Well, I'm really pleased to be invited here and I think all the organizers and here in Palo and anyway, great crowd and thank you very much for the opportunity. So I'd like to talk about a set of topics that I find very appealing and I'll spend a certain amount of time today motivating what I want to do and I'm not quite sure how far I'll get to by the end of the week, there's sort of an infinite amount to say, but I hope to give you a bit of a flavor of several different approaches. So I'll describe the problem in some detail as we go along, but what I find appealing about it is that many different parts of mathematics come in. So I mean there's really input from analysis and geometry and also geometric analysis, synthetic geometry, algebraic geometry, low dimensional topology. I mean it really has a lot of lovely connections. So I'll try to impart some of that but I'll sort of obviously have my own bias perspectives and so today I'm going to give, I think it'll be some background and I'll try to talk about the more classical parts of the subject and so it'll be, well, we'll see, a rather different flavor than what I'll be talking about later in the week but I think it's important to cover various different grounds on this and I'll try not to lie too much about how I describe things but of course if I lie too much, the sun's going to go out. Okay, so let's see. So I want to say a little bit what I mean by a metric with the chronic singularity so fortunately Alex gave me the perfect introduction and so you have a bit of intuition about these but I'll describe what I mean a little bit more specifically and as he mentioned I'll be interested in surfaces with more general chronic singularities. So okay now I got to get used to this chalk which I was told there's some pricks too. So let's see, so hold on. No, Scott you're wrong. I like corners but that's still not working. No, here we go. Okay, sorry. Okay, so chronic singularities. So one can talk about these in sort of generality in higher dimensions and in surfaces and I'll define what I mean by a chronic metric initially and then I'll specialize to surfaces. So chronic metric. So the model cone is just a warp product so it looks like g which looks like dr squared plus r squared times h where I have a cross-section zh and this is a compact let's say compact smooth manifold and what this looks like is just exactly this. Here's a copy of z and it's whatever Riemannian manifold you wish and you just sort of cone it off so as r gets large this is going out and as r gets small this is coming down to a conic tip. Okay, so nothing too mysterious about this. More generally we'll talk about, I'll just refer to them as conic metrics, but asymptotically conic metrics it'll be some sort of perturbations of these. So there'll be some sort of metric perturbation and I might think of this plus higher order term so if you like just adding on extra tensors that decay as r goes to zero. Okay, so that's sort of the general class of things. Now when we talk about surfaces there's only one choice so of course if this is supposed to be n dimensional then z is n minus one dimensional and if the ambient space is two dimensional the cross-section is called the link is a circle and the only invariant of it is well you know all circles are isometric once you fix the length so there's exactly one real parameter which is the length of that circle which is the cone angle. Okay, so in one dimension in two dimensions for surfaces I can write these in a special way I can write them as dr squared plus and maybe I'll write this as alpha squared r squared d theta squared where here alpha is any number greater than zero. Okay, so theta is on the regular circle of circumference 2 pi and alpha is greater than zero and so the cone angle here is 2 pi times alpha so I'm allowing cone angles of any any positive number. Okay, so this picture looks exactly right so a cone angle that's strictly less than 2 pi I've made the circle a little bit small and when I take the cone over it it looks sort of like a decreased version of Euclidean space when the cone angle is exactly 2 pi well of course Euclidean space itself is a conic space it just the cone singularity is kind of invisible and it's doesn't really play real it's arbitrary and then I can let the cone angle be greater than 2 pi and then you can't visualize it anymore you can think of it as something which is wrapping around a bit but in general I'm interested in in real cone angles and so Alex talked about some of the milestones where we have cone angles maybe pi 2 pi 4 pi and so on those play very interesting roles they actually one has a little bit more rigidity there but those are just special cases of this general construction okay so that's okay so now here's my general problem so given let's say M and I'll maybe put a conformal class on it see this is a compact Riemann surface okay I want to choose a divisor I'll just be a disjoint collection of points p1 through pk be a disjoint collection of points set of points on M and I'm going to choose parameters alpha 1 alpha k and then what I want to do is I'd like to find a metric G so this is in the conformal class C such that G has a conical seniority at pj with angle 2 pi alpha j j goes from 1 to k okay so in other words whatever the surface I'm always looking at compact surfaces by the way I have a collection of points and I want to find somehow metrics that have own angles well if I just specify this problem that's not very interesting or hard namely it's easy to find metrics in a given conformal class and to modify them a bit to have a certain cone angle is not very interesting either not very hard so I want to say G has constant curvature so in other words I can think of this as the uniformization problem in the category of of these types of cone metrics okay so that's a problem it's been around for 150 years I mean in some sense people thought about this for a long long time I'll talk about what's known and and so for certain cases we're going to see that there's a radical difference when the cone angles are less than 2 pi or greater than 2 pi things behave very differently geometrically analytically and so on the story is understood and a little bit surprising in some cases when the cone angles are all less than 2 pi when the cone angles are greater than 2 pi then there's well like in all of these problems you have to specify are you looking at negative zero or positive curvature negative in zero curvature is in fact easy there's still some interesting aspects but the existence problem is pretty easy and I'll give you the full solution today the spherical case is hard and in some sense the answer is still not quite known in fact it's rather far from known surprisingly so I'll be describing that this week you know what's known and what are the various approaches that people have used to study this but anyway there's the problem okay now let me just say a little bit about motivations so why spend so much energy well it's you know there's an infinite amount you can say about surfaces and this is you know yet another whole class of problems I want to motivate it a couple of different ways so first of all motivations so one is is that as I already mentioned the words there's sort of a general class of you know problems of finding canonical metrics and I want to think that this is a problem of finding canonical metrics on singular spaces okay so a canonical metric I mean that's a little bit of a vague term but you know we're interested in constant curvature metrics or you know Einstein metrics or something like this and you can look in every dimension in every category of space you know what's the most reasonable natural metric that you expect to just find a few of them or you know a finite dimensional family of them now you know why look on singular spaces and you know I think the argument my kind of message is that singular spaces are pretty natural they come up in some sense just as often if not more than smooth manifolds and so even if you're just interested in smooth manifolds you often encounter singular spaces as sort of and when I say singular I'll I'll specify what sort of singularities I'm not looking at sort of candor sets or you know wild singularities I'm looking at spaces have stratified sort of well regulated singularities so you know these come up just as often as smooth manifolds and for example if you take a you know a more stratification take a Morse function you look at its various levels most of its levels are smooth manifolds but every once in a while you have to cross a singular level and it has a nice stratification and understanding that stratification is you know classical well-known thing but you get into a new category of spaces so you know there are many similar constructions in topology where you just sort of naturally led to singular spaces algebraic geometry obviously algebraic varieties compactifications of a locally symmetric spaces you know I can give you many many natural examples of singular spaces and so part of the goal is to you know develop tools of geometric analysis that work on singular spaces just as you know we have a lot of familiar tools that work on smooth compact manifolds okay so that's one sort of thing now this is kind of the lowest dimensional version of a of this problem so singular spaces I mean well there are actually some interesting singular one-dimensional objects namely networks of curves and there's some nice problems there but this is sort of you know in terms of bringing in a surface geometry this is the first interesting case now this particular problem actually has two interesting extensions you know let's say it's part of sort of several interesting problems so one is is it's the lowest dimensional example of a problem a well known problem in a complex or algebraic geometry of a problem in complex algebraic geometry which is a finding Kaler-Einstein metrics and so many of you may know this has been a subject of you know great interest in and work over the past many decades and there was a big success not long ago sort of finally completing the last case and it's you know unlike some other problems you cannot always find a solution and we're gonna see that here too so there's gonna be a certain and there was emerged as a called stability condition algebraic geometry case stability and it turned out that you can only find a solution of that problem when you're a case stable so this is actually the very lowest dimensional case where you see that phenomenon and from that point of view it's interesting there's a complicated obstruction okay so here's another example that goes back to certainly the 70s with Thurston so a class of what Thurston called a cone manifold and so I know it's kind of bad terminology because he wasn't thinking of just isolated conic singularities but he was thinking of constant curvature spaces which he built synthetically sort of this GX type construction and you have spaces that have stratified singularities so he was thinking obviously of the three-dimensional case primarily but one can define this in higher dimensions where you know you might have a hyperbolic structure away from some singular locus and there's some sort of branching or some sort of conic behavior as you go around these co-dimension to singularities and you might have higher co-dimension singularities okay and it turned out that these were a rather important tool in sort of his approach to sort of geometrization and putting hyperbolic metrics on certain manifolds and so he had an approach which again was synthetic but these have emerged as sort of interesting tools and then slightly more generally you can just ask for some sense of a very natural problem of you know find singular Einstein metrics and when I say singular I mean sort of with very specified class of singularities these often arise as sort of boundary points on the space of smooth Einstein metrics okay so I don't want to belabor this too much but just to say that the types of techniques and ideas I'm going to introduce for this problem can be used in all of these things or generalizations that you can be used and this is sort of a good test case for that okay okay so now let me point out that you know some examples of these things just off the bat and in fact well Alex gave us a lot of examples in the last lecture let me give you one other sort of general class of examples which is very closely related to what he did so let's suppose I take m2g2 to be any smooth compact surface compact constant curvature so it could be flat spherical or hyperbolic right and so obviously with the hyperbolic case there's a lot of examples if I take f from m1 to m2 which is a branched cover and I pull back the metric I let g1 the f pull back g2 then I'm going to get a metric on whatever m1 is so m1 might have very high genus because it's a branched cover and it'll have conic points exactly the ramification points of the branching and if I am winding around k times then the cone angle here is 2 pi k so in other words my my 2 pi maybe I should call it a l okay so if I've wound around if I've ramified l times then it's opened up to a cone angle 2 pi l okay now this already gives you a little bit of a hint that there's going to be some complexity so if I if I've asked over here can I find a constant curvature metric with specified points and specified cone angles right already we know that that's going to be a bit dicey because if I were to say I want to sort of find a hyperbolic metric with arbitrary points and arbitrary integer multiple of 2 pi cone angles I'd be asking for a branched cover with sort of a you know specified ramification points you know and so that's not such you know that's a constrained problem okay so okay so I'll mention various people that have worked on this as we go along but so I'm going to now go to the you know sort of the classical uniformization theory so first I want to remind you classical uniformization from a PD point of view and so it's exactly this one which I'm going to be using so I'll be talking about other perspectives as we go along as I said synthetic geometry even has sort of a a very important role to play here and has provided some answers that we can't quite get other ways in some cases but I want to talk about this problem in the case where I've no cone angles and this is you know really ancient history but just to remind you how the how the PD perspective goes trying to solve the uniformization okay so the problem here is that I have mg this is smooth compact and I want to find a metric g which is conformal to g not so remember I said I wanted to be in the same conformal class so g looks like e to the 2 u g not the only reason for writing the conformal factor as an exponential is a I can be sure that it's never negative or zero just by writing it this way and secondly it makes the equation look nicer right otherwise I could write this in I could write it as you to the fifth or you to the you know whatever log of you or you know you can write anything here you get a worse equation anyway if you write it as e to the 2 u then there's a classical formula which is not very hard to derive that if I take the Laplacian with respect to g not of u minus the Gauss curvature of g not plus the Gauss curvature of g times e to the 2 u that's equal to zero this is just a relationship between any two conformally related metrics it's pretty easy to derive and it's called the the Louisville equation okay so okay there we have it and from our point of view what we're asking is we have a smooth compact surface I want to find you such that k is equal to a constant okay so in other words one way you can think about this is somehow find to you so that that coefficient is constant which if I write k is equal to e to the minus 2 u times this stuff then that's like solving some PDE the other way to think about it is I can sort of fix k and then I can ask well can I find a solution and you'll pretty pretty quickly see that you're gonna run into some constraints so I'll just remind you there's a constraint which is the Gauss-Benet theorem which just says that you know the integral so whatever this curvature is so the integral of k times da so this is case of g da sub g which is the same as k times e to the 2 u da g naught this this has to be 2 pi times the Euler characteristic of m okay so very classical formula and all that tells you is that if you are in a surface with negative Euler characteristic then this k it better be a negative number okay similarly Euler characteristic zero or positive you correspondingly have to choose that to be zero or positive now the exact value of k is not so important so in this case if I replace u by u plus a constant I can rescale this so u plus a constant the Laplacian is the same u plus a constant it gives you an extra factor out here a positive factor so in other words I can always assume that this k if it's going to be constant is negative one zero or plus one okay so okay so I know what constant to put in there now I can ask can I solve that problem you know once I put in the right sign here okay so there's this old theorem of course that was obviously really a theorem in complex analysis you can think about that about this many different ways but I want to state it purely this way as solving this equation I can ask you know can I can I solve this so the well-known answers of course yes this always has a solution if the sign of k is chosen appropriately and I'll just want to remind you how you do that okay so there are three cases so the easiest case maybe I'll call case zero is k is equal to zero why is that easy well simply because the nonlinear term so anytime you have a nonlinear equation you're gonna presumably have to work harder when k is equal to zero that exponential is gone and so I'm just asking to solve the equation Laplacian Laplacian of u plus interview is equal to k not now one thing which somebody should a squawk to notice of course I certainly expect questions is which Laplacian to my do I mean so there's always a normalization and here I mean the Laplacian which in the standard Euclidean cases d squared dxj squared some j goes from well in this case one to two so in other words it's little plus in with the plus d by dx squared not minus so that's my normalization I use here okay well when does this problem have a solution so this is classical fact this has a solution if and only if the integral of k not is equal to zero but the Gauss-Benet theorem says that if I have any metric on here five any metric on the surface g not then its Gauss curvature has to integrate to zero so that just sort of built in if that's the Gauss curvature function for any metric whatsoever even if it's a variable function it has to integrate to zero and that's a necessary and sufficient condition for this this problem to have a solution okay now you can modify you like I did before I can modify by a constant you up plus or minus a constant well that would correspond to just scaling my surface okay so there's not a natural way to normalize in zero curvature but in the other cases there is okay the minus one case is more interesting okay so let's assume that Kyve m is less than zero and what I know now is simply that the integral of k not is negative and I need to solve that a problem so I need to solve Laplace not of u minus k not minus e to the 2u is equal to zero please sure well why would I compute it with respect to a different well I mean so I didn't mean this being I mean I just mean that I mean the Laplacian with a positive sign here but the curvature I mean so what other normalization for the curvature do you mean well okay so I'm of course this is not the Laplacian with respect to a general metric I've written the Laplacian with respect to a flat metric and you know when I write Laplacian sub zero I mean the Laplacian with respect to the metric g zero okay which looks like you know one over the determinant of gd by dx i root gd by dx j okay yeah so you're absolutely correct there okay so I need to solve this and so I want to introduce you to the method that's been used and I'm going to be able to solve this conic metric problem with exactly the same method so I want to go through it carefully it's not a very sophisticated method in fact if you're trying to solve elliptic nonlinear elliptic equations it's as they say the most boneheaded method you could try and you know you're in luck if it works it rarely works but in this problem it works okay so this is called the method of sub and super solutions and it feels like cheating because you don't have to do very much work so the method of sub and super solutions okay so you may have seen this so bear with me I just wanted to review it here because we're going to want to modify this in the singular context and something that I maybe didn't go you know there's a whole yoga a whole methodology of trying to do elliptic theory and parabolic and hyperbolic theory in on singular spaces and you know this is sort of the first example where you see sort of the new features that come out when you try to sort of understand the Laplacian on iconic manifold okay so the method of sub and super solutions is well let me look at the equation in the following way Laplacian you is equal to f of x and you so here x is a local coordinate so that just kind of a symbolic for local coordinates so this is some it's called a semi-linear elliptic equation and so the method says the following suppose that I can find what are called sub and super solutions so definition is that you lower is a sub solution if Laplacian of you lower bar so there's this stupid sign convention which is what leads people to sort of talk about the Laplacian the other direction is greater than or equal to f of x you bar and it's called you upper bar is a super solution if Laplacian of you upper bar is less than or equal to f of x okay so that's just a definition for function so maybe you can find them maybe you can't obviously if you found an exact solution then it's both a sub and super solution okay so this method is based on just the maximum principle which is a very powerful technique which again rarely works but when it does it gives you very powerful estimates okay and I say it rarely works I mean if you have a system if you have you know I mean all sorts of examples it doesn't all sorts of cases it doesn't work okay so the theorem in this general sense is that if there exists you bar which is less than or equal to you bar so sub and super solutions then there exists an exact solution a true solution which is sandwiched in between them you lower bar is less than or equal to you is less than or equal to you okay so in other words if you can somehow find a sub and super solution and they have to satisfy this equality this sub solution is less than the super solution then you're done okay now you know these functions don't need to be very complicated you can just build them out of you know functions that you know and love and that's what we'll do here but it turns out that that's enough so the claim is is that we can do that here so what I want to do just briefly is first of all apply it to this case to this theorem this level equation and then secondly just remind you of the proof of this theorem okay this will look very different than the sort of analysis I'll be doing later on but it's sort of a good basic technique which is very worth knowing and reviewing even if you know it well okay so apply this to our problem well so the functions that we know and love the best are constants and we'd like to say that we can just apply constants as sub and super solutions right which is what we want to do however the problem is is all we know about k not as it's integral is negative we don't know that itself is negative so there are various ways of handling this but the first thing we can do is prepare ourselves by finding an initial conformal transformation namely an initial function so that we can transform to a function that has strictly negative curvature okay so here's my first step so my preparatory step so I wanted to find e to the 2 u0 times g0 I'm going to call this g1 and I'm going to choose this so that k g1 is strictly negative everywhere in the surface okay now how do I do that well what I'm going to do is I'm going to solve Laplace in with respect to g0 of u0 is equal to so this k not now unlike in the flat case I can't solve this problem right because this does not have integral zero so what I want to do is I want to subtract off its average so k not bar is just the integral of k not divided by the area of the surface okay I know I can solve this problem by the same general criterion I talked about and so then if I take this solution so now I'm going to define g1 by this formula here and now if I look at Laplace in u0 minus k0 and I want to write this is plus k1 times e to the 2 u0 we know that this is always the transformation rule between the Gauss curvatures of g0 and g1 right but we know that this thing is equal to well by the way I've done these things this is equal to k not bar right so therefore k1 the true Gauss curvature is equal to k not excuse me yeah k not bar times e to the 2 u0 k not bar was negative so that's strictly negative okay it's variable it's not a constant right we don't know what it is but it has the advantage of being strictly negative okay so that's a nice preparatory step okay and now we can find sub and super solutions really trivially namely we just take very large negative or positive constants consider u lower bar which is equal to minus a and u upper bar is equal to plus a so if I plug this in I'm going to get you know Laplace in not of minus a minus k0 minus e to the minus 2 a well this part is gone this is strictly positive by how I chose things and this is extremely tiny if a is big enough so that's going to be greater than zero everywhere okay so there it is there's my sub solution and then similarly you can check that this is a good super solution okay okay so very easy I've done no work at all doesn't seem like I'm doing analysis and everything kind of rests in this this theorem here okay so let me sketch the proof of that so that's a you know a very useful tool classical fact it's you know finding this sort of the proof written down is not everybody refers to it without writing down the proof so let me just give you the proof okay so this is done by an iteration method okay now what I want to do is first change the equation a little bit so I have this equation Laplace in a view so what I'm trying to do is proof of this general theorem general PD though this thing over I've written over here okay so what I want to do is I want to rewrite the equation as Laplace in minus u Laplace in minus lambda view is equal to f tilde of x and u which by definition is just f of x and u minus lambda u okay now why do I do that because what's not to make this thing invertible though when lambda is a positive number that's invertible right so that's sort of a nice fact but in fact it's going to turn out that you want a certain property of this so what I want to choose is I want to choose lambda so large so that this function f tilde of x and u is monotone decreasing in you in other words I want du of f tilde of x and u to be less than or equal to 0 okay so why well first of all how do you why should we be able to do this because of course you can take on any value could be up to plus infinity minus infinity well in fact all I care about is that this happened for values of you which are in between the sub and super solutions okay so suppose that we have our sub and super solution so me just do this over here so I have these sub and super solutions which are you know let's say c2 functions so that I can define this point wise so I have u bar and u upper bar that gives me bounds on where f is and so it's certainly true that you know just by compactness of the surface and so on that I can make this function I can choose lambda so large so that this is true for any value of you in between you lower bar and you upper bar because I'm just dealing with a compact set of values there okay so I so I want du f tilde is less than or equal to 0 if you lower bar let's say the infimum of you lower bar is less than or equal to you is less than or equal to the supremum of you upper bar okay so that just a bounded range and I can easily arrange up by choosing lambda large enough okay so now what I do is I'm going to define a sequence I'm going to let let's say u0 be you lower bar now I'm going to define uj plus one iteratively is equal to Laplacian minus lambda inverse f tilde of x uj okay so namely I just solve the equation Laplacian minus lambda uj plus one is equal to something that I already knew at the previous step okay so that's called an iteration and see let's go back here so the fact that we can always do this is just the statement that Laplacian minus lambda is invertible because lambda's a positive number and so the real interest is inequality so the claim is that all of these things stay sandwiched in between you lower bar and you upper bar please okay so it doesn't really matter here because so and I'm definitely sloughing over the regularity issues so you know I can assume everything is smooth so I work in you know ck alpha so let me defer that for a little moment but let's say you do it in LP or L2 or something like that it doesn't matter very much the important things here of course is that at the end of the day because this is an elliptic equation if let's suppose that I found you lower bar and it's smooth right then the solution of this is good and as long as capital F is a smooth function then this guy will be smooth so you solve it in some space and then apply elliptic regularity okay now I'm assuming and I'm happy to sort of talk a bit more about general elliptic regularity I'm going to be talking about it specifically in this context of conic singularities but anybody want me to say a few more words about that be happy to okay so so you do this and these all follow by the maximum principle so let me just sort of do one of these just for for instance right so if I take uj plus 1 minus uj right this is going to be Laplace minus lambda inverse of f of x uj minus f tilde of x uj minus 1 now this thing here so I carefully arrange things so that's monotone decreasing by induction so I'm going to do this by induction by induction uj is greater than or equal to uj minus 1 which means that this is less than or equal to 0 okay and then well here's the point is that Laplace minus lambda and its inverse are you know what's called well positivity preserving but of course that's a misnomer it makes positive curvature and positive signs and negative signs so I'll remind you why that's true in just a second so this whole thing so I claim that this whole thing is greater than or equal to 0 and that's the inductive step okay so why is that true so what I'm really claiming I can see from two points of view and I'll remind you both points of view so I'll remind you two points of view the first is that if I have let's say Laplace minus lambda v is equal to f so if I have Laplace minus lambda v is equal to f and let's suppose the f is greater than or equal to 0 right then I claim that v so I was the hard part here is remembering what happened so if f is greater than or equal to 0 then I claimed that this means that v has no which way do I want to get here so it's a sub solution so v has no man so v does not reach a positive maximum v does not reach so the met the super v is less than or equal to 0 okay so that's the maximum principle well so that's just noticing that Laplacian v is greater than or equal to lambda v right and if the supremum were strictly positive then lambda v would be strictly positive but the Laplacian at a maximum has to be less than or equal to 0 okay so that's one point of view the other point of view is that if I take the integral kernel for Laplacian minus lambda so suppose I write Laplacian minus lambda inverse applied to any function f is equal to the integral of g maybe I'll call it g lambda of x y f of y d y okay so this is the green's function at energy lambda or however you want to call it then this g lambda just point wise is less than 0 of course it has a singularity when x equals y but point wise it's less than 0 okay well you have to prove that one way of proving that is sort of an analog of this result or you can relate it to something that's arguably even more elementary which is the heat kernel so I can write g lambda is equal to minus the integral from 0 to infinity of e to the minus lambda t times well right sort of provocatively e to the Laplacian t dt so this is the heat kernel h of t x y okay that's the integral kernel if I put a dollop of heat at a delta function of heat at y and I wait for time t then that's how much heat persists at the position x okay so again there's a maximum principle at work this is a strictly positive quantity I'm integrating against a exponentially decreasing but strictly positive guy from zero to infinity and I have to take a minus sign because the zero is on the lower limit of integration so this just looks like thinking of the Laplacian as a number so this is really a functional calculus statement but it's true okay and it's true pointwise in fact you know if I put in arbitrary values x and y not equal to each other this is a this is an equality yeah please yeah well so what I'm doing is I have a different right hand and left hand side right so you know I'm solving this equation Laplacian minus lambda uj plus one is equal to f tilde of x uj of course this is the same as what I just wrote down but I assumed that I've already figured out what uj is inductively and then I find a new function uj plus one okay so they're different and I keep on stepping up and I find new ones okay okay good okay so I find these functions and they satisfy this chain of inequalities so what well yeah okay so they satisfy this chain of inequalities I'm just applying arguments like this again and again it's an easy exercise and so what I get is that you which is equal to the pointwise limit of uj so I'm doing the most unsophistic kind of kind of thing that we teach our real analysis students not to do with functions right it just take the pointwise limit that's usually a disaster right because we know nothing about the pointwise limit except for that it's bounded okay it's an l infinity function that's all I know about it because it's bounded between u lower bar and u upper bar okay however what I know about it is that so Laplacian minus lambda of this u is equal to f tilde of x and u well this right-hand side it just I mean f is a smooth function so I can just take the pointwise limit of f of x uj and it converges to that how can I differentiate this thing well you know if you're like me think of this as a distribution and you can differentiate anything right so you can think of this as just a distributional equation okay so I think of you as this as a distribution on m and the right-hand side a priori is just an l infinity and then I apply iterative elliptic regularity so the first thing I do is note that this is going to tell me that this equation says that any distribution which satisfies that equation has to be better than it looks it has to have two derivatives in l infinity so so u is in w to infinity okay so that's kind of a weird function space it just means two derivatives now infinity but in particular that means that and in fact let's just be safe call this two derivatives in LP for all p less than infinity right this is going to say that u is in c1 alpha the holder space c1 alpha right and now the right-hand side is in c1 alpha and u has two derivatives as it's Laplacian and c1 alpha that's an a space in which I can apply regularity so now I'm saying that you know Laplacian of u is in c1 alpha that's going to tell me u is in c3 alpha etc so I keep on bootstrapping and I get eventually that you is in c infinity so I found a solution okay and moreover the solution is unique so if I have two solutions Laplacian u is equal to f of x u and Laplacian of v is equal to f of x v and maybe excuse me I want to do this with the monotone guys just to be safe Laplacian minus lambda and what I mean Laplacian minus lambda v is equal to that the difference Laplacian minus lambda of u minus v right is equal to f tilde this is f tildes now of x u minus f tilde of x v and I apply the same sort of ideas that if u is ever bigger than v then I look at the supremum of u minus v this right hand side is positive and then that contradicts the same sort of argument that I did before that the Laplacian has to be less than or equal to zero at a supremum of u minus v okay so u minus v is both less than or equal to zero and greater than or equal to zero okay okay so that's like I say a old argument going back 50 years or so but it's still a very powerful one when you can make it work so we've completely solved the uniformization theorem in in a in the zero and negative cases now okay in the contact case okay so let me say a few words about the spherical case but quickly move on to the iconic situation which is more a bit more challenging so the spherical case so the reality which we know from complex analysis is that any metric whatsoever on the sphere is conformal to a constant curvature metric there's only one conformal class okay and you can prove this by you know finding a certain sort of holomorphic function from so if I have M what I really would want to do and there's this is sort of a task thing is if M is looks like the sphere then well you see what do I want to say so I'd want to find a holomorphic function from this on to the complex plane so holomorphic that has a simple pole at P and you can find such things okay I mean there's sort of a standard tricks to do if we were to look at this from the PDE point of view we'll plus in you minus k0 plus e to the 2u so this is one of these things that you know it's sort of good pondering you do something incredibly minor looking like change a minus sign into a plus sign and it just screws everything up okay namely this method of sub and super solutions just fails you know you can try to apply it and things don't quite work and you think well I could just work a little bit harder and make them work somehow it just won't work and the reason is is that there are families of solution well it's a little bit contradictory but there are too many solutions here is the reason it doesn't work so in fact it turns out that there are solutions that so here's the sphere it turns out there are solutions to this problem which kind of bulge at one point they kind of bubble at one side and you know get very small at another side I'm not going to say too much about this point of view but the fact that you have a non-compact family of solutions so where these solutions come from is pretty easy I just take a conformal transformation just a dilation on the sphere that sort of sucks the south pole to the north pole or you know any pole to any other pole and I pull back the round metric with that I get that's always conformal to the round metric and I get a family of functions which get larger and larger as I take a stronger and stronger dilation they all satisfy this equation and though that's not directly contradicting the fact that I might use sub and super solutions that turns out to be you know really it really screws things up and so the method just doesn't work now I'm not going to sort of give you full details of the proof that you can just solve this equation by hand for any function k not that integrates to the right thing namely 2 pi times order characteristics namely 4 pi and but the way that you do this is that you do this variation way so you use the calculus of variations so variational ideas so namely what I want to do is I want to look at the energy one half times the integral of grad u squared so then I'm going to have to do plus minus the integral of k not times u and I want to find the I want to find you such that this thing is minimized but I want to do this subject to the fact that the integral of e to the 2 u is fixed let's say is always equal to one this is the case so this is a variational problem so in other words I look at this functional here and I look on a nonlinear constraint set so this is some big infinite infinite dimensional manifold in the space of all use in fact it has co-dimension one so it's this huge hyper surface in the space of all use and somewhere on there I want to find the minimum of this functional now why does it do the right thing well if I take the Euler Lagrange equation of this which I'll compute for you tomorrow and use this is Lagrange multiplier I get exactly that equation okay how does the plus sign come out well it's this minus sign here and that minus sign means that this is possibly unbounded below and you run into all sorts of problems about why can you even find a minimum and so on whereas if this were a plus sign which would be that being a minus sign the negative case this would be easy okay so there's a whole line of work that sort of goes here and this turns out to be a somewhat delicate variational problem precisely because there are too many solutions right it's always this sort of a normal thing if you have this big non-compact family of minimizers it's very hard to find them variationally because you know you take a minimizing sequence and it can go off to infinity okay so there's sort of a whole line of work there which I am not going to be pursuing very much in these lectures but I just want to mention this okay so the last couple of minutes let me just set these problems up now for the chronic case what I'm really interested in and why I'm you know so the first thing is a theorem kind of completely settles the problem I stated in one specific case so here's a theorem by Bob McEwen and it was proved about 1981 I think and what he proved is that the the main problem is solvable if the corresponding curvature is less than zero or equivalently there's a number which is the chronic Euler characteristic I'm gonna call this kind of a less than zero okay so there's a condition which is just like an Euler characteristic condition and provided that simple number is zero as negative this is just a combinatorial thing then I can always solve this problem for any cone angle okay and the method of proof is going to be just that we have to set it up properly and there's some new details which I want to describe but that's it the whole solution okay the k equals zero case turns out to be even easier as usual it's just a linear problem and the k positive case is the one that's still open and unsolved and it has a lot of subtleties okay and that sort of relates to what I'm talking about over here now I want to sort of conclude today by talking first of all what this chronic Euler characteristic is and then secondly sort of why they're still interesting things to say even in the negative case okay so what is so suppose I have any conic surface so maybe there's some handles in here but there's a finite number of conic singularities okay I'd like to compete the conic Euler characteristic so what that means is I want to take the integral so I want to take m minus the union of balls j goes from 1 to k and I just want to take epsilon balls around these points pj case I'm just excising balls here okay now on this surface I want to apply the ordinary garden variety gas beneath there I'm on a man on the surface with boundary okay so I'm going to have the integral of kda on this m sub epsilon I call this m sub epsilon and that's going to be the integral over the boundary of m sub epsilon of something plus the Euler characteristic of this thing so 2 pi and then I'm going to have chi of m and every time I've removed a disk it subtracts one from the Euler characteristic so I have 2 pi of chi of m minus k okay so so the whole issue is what is the boundary contribution now the proof is you know the various ways of doing this but so this this boundary contribution the integral of boundary vm sub epsilon so this thing so there are various ways of how you might write this what this thing does so this is going to be asymptotically the same no matter what this conic metric is to the corresponding contribution on a flat cone okay so this is going to be a sum j goes from one to k of integrals around little circles and I'm going to have this corresponding so if you like this is a circle with cone angle 2 pi alpha j okay now there are various ways of writing down this contribution but it's essentially the whole ennemy around that it's the parallel transport around there so this is just going to be the sum of 2 pi times alpha j okay so what I'm left with all together is that the integral of kda is going to be 2 pi times the sum j goes from 1 to k of alpha j minus 1 plus 2 pi chi of m and so I call this number chi of m beta so I'm going to switch parameters now is equal to chi of m plus the sum of the beta j's where each beta j this just turns out to be a more convenient parameter alpha j minus 1 okay so it's a way of writing it and so the theorem of macoans is that this is a necessary condition if I'm going to have negative curvature you know if k is everywhere negative right then this right hand side better be negative and he says that's a necessary and sufficient condition okay proof is same kind of boneheaded stuff though we have to worry about how do we solve laplacians and conic manifolds and I'll be telling you about that in negotiating detail okay so that's one thing and then and then I promised you that there are some interesting questions here and so here are the questions that I want to get to by the end of the week so I can find a solution with singularities now sort of the the frontier of this problem is the frontier of what happens here namely what happens when the points coalesce so what we're after is sort of a very accurate analysis of what happens when you have clusters of conic singularities that that vanish I mean that that coalesce into one point okay so we know that there's always solutions by macoans general theorem but something horrible happens to them I mean they're gonna converge to something with cloning singularities I'd like very precise information on how that happens right now why do I care about that so much well it's emblematic of a lot of problems where you are have clustering singularities and you're interested let me give you two examples and I'll stop there one example is suppose you have a holomorphic quadratic differential on the surface and you look at a family of such things where the zeros are clustering together okay how do you sort of understand the fine asymptotics of that now there's old work on this that actually Alex went out to me was solved by my colleague Halsey Royden you know forty years ago but in fact we have a very interesting new approach on sort of how to understand those asymptotics okay another problem this comes out of quantum mechanics really suppose you have a bunch of Coulomb potentials you have a shorting or operator with Coulomb potentials you know one over mod X singularities and they're all clustering you know you have a bunch of particles which are very close together what happens how do you analyze the fine spectral theory of that so the tools I'll be describing give you a way of understanding that and this is just sort of the easiest kind of model case to understand things like that okay so I'll stop there so yes I mean so the the the right question is given a collection of k distinct points in the surface how do you compactify the space of k distinct unordered points in the surface now there's complex ways of doing that I mean you know complex geometric ways of doing that which I don't want to do so what I want to do is define a real manifold with corners which describes the compactified configuration space so that turns out to sort of describe all possible ways that these points can come together right and then sort of the theorem and after is saying that on that compactified configuration space I have a universal curve which is over every divisor I have the curve M you know and it's sort of changing as the points move around and that itself you know as the points cluster you know that that has itself kind of a complicated resolution is a real manifold with corners so I have some very complicated geometric object it has a lot of boundary faces and so on but it describes all possible ways that points can come together and I'd like to understand this family of metrics on that space


Alrighty, I have a construction update for you guys. Again, nothing too much going on over at Tundra Twister. That area was flooded for a lot of the week that just passed. So they're probably going to start work up again this week. But if you look closely around Vortex, there's the service roads. So Wonderland actually sneakily built some service roads around Vortex's circuit over this last season. Actually, the last off season they did it. And if you paid attention this season, all of the support columns are marked with numbers. So again, it is speculated that Vortex is to receive a major overhaul, including a paint job. It was supposed to be this off season, but because of a lie, then we're not quite sure when that's gonna take place. However, there are some fresh tire marks that I saw in and around, and it looks like there was a little bit of an extension of the service road. So they could still be working on getting ready to do that upgrade. It might take place now in the 2023 season off season, but hopefully it still takes place now because it would be awesome. Vortex is a beloved attraction and I would love to see some work done to it. Snoopy's Racing Railway had a lot of its electrical installed for the station. As you can see that massive orange square tarp there, that is going to be the launch and station area. Again, it looks a lot smaller from where I'm flying. You have to remember I am so far away. So perception sometimes is really off. It is really hard to tell the size of things and the scope of things from a drone that is as high as it is and as far away as it is. The drone has a significant zoom, again, which will distort some things when you're trying to take things into account. But nonetheless, nothing really going on in terms of Mind Buster. There's just been continuous work, very small, nothing like they are doing at their other parks. Sad face, tear. I would love to see Gravity Group come on in and give this coaster some love. But yeah, who knows? Maybe there's a reason that Gravity Group isn't coming in and giving the coaster some love. Wild Beast is still showing no signs of love either yet. Again, this is really early in the off season. Retracking sometimes takes place literally a month before opening or a month and a half. But yeah, again, Wonderland's really ahead of schedule maintenance wise by the looks of things. And Behemoth has two trains on the track. I don't know if there's anything else of importance. I'm not ready to talk about a specific coaster that is showing signs of a potential removal coming up. So we'll keep that on the down low until we get some more concrete information. But nonetheless, I thought I'd show you guys around.  And I was like, I don't know, I kind of just got used to just making the videos really quick. But here, I'll show you guys some shots around the whole park so you can browse see what's going on. Maybe sometimes people do notice things that I don't notice. Especially some people on my team. But yeah, more supports have arrived for Snoopy's Racing Railway. It doesn't look like any more track has arrived. And I'm not quite sure if they're done 100% of the footings. They are getting close. Leviathan statue is still missing. As you can see, I think one train might be on the track for Leviathan, but it is really hard to tell. I can't get an angle from a legal place of flying. So I won't ever be able to tell until all three trains are on and I can see the one closest. But yeah, nonetheless, not too much going on. There is a weird dumpster over by Maple Treehouse Park. I can't tell what's inside the dumpster or what area they've been gutting. I know that they're working on upgrading restaurants. So it might have just been moved there. And that's the innards of one of the restaurants in the area, like maybe the Pizza Pizza and the Fry Place. Again, those need some serious upgrades. So it could be that and it's just its weird location. I don't see any signs of Maple Treehouse Park being dismantled yet. I looked pretty closely and there were no signs. It's definitely not Action Theater. But yeah, so we'll keep an eye on that. Dumpsters are always a good sign for potential infrastructure upgrades of the restaurants and stuff like that. So yeah, trying to think if there was anything else of importance that I'm missing here. Again, we are keeping an eye on this area that I'm showing right now. There were a lot of markers at the end of this last season, in and around flying canoes, the gift shop, the seating areas, Action Theater, the restaurant going all the way into Kids Zone. And yeah, all in this area. There's an orange marker right around there. So we'll definitely be keeping an eye on this area for 2025 2024. So yeah, hopefully you guys enjoyed today's construction update. I know it wasn't too exciting in terms of the new rides, but there was still some decent information to share with you guys. But yeah, thanks so much for always watching. Thanks so much for watching the podcast. And I hope you guys are having a great long weekend. And hopefully the weather gets a little warmer again. I was enjoying that. Oh, Clockworks! Clockworks is still having its little, as you know, Cedar Fair does little like upgrades to their older attractions to just keep their service life going. That's all that's going on with that is what I'm predicting. Nothing, no sign, significant signs outside of Wonder Mountain. But we'll keep an eye on that as well. Thanks so much, guys. Have a good one. Bye.


Well, good morning, everyone. In this session, I want to introduce you to the foundational concept that provides us an understanding of the working of the market economy. And Ludwig von Mises called this concept economic calculation. And the cornerstone of this foundation of economic calculation is the theory of market prices. And so that's what we're going to do in this session is talk mainly about market prices and how market prices form the basis for engaging in economic calculation. And then, in turn, economic calculation is what provides the possibility of a social economy, a market economy, a division of labor economy. So we'll proceed in four steps. The first step, I'll just talk about some of the basic principles of human action as a ground on which to proceed. And then we'll talk about what we call the personal economy. And this is the first part of our title, subjective value. We'll show how it is that you and I and every person formulate a personal economy on the basis of valuations we make. And then we'll move to the social economy. What happens when we want to integrate our personal economies? How do we do this in an effective way? And this is where we get to the principle of economic calculation and what, again, Ludwig von Mises calls appraisment. So we'll talk a little bit about those concepts. And then the fourth thing that we'll do is talk about the theory of price. We'll talk about the determination of market prices. OK, so you've probably heard already that we begin economic analysis with the concept of human action. Human action is just purposeful behavior. It just means that it's human behavior directed toward the attainment of an end. It's quite obvious, upon immediate reflection of this idea, that having an end in and of itself does not constitute action. We can have all sorts of ends that are unmet that we have not yet attained. That it's human beings because we're finite beings. We have to first identify and then acquire and then control the use of means in order to act to the attainment of our ends. If we were infinite beings, we could just attain our ends by willing them. So we have unmet ends at any moment in time. Or to say this in the more traditional way, the means available to us are scarce. And because the means available to us are scarce, we have to choose in action. We have to choose in two dimensions. One is whatever means we happen to possess, we have to choose which ends we'll pursue with those means. And for any given end that we select, we'll have a choice between different combinations of means to attain the given end. So we have the ends means framework of action and the way in which we perceive this as actors. So this is where we get to the principle of the personal economy. This is, again, what Ludwig von Mises likes to call economizing in action in these choices we economize. That is, we take given means and we choose to apply them to higher valued ends. And we set aside ends that we value as less highly. For a given end, we choose to arrange the means in a way that the combination is less valuable than alternative combinations of arrangements of means, which are more valuable. We economize. This is the mode of human action, we might say, or the logic of human action. Since we already mentioned, though, that when we value in this way, we're always choosing, we're always then comparing the value of one alternative to the value of another. And this is what in economic theory we call preference or preferring. And since we're talking about the real action of real human persons, when we talk about preference, we're talking about what Murray Rothbard liked to call demonstrated preference. We're not talking about the thought process of things that we like or don't like. We're talking about what we do in action, how we choose between the alternatives available to us in action. So our preferences, our valuations, in other words, are always demonstrated in action. They're coincident with action. There are two basic features then of value in this logical sense. One is the subjectivity of value. And by subjective, what we mean as economists is that value is nothing more than an intensive state of mind. Value exists only in our minds. It's a state of mind. It lacks an extensive property. There's nothing outside of our mind that is related to the value that we're assessing in our minds. It's a judgment that we make in our minds. This is the idea of subjectivity. Now, if this is correct, if value is subjective, without an extensive property, then we cannot measure value. Value is not a substance or something that exists objectively outside our minds that we can have a shared experience of and come up with a unit of value. We can't say I get 10 units of value from eating a chocolate ice cream cone. This is just like a nonsense statement. I mean, you could never make sense of this because we don't have a shared notion of what a unit of value might be. It's only in my mind. And then you can have value, of course, in your mind, but never the twain shall meet. We don't have an objective way of comparing value. Hopefully, you can see right away the truth of this proposition is what presents the problem of arranging an efficient or an economizing social economy. How are we supposed to do this when we can't actually know what other people value by our own experience? We can only experience our own personal assessments of things. Now, the other feature that I want to mention of subjectivity, which is sometimes confused with the principle subjectivity, is that value is not constant, or at least we cannot assume that it's constant. And by this, we mean in particular that there's no quantitatively constant relationship between external circumstances that surround our action and the subjective value that we place upon acting in these circumstances. There's no known quantitative relationship here. And hopefully, you can see that what follows from this, of course, is that we can't formalize economic laws in mathematical style, because there are no constants. Everything is a variable. And if there are no constants, well, OK, certain things we might see are related, but we can't have formal expressions of this relationship that are quantitatively definite. That is not a principle of subjectivity per se. That comes from the fact that we simply don't know what the connection is quantitatively between external circumstances, our judgments, and then the way we act, like the price of the product that's external to my mind and how much of the good I buy, because that action takes place through my mind. And there isn't any known quantitatively fixed relationship between the external circumstance, my mind, and then the consequence, my action, that's a consequence of this. All right, so let's move on to another, a final principle. And I want to do a little comparison here. And this is the idea that economists call imputation of value. So all we set up to this point is that we value in action the attainment of the end. But does this mean that the means that we use in action do not have value, or what's the association of the value of the means to the value of the end? And you'll see in the first line, the top line, is the Massessian view, the Austrian view, which holds to the logic that we've mentioned already. If human action is purposeful, if the point of human action is to attain the end, then the value of the end drives everything else in action, logically. The value of a consumer good is valued by a person only because it's an aid to the attainment of the end, whatever the end is. The value of producer goods are only assessed or valued by the person with respect to the aid that they give in producing the consumer good. So here we're not talking about some sort of philosophical notion of the value of things. We're not talking about the intrinsic value that God sees in the created order, or something like this. We're just talking about human beings engaged in action to attain ends. And here the logic must flow in this fashion. So this is a way, of course, that the price structure in the market will then flow. Now, the second line is the British classical way of thinking about the relationship between these things that culminates in Karl Marx and the labor theory of value. Value is intrinsic in the producer good. It's transmitted to the consumer good by production. Our minds are sent to this value somehow. It's consistent. We see it as consistent with the labor embedded in the good. Now, we're not going to go through a long demonstration of why this is false. I just want to point out that this can't be true if our discussion of the ends means relationship is true. This can't be true if human action is actually, the value of it is centered upon attaining the end. And it can't be the case that there's intrinsic value, values separate from that, from some other source, in the means themselves. And then the final line, the third line, is the neoclassical view from, say, Alfred Marshall. That the consumer good is valued as a mutually determined process by the independent value of the producer goods, the cost of production, and the demand for the good, the subjective value. But again, this cannot be true if our ends means discussion is correct. If our ends means discussion is correct, there can be no such thing in human action as independent value of means. The value of means must be as they are aids to the attainment of the end. The logic of it necessitates this top line, right? Okay, so now let's go to the next step. For our purposes in thinking about market prices, we wanna take one more step in valuation. So up to this point, we've just said, remember that, each one of us engages in human action. We organize our human actions as a system based upon the way that we value the various ends that we can attain with the various means that we have or that we can acquire through activity, right? Production or trade or whatever the case may be. But there are a few important laws of this valuation process that are useful in thinking about how we integrate once we start to integrate our personal economies. And so here's an illustration and you can see there are two separate sets of laws. On the left hand side, let's start there with the laws of utility or the laws of value. Utility is just another word for value in the context of economics. So here's a preference rank on the left hand side of equally serviceable units of a good. And the good that I picked is an iPhone SE. So the entry level iPhone. And if a person had just one iPhone SE, the person would allocate it to the most valuable purposes to which they put a smart phone, right? And so the value of that iPhone would be imputed to it according to the value the person placed upon doing those tasks, you know, attaining those consumptive ends. If a person though had two iPhones, a second iPhone, that second iPhone would have to have less imputed value than the first. Because the second iPhone would have to be applied to the person to a less valuable set of activities because the first iPhone is already available to him to accomplish the most valuable. And so on. And in fact, we can even see that it's possible that a person would have so many units of a given good that it would no longer be scarce to the person, right? And therefore they wouldn't value the iPhone at all. You imagine Tim Cook if he suddenly embezzled the 100 million iPhones that are in warehouses someplace and went off to Tahiti. I mean, what value would he place on losing one of them? Right? The answer is zero. So you can see again the logic of this structure. And hopefully you'll notice that this will apply to the law of demand when we get to the market. But, and then on the right hand column, I just got one other law, the law of the allocation of consumer goods. So we know that we don't act with just one consumer good, right? We don't have just a smartphone. Although sometimes I wonder about some people. We do other things. So how is it, is there a law? Is there a logical principle that we can apply to the understanding of how each person allocates their acts of consumption across different consumer goods they have? Is there something that's always in place logically there? And the answer here is that, as I've suggested, is that a person will allocate different tasks with different consumer goods so that there are no big value differences left after this is done. So in my example, this person uses the iPhone SE they have in order to FaceTime with their family. I know that that's what you're primarily doing with your phones, right?  Yeah, so you do this for two hours. Why is it two hours? Why isn't it four hours, right? Well, it isn't four hours because the marginal utility, the extra value of extending activity along this line would be a lower ranked, right? This means that other activity, which is higher ranked in marginal utility would then come into play. Let's say my example, this person goes for a run and takes an hour to do this. Why doesn't the person run for two hours? Well, okay, so you see the principle, right? So we're always allocating across different consumer goods in this way so that we don't see any big value differences that would allow us to shift out of the low valued activity into the high valued activity and gain overall. That's the logic of it. Okay, now let's think about producer goods or the laws of producer goods. Remember when we talked about imputation, we said value goes from the mind to the consumer good and from the consumer good to the producer good, right? The person's imputing along this logical line. So what about producer goods? Are there laws of the use of producer goods? And here, again, there is an important law on the left hand side, you see it, called the law of returns. So here my example is suppose this person who has the iPhone lives out in the country and through the property, he's got some land and through the property in the woods, there's a stream and in the stream he can go fishing. So it's a natural resource and they're fishing the stream and he's got all the consumer goods he needs to go fishing. He's got his pole and his line and his flies and whatever, whatever, he has waders to go into the stream and whatever. He's got all these complimentary factors of production, right? And then he goes into the stream with his labor and one unit of labor, let's say it's an hour of labor, he fishes and he gets four fish. But hopefully you can see right away that if he continued to do this, if he continued just to apply more and more of a variable input to a fixed input like the stream, he would fish out the stream, right? He would exhaust the physical capacity of the stream. And that's a general principle of all production. If we have a fixed complimentary factor of production and we add more and more of the variable factor to it to produce more and more, we'll eventually exhaust the finite productive capacity of the fixed factor, in this case, the stream. It could be a factory, it could be a farm, it could write anything. This is what we call the law of returns. So the law of returns is a necessary feature of human activity because again, we're finite beings, right? We live in a finite world. Okay, this has the same implication then as the law of utility that we talked about that diminishes as we engage in having extra units of a consumer good. It means that as we apply more and more of our effort to produce something, we get less and less valuable results. And therefore we switch to other productive activity because it becomes more valuable as we extend production in one line, right? Then the marginal value product, as we say, the MVP of doing something else becomes more valuable and so that's how we allocate across different productive acts by this logic, right, of thinking this way. So in my example, this person fishes one unit of fish, one hour or whatever the unit is, one hour of fishing gets the four fish, produces the consumer good, fries the fish, eats the fish, right, for dinner. Whatever is gonna mount some on the wall, whatever he's doing with the consumer good. And then instead of going fishing for another hour, he makes a doghouse. The doghouse takes him two hours, but he switches and he goes to the doghouse. Now, would he produce a second doghouse? Would he take another two hours in the day to produce another? No, he doesn't do that, why? Because the marginal value product of using his labor to produce a second doghouse is not worth very much. He only needs one, he's got one dog, he doesn't need two. So you see the logic of how this works. Okay. So just to sum up on the personal economy, if we look at any person, this is the way economists would analyze your behavior of mine, we look at any person, we conclude that their personal economy is integrated by them according to their perceptions and intellect, according to how they perceive the world and the possibilities of acquiring means and using them and so on. And then by the way that they value the attainment of various ends. So this is the way that we're acting. This is what gives structure and the possibility of economic analysis or any kind of analysis of human action is because we had these principles upon which all human action is based. Now let's turn to the social economy. In the social economy, as we suggested already, each one of us wants to integrate our personal economy with the personal economy of others. That's what the market economy is, right? The voluntary integration of our personal economies. This person doesn't want to produce his own fishing pole. Maybe he could do it, but it would take him weeks or whatever, it'd be inferior. He wants to go down to Agway. You have Agways down here? Agway's like a hardware store back in Pennsylvania where I'm from. And buy a fishing pole. He wants somebody else to produce his fishing pole, right? And he's just gonna buy it. He's integrating his personal economies, acquiring things now that other people have produced. Why does he do this? Why do you and I do this? Well, it's because of the greater productivity of the division of labor. Because I can buy a fishing pole for whatever, 50 bucks and I can teach at college and earn 50 bucks much more easily than I can produce my own fishing pole. And I'll get a better pole and so on. Right, you see the principle involved. So then the question becomes, can we as, is it possible to arrange an economizing social economy by valuation alone? And to think of the problem, there are two possibilities of whether this is possible. One would be we could appoint somebody, we could appoint a representative to make the choices for us. Somebody who's objective and independent, because if we try to do this personally, right, we're gonna be biased and we're gonna want things done. I'm gonna want the fishing pole for five cents or whatever, right? You know, whoever. We'll have this interest. So maybe we could appoint an unbiased person who would make the decisions for us. But hopefully you can see right away, we alluded to this before, right? The subjectivity of value does not permit this. There isn't any way that a third party can objectively compare the subjective value of someone who wants fish with someone who wants a doghouse with a given person's labor. It just can't be done, right? It's conceptually not possible to do. This isn't a information problem. This is what Mies has called a calculation problem, right? Well, so in any case, it simply cannot be done and we can't do this, we can't even do this by cutting out the representative and having direct democracy because we're in no better position to objectively compare the subjective value between different individuals than some third party, right? The problem can't be solved this way. And it's similar, of course, with the use of inputs or the use of means. If we're trying to, you know, who's gonna be the fisherman? Who's gonna be the baker? Who's gonna be, if we're gonna try to assign different tasks to different people, just on the basis of valuation, we can never do it in a way that's economizing. We can do it, right? It's easy to assign people and, you know, if you have the coercive power to force them into different jobs or whatever, or to give them monetary incentive to take a job, that's not the issue. Of course, you can do this, but it's not economizing. It doesn't, it's not human to do it this way because it isn't economizing. It isn't choosing the more valuable alternative over the lesser value because we can't know what these are, right? They're subjective, and so we can't know what they are. Now, as Ludaclan Mises famously showed us in his 1920 article on economic calculation in the socialist Commonwealth, the solution to this, of course, is a market economy. The solution to this is, if we have a market economy with private property, then we can engage in voluntary exchange in a monetary economy, and you'll learn more about the details of this today and tomorrow. Then we know who values things, consumer goods, let's say, relative to money more than others. Now we have a means for registering people's subjective value in a way that generates an objective evidence about one person's valuation relative to another. This guy goes to Agway, he buys the fishing pole for 50 bucks. Then we know he values the fishing pole subjectively more than he values the 50 bucks subjectively. I go into Agway, I see the fishing pole, I don't buy it. Well, then we know that the guy who buys it values it relative to money more than I do. And so that's the next step for us to take, right? Is to give an explanation now of money prices. Now, because I probably won't get through the whole discussion of money prices, I want to give you just an overview schematic. This is the logic of how it flows out, just in case we don't finish the whole thing. You can take this as a homework problem. So this is how our understanding of a market economy exists in the Austrian approach, an Assessing approach. So as we've said all along, we've got preferences of different persons. We've got this guy who wants the fishing pole and he goes to Agway and he's got preferences. And we've got the owner of Agway who has preferences as well. Preference, he's willing to obtain the $50 and sell the pole. And so we have a demand for the consumer good and supply of the consumer good. And that's the interplay between the two, which we'll talk about in a minute, is what determines the price of that fishing pole. It's $50, not $100, not $5. It's determined by the voluntary exchange activity of people expressing their preferences for things willingly in the market. Now once their price is a consumer goods, then two effects come from this. One is expenditure for consumers. So the guy who buys the fishing pole, he expends $50, right? He's making expenditures. But the other is the revenue for the entrepreneur. So the entrepreneur, Agway, earns the $50, his revenue. So every time there's an exchange of a consumer good, the seller gets the revenue, the buyer gets the good, right? The seller gets the money price, the buyer gets the good. So this revenue that's earned by Agway allows the entrepreneur at Agway to fund the purchase of producer goods. So that's the revenue stream that he needs to fund his expenditures to buy the inputs to, you know, he's just a retailer, but he's buying the pole, right, from a wholesaler and the wholesaler's buying inputs to produce the pole so on and so forth throughout the economy. So that's where we get demand for producer goods. The supply of producer goods just come from the workers and the other produce, it's just preferences, right? And again, we'll show this if we have sufficient time. So that's where we get the wages of the pole making workers, the, you know, the price of the fishing line and the component capital goods that go into making a fishing pole or whatever the product happens to be. The prices of the producer goods. And then the prices of producer goods generate two effects. One is the income for the seller of the producer good. So the worker who makes the fishing pole gets, the wage is not a cost, right? It's an income. It's a cost to the entrepreneur. And so on the left-hand side, we see the first principle of economic calculation. The entrepreneurs are going to try to generate production of goods where the revenue from the purchase, from the voluntary purchase of the buyers will more than cover the costs that the entrepreneur incurs in buying the producer goods, right? That's, that then would be economizing, right? Because the producer goods like this worker who helps to make the fishing pole, the wage is set by the general market for that kind of labor, which means that other entrepreneurs in other fields are also paying whatever the wage is, $25 an hour or whatever the wage is, to hire workers like this. And so if Agway or if the manufacturer of the pole is hiring this worker, what's being given up by other consumers is the value of the things that could have been produced if that worker would have stayed in a different industry. Why doesn't the worker stay in a different industry? Because the entrepreneur is paying a better wage or giving better working conditions or whatever more favorable job opportunity in fishing pole production. And so you can see how the economizing structure of the market is formed just by our own attempts to integrate as best we can our own personal economies. Okay, whoops, so let's go to the, let's go to the first part, sorry about that, let's go to this part, this top part and give an explanation of the prices of consumer goods. And if we have time, we'll do the producer goods. So let's start with this case. I'll switch my example a little bit and go back to the iPhone SE. And here, let's start with a simple case. And the simple case is, suppose we have an iPhone SE Gen 1 that was sold back in 2021. So it's used. So it's owned by just another guy or gal, right? It's not owned by Apple. And suppose we have this potential seller here on the right hand side who owns the used iPhone. And then we have somebody who comes in contact with this person who sees that the person has this phone and who's interested in buying it. So the buyer on the left hand side. And the parenthesis around the iPhone SE for the buyer means the buyer doesn't have it yet. The buyer's only sort of anticipating what it would be worth if he had it, right? The seller has the phone. And you can see by the numbers I've used here that a mutually advantageous trade is possible, right? Because the maximum that the buyer's willing to pay is $340 and the minimum the seller's willing to accept in trade is $300. So they can certainly form a mutually beneficial trade. Somewhere between the maximum price the buyer's willing to pay, the minimum price the seller's willing to accept. If that condition exists, then they can have a mutually advantageous trade, right? They can integrate their personal economies. The seller would rather have the money as opposed to the phone to do something else and the buyer would rather have the phone instead of the money. And so this is how they would integrate. It's not a logically difficult issue to notice that they can negotiate the price. Somewhere in between $340 and $300, they can higgle and haggle and come to a mutually agreeable price. I mean, they may not do this, but it's certainly logically possible that they could do this, right? And then if they can, then we've had a voluntary exchange at that price. And then of course, all we need to do in addition is add competition. We need to have, we have a market where there are lots of buyers potentially and lots of sellers. And then we find willingness then and ability to make trades within the group of people in the market. So that would be the next step. Now before we go to that step though, let's take an intermediate point, a side point which becomes important too. Because even my simple example, illustrates the fundamental principles involved in all market exchange. And so this is important to keep in mind.  On the demand side, the preference of the buyers then always looks like this. The value of the good obtained is greater than the value of the money given up. If they buy the good. Now it could be of course that they're just looking and potentially buy it and they don't buy. But if they buy the good, then at that moment they value the good, the acquisition of the good more than the surrender of the money. And as far as the value of the money given up, there are two categorical possibilities, right? They could just keep the money for their own personal use, do something else with it, right? Or they could surrender the money to a more eager seller. Maybe our iPhone buyer negotiates a little bit with the seller that we mentioned and the seller holds out for $300. Well, maybe that puts in the mind to the buyer. Maybe I can whatever go on eBay or extend my circle of friends and find some other potential seller of the iPhone SE who would take it at a better price. So they may refuse to trade on the basis of that with any particular seller and seek out another one. And if so, then, well, they're giving up their money then to the most eager seller that they don't deal with. That would be their alternative. By the way, we call this, if we had a case where the, and we do lots of times, right? A case where the buyer holds out and doesn't trade. He doesn't give up the money. We call that reservation demand. But we'll see that term is sort of important in further discussions. And then supply is just the flip side, right? In every act of supply, no matter what the circumstances are, the seller is just engaged in valuation, is just saying there's the value of the money obtained which ranks above the value of keeping the good and the value of keeping the good could either have a personal use or it could be sold to another eager buyer, right? Those are the two categories. So again, Tim Cook has 100 million iPhones in warehouses all over the place. And he's perfectly happy to sell one of the iPhone SEs right now for $429. Do anybody who puts up the money? Because he has no reservation demand, right? He has no personal use for the 100 millionth iPhone. He just wants to, he wants to sell, he's produced it to sell it, not to use it personally. That doesn't mean he couldn't, right? That doesn't mean in the abstract, a person couldn't have reservation demand for the good and hold out, just keep the good and not sell it. Okay, so those are the general principles. It doesn't matter if it's Tim Cook or if it's this used iPhone that's just owned by Joe Smith or whatever. These are the general principles always involved. Now let's go back to the market analysis. So this is my example of the market clearing price. Suppose this is July 1st, 2000, excuse me, yeah, 2021. Suppose this was the market a year ago, roughly. What we know if we look at the empirical evidence about the market a year ago for used iPhone SEs is that data point, right? That's the only actual real empirical evidence that we have in markets. We have the price and the quantity traded. The supply and demand curve are just conceptual structures for economic analysis. There are no data points on the supply and demand curve. That's all just our concept of what would happen if the price were higher, what would happen if the price were lower. But the data point is always the argument is it's always at the intersection if you will at the market clearing point for the conditions of this market. And so our data point, my example, the data point is $300 price and 2,100 iPhone SEs used in July 1st, 2021 were bought and sold. I don't know what the actual data was, but something, right? But we could get the actual data and see what it is. Now the other point about this is what about, why is it that, what's the argument that there aren't data points up here on the demand and supply curve? Well, the argument is that if the actual price in the market is 300, if the price instead would have been 320, let's say, then there would be more sellers who would come into this market eager to sell. They're holding out again because they think the $300 price is unfavorable. They have reservation demand for their iPhone. But at $320 or $350 or $400 or $500, they come into the market and try to sell, right? At some higher price they'd be willing to sell. But on the demand side, as the price is hypothetically higher, the number of buyers would be reduced or certainly not increased, right? And so we wouldn't have a matching of buyers and sellers. We wouldn't have a full integration of the personal economies of people. We would have, as economists like to say, excess supply at prices above the actual price. We would have some sellers who can't find buyers and therefore they can't integrate, right? They can't get involved in a voluntary exchange. And if the price is too low, below the market clearing point, we would have excess demand. In other words, we would have lots of buyers but not enough sellers. And so we don't get full integration. So what's happening with the movement of prices, and we're talking just here about market prices, right? Not interfered with by state coercion and so on. The movement of market prices is that prices are moving to accommodate the integration as fully as possible of our personal economies. We get the best result in that sense, right? The fullest integration of our personal economies. To put this a different way around, given that there are a certain number of iPhone SE 2001, Gen 1 on July 1st, there are a certain number of them, 20 million or whatever that exists in the world. When people engage in trade, the good is moved from people who value it less with respect to money, into the hands of people who value it more with respect to money. The market is continuously moving goods away from people who value the goods less and into the hands of people who value the goods more. Again, relative to money, right? That's the accommodation that's occurring or one way to put it. Okay, now let's move to the case of new iPhones. So we haven't explained that case, right? We have only given you the general principles of how we would start the analysis. So let's suppose we go back again to July 1st, 2021 in order to think about Tim Cook and his entrepreneurial group who are anticipating the beginning of the selling of the 2022 model, the Gen 2 model, that started, I don't know exact dates, but started let's say on October 15th, 2021. And now they're selling this through October 15th this year, right? This is the Gen 2 iPhone SE. Okay, so how would Tim Cook think about this or how would any entrepreneur think about pricing the product? And the answer is if they don't have reservation demand, which is the way I've drawn this, just again for that particular case, if they don't have reservation demand for the good, they're gonna try to sell the good to fetch the best price possible, right? They're gonna sell it for what the market can bear. And so the way I see this particular circumstance of the iPhone SE, I'm just looking at their behavior. So they started selling it back in October and the price was 429. And as far as I know, the price has been 429 every day since then. Maybe they have a sale every once in a while, right? They do whatever, a Christmas sale or whatever. But they've decided in other words that $4.29 cents is the price. That's the price we're going to sell out. And then we're gonna accommodate people as they come day by day to buy at that price. So where did they get the 429 price? Well, they get it by, you know, if we diagram it this way, they get it by their assessment of the fact that they think that if they charge $429 over the production run of this iPhone, the one year, right? Selling run of this iPhone, that they'll get the best, most revenue possible. If they charged a price of $500, their revenue would be less. If they charge a price of $300, their revenue would be less. And then as I suggested, they can do different things, right? The entrepreneurs can do different things day to day. Once they decide, okay, the sort of price, we think that the overall market through the selling of this good is going to command a price of $429. They can do what Apple seems to be doing, which is we're gonna keep that price the same. No matter what, there may be days where you don't sell a single iPhone. Then other days where you sell $50,000, but we're gonna keep the price at $429. And how are we gonna accommodate the people who come to us and wanna buy iPhones? And we're gonna hold inventory. We're gonna produce the 20 million iPhones, and we're gonna put them in a warehouse. And then every time we get in order, we're gonna ship, right? That's how they do it. Or at least it seems that that's what they're doing. Now, they don't have to do it this way, right? They could just say, okay, we've got 20 million iPhones and we're gonna sell whatever, you know, say 50,000 each day. And if demand is shifting around day to day, we're gonna move the price so that the market clears. They could do that, right? They could let the price adjust. Or they could do some combination. The supply curve doesn't have to be vertical. It could be horizontal. It could be between the two, right? This is just an entrepreneurial strategy. But the point is they're always going to ask the price that they think will generate the revenue that they anticipate, right? Will generate the best revenue for them. So that's how the price gets to 429. Let me say one last thing. I won't go through the rest of the slides, but about the prices of producer goods. But I'll say one last thing. I put the revenue amount at the bottom, right? $429 per iPhone. They sell 20 million. That's just my made up number. I don't know how many they'll actually sell or plan to sell. But if that's the number, then it would be $8.58 a billion in revenue. And they'll earn that revenue over the year, right? Every day they'll sell some iPhones and that revenue will come in, and just like we explained already, they're earning that over the year. But they incurred the production costs earlier, right? In fact, they incurred the production costs, probably most of it, at least before the summer of 2021. They're incurring all the production costs first. And so notice when they're hiring workers and they're buying materials and they're amortizing the value of their facilities, the Foxconn facility in China and so on, the things that they own, right? When they're assessing the expense of all this, they're doing so only in anticipation of selling the iPhones to generate this revenue. And this is the entrepreneurial foresight dimension, right? That's expressed in the market, economy that you'll talk quite a bit more about today and tomorrow. So thank you very much for your kind attention. Thank you.