forward then our choice is cute to the more impatient consumption stream. So these eight axioms give us the following result the following form of discounting. Okay, so I'd like, I think I'd stand to some yapsher. If you reverse your order on the last axiom would you axiomize something where vases more than one? Well, yeah, I think so, yeah. So as a matter of fact, this present bias condition is necessary in order to state that better the ones to this interval from zero to one, yeah. I think so. All right, so summing up. So the key idea here is that application of Anscombe and Alman result representation theorem allows us to obtain this simple axiomatization of exponential and quasi hyperbola discounting. And I demonstrated it to you today but there is also one more general class of discounting which is called semi hyperbola discounting. Its form is like that and it's also possible to axiomatize this with the same instruments. However, there's experimental evidence that in fact individual type preferences they look more like generalized hyperbolic functions and it's not at this very stage it's not quite clear how to axiomatize this. So thank you very much for your attention. Why is it hyperbola discounting? Hyperbola discounting. Well, generally it means that it's a steeper decline in comparison with exponential function. So that's the whole hyperbola discounting is the whole class of discount functions. Yeah. That's the last line. The last line, yeah, that's the most general form of hyperbola discounting, right?