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Math for Economists - Lecture 11

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It It started last time we started taking 2nd derivatives of functions of 2 variables but the 1st thing I wanna do is I just wanna help you realize that all of this theory here can be done with more than 2 variables typically we start off with 2 variables every practice with that but occasionally will Will Cut have expanded to 3 variables should always keep in mind that you could do this with many variables and the reason it's important for you guys is because when you look at economic models they usually have a general model which has Many different goods and good Something like that said to goods treated so there's always a generalized case the theory works for everything but have you could only see pictures ever when he had 2 variables You can earn limited Friday will you could see in your mind if I have many variables that Hardaway described that I wanna say will have had 1 specify how many variables that just They have 1 3rd ahead of them so that's a function you just wanna think If if that's hard for you look at which you want do exist they got it will this is just like at the extra from single variable calculus that when you have 1 variable And then if I have to that that's what we've been studying but we really could extend on And then there's an alternative notation that goes with that so have a function of n variables what that means is it's a function and The input from R and that and variables is a stack of numbers here and number I take those of put a mall in air and that gives me a function value as as a signal that they put in a vector in and dimensions get out a single number so that means it's a function from R into are Our case The deal in general and my well have a simple case The 1 I always years Obama's 1 x x 1 square plus X 2 squared The author paraboloid here next 1 Next to those variables and if I wanted to this function and in the same notation then I'm gonna say will you see in any year and their habits to hear This is this is the 1 comma I added to here so that means I'm gonna be going at goes on hard to when you add it this way you can the picture so reveals that that's what this is because here's the input fears that copy of Archie began there on the floor Take points in our too That's a point here and then you call and the functioning he function value that the number and that's defined by that access here so they're here 3rd dimension here so you start off with 2 dimensions new map that in the 1 that's that those of the functions that were studying here OK so we saw the derivatives were complicated 2nd derivatives especially because if In this case here But during the 1st period of a dead end 1st derivatives I get at 1 of my derivatives all the negativity that someone 1 had said all the way down At the other end and then now what do you do for 2nd derivatives for 2nd areas while stake F-1 1 donor needed do at 1 end and different 2nd derivatives just for this guy and then I go this 1 I had end 2nd derivatives that's gets really tedious and complicated so just symbol elegist saying these are off that I J With all the possibilities and then what I say but i and j just so that I know we all know we're talking about I have to say that But I between wanted it That's how it's that's how we see how I talked about infinitely many possibilities on 1 little compact notation that's the nice thing about pack a lot information in the 1 symbol OK and then 1 thing that we noticed last time is that if we took the derivative with respect to 1 variable and then we did the other variable and we switch the order we got the same 2nd derivative so we can write that down that's called young swam in young serum says it doesn't matter what order you do these derivatives A case I say English it doesn't matter what order than write that down and math symbols I could say that I Jiang hedges wrote down every possible 2nd derivative but if I were a switch order that would switch the indices and so what I say FIJ is eagerly that JI that's the symbolism that goes with what's called Young's and I read it in English I'd say it doesn't matter what order you take the jury is always get the same 2nd derivative and I say always that's not quite correct you have have derivatives their continuous In this class it's always happen so you not examples where it doesn't this doesn't occur sooner feel like it's always true and it probably always true in economic models that you create but in the math world there is this 1 stipulation that you really you have that have all your derivatives have to be continuous at least the 1st to be continued a level that we don't get to hear but I'm making a video I can't say something that's also not always true but for your purposes OK so see this in action this is actually helpful when you have several variables because as soon as you have even 3 variables there's 19 2nd derivative and there are 2nd derivatives so this makes it shorter to do on line because some of the same pockets What's I was just a couple of examples here just play around with it warned that they a function to variables son in this setting here can't necessarily see with this looks like
You put it in the away from our further something you have some tool that shows it but we can still do all these things have won the the derivative with respect to the 1st variable that's 2 x 1 in the neighboring the X The 5th on along for the ride and then end to the derivatives with respect to the 2nd variable pages do the derivative yet 5 x 2 the 4th and then bring along that X 1 squared for the ride its behaving like a constant in that case It is command of the problem refined the greedy Internet and that 1 native OK so I've got the 2 things that need and gradient that's what do the proper notation him ahead of the gradient At that point letters That but this points in here I get to times negative ones negative For the 1st component and then when I put that in here this stays positive positive That's the gradients at that particular point and then has that I can take the derivative with respect to the Variables at 1 1 his me 2 times executed But he like a grid there see don't lose track of what's in 1 location when you have a lot of stuff to write down sometimes you might have pluses in there so might be a good idea that grid look-in derivative of this with respect X Q Is And attacks 1 x the 4 Take Maliki museums there be the 2nd part because if F 1 to goal at 1 2 1 1 2 equals at 2 1 this year is F 1 2 that means this guy down here as been saying that just break down but Probably better than his writing it down interesting it you muzzle take the derivative meanders to make sure you got it right stirs is really a check to see that you're doing everything right you should get the same thing so not as old as mentally check that you do just to make sure you did that focus so if I do the derivative of this with respect X 1 I get the 10 in front X 1 X 2 force that is what I have than the last 1 derivative with respect to x 2 20 extra 1 square X Q the 3rd and then we can plug in that point The plays the point I get to a negative too Man tens and those to be the same as the last 1 is A negative 20 So OK so let's move on and get a little taste of 3 variables 3 variables that X 1 next to 6 3 in the of what's of Texas can messy looking out all very was yet to be really careful you go up a little subscripts exponents of stuff Look around and you guys are deal with Matt constantly on Iike I'm always deal with masses of that big a deal but I remember what it was like when I was 1st dealing with notation It took a lot of concentration and to get that right copying it down takes come concentration you're probably noticed OK so what we wanna do it is that I wanna find that That's fine gradient ahead at its and I have to say it's at a point but now I have 3 points the quality and so it's gone 1 of 3 differed X 1 X 2 X 3 summer plug that poignant once finally get my head gradients and my passion for caseloads get that 1 has to have 3 that area agreement boats go get those thing cast at 4 1 that is needs Start 1 x 2 plus 2 x 3 plus 0 this part 0 . is driven with respect that's 1 thing that's it with respect to x 2 I have forecast on ask where there is no X 2 years this derivative is 0 And get 2 x 2 times And the last 1 derivative there is no x 3 in this term so the derivative 0 here the derivative is 2 x 1 and the derivative here is X 2 square Frost off off at art Right now the gradient at this point is Um at 10 . 6 3 which is fairly compote computational that's the 1st time we've seen 3 variables sulfur 3 variables that 3 derivatives and just go about our business the same way we were before But there's 1 more variable to keep track of how God himself what we take our 2nd drill is there's a lot more to keep track of what had 1 1 peso
The top line up there the top-line that's the F-1 anomaly to the derivative of that with respect to x 1 again there's only 1 tournament has an X 1 that's the first one by a items as I'm doing this you should see few believe me OK but now it's a mistake with that 1st function F 1 another do on derivatives that at 1 1 1 2
That would be a someone there is no way to determine the 2nd part and then had gone through F 1 3 will be
From now on the 2nd function as to the derivative with respect to the 1st variable PART had won the luck he see these 2 are same that was young steer this 1 Eddie knew this 1 so we could it is written it down but I recommend doing it just check your work make sure the same of highlight that that's young spinner Add 2 2 at 2 2 0 It's add 2 x 3 I had to really the derivative of this 1 year with respect x 3 years me just 2 x 2 It onto the 3rd 1 I had 3 1 Fayette 3 1 Madrid adheres to voting is secondary U.S. 1 3 box are those that the match at 3 Clearwell Irene lf 2 3 of others on past agreed to act until have 3 3 but there is no all 3 in the 3rd
Line there so that 0 any CD Indices tell you exactly what a put these things in the matrix this I just couldn't exactly the right spot here's the 1 1 0 1 2 1 3 2 1 2 2 The rather proper locations when put that all in The Hessian by just stay in the same location and actually lets a close to the 1 Warwick plug-in 1 1 1 Common evaluate that had 1 1 1 OK so Playing Arlette NATO 8 8 2 8 2 2 2 2 0 there's the Hessian at that particular point now where this is headed is this Point is in and be our critical point where and when we do our calculus would do with calculus you have a function data derivative the 1st derivative he said equal a 0 get were critical points right so that to be the setting that when Indian we will we want try find Max's isn't millions want you have your critical point in you in the 2nd derivative to determine the can't cavity and then I'll tell you others concave leprechaun came down we don't know yet had a look at this and say whether this means concave upper concave down The we'll will get that but that's That's where we're headed his their 2nd derivative This is to tell us about Khan caddie at their 1st derivative that in a tell us about slopes of teams of the plane and that gave us the critical points from city of 0 If Lara battled We 1 another topic to cover before we get to that Max Min analysis Totally this for a little bit With the house Now say This is the new topic differentials soap this is a topic that usually people forget its there's just 1 week in your Calculus glass maybe it was on the mid-term maybe it wasn't made was on the final but this was way back when you take your 1st
The master 1st quarter of calculus And it's not a topic that people keep in their brain so we need to do is go back at me how comedies and disaster Selby remember what this is about so we had to go back to map to a just we do do this just a little bit to find out what this sets by Sold 1st of all what kind of word trick is triggered in your mind when you see this Differentials lower differential in the in the English language There's a differential on a car It's that big ball 50 slid under the car year head on May the differential that ball on the back so it is that with the different So what it is is not here Here is your wheels and we have that In the middle already So when you're driving if you go around the turn of this wheel is traveling a further distance Who actually spins faster than the way it uses the physics of dates if they go from here to here this is a further distance than that of the army outside the are so this we'll travels faster so what happens if you don't have this little mechanism to counteract that the we'll start by bouncing off the ground it will build a stay on the ground so you have this thing here that makes it even lets This will go faster And the differential is the difference in speed between the 2 tigers
So it should anyway it's a word in English but what it really means is is just it's there is the differences is the differences that we should think of it so high that apply to the functions in math We have multiple variables we said buzzer say we have less just after the next British
At OK so here we have a function of that And the question is a gag odorless at that particular point like X not there and I let x not just wobble So I mean by wobble what I allow for to do is just be off by a little bit and infinitesimal that's the notation infinitesimal but There's another point them if I'm just a little while ago 2 points x next not And this little distances DX we really wanna think it is really kind so here's the question if I let acts just very villages while a little bit How much does the function well that's the question that comes up in the concept of different Sophie look at how the basic idea is and I wanna know how much at changes if I just let acts change a little bit The what I do is I go appear at X nod and I hop on the changes were and I just see how much y changes on the tangible basically a disease that so what happens as I get a little wiggle here by plug-in exercise that gives me a death of X not our why not And so when I let it with a little bit here and if I wonder how much it will those appear I just approximated by using the tangible and that's gonna cut a little wiggle here And we can't IDY so the question is if If you get a little wiggle On X how much of the way Scores on war that the basic question and it all comes from the equation of the tangent usually get memorize this is a formula that's why doesn't stick with you But if you realize that it's really comes from the team's Atlanta's formula might stick with also make a little more sense multiple variable so what is the equation and candlelight Willy Claes Any line here's why Minus why not equals the slope m times next minus X the equator value But then we're talking about 18 July and we replaced With a crime that gives us the slope That point So that's the equation of any Tianjin line in the universe right there now look back at our situation here it is isn't it this is a little bit too complex what What's y minus why not well let's just get little change in white soldiers Khaled is 1 when it when he's a really really close we have notation for that to to to tell you that we have an infinitesimal than over here and a promise that Not and then what's this X minus X not that the DX that's the infinitesimal DX and so they're here for me if I wanna know how much does the y wiggle given that the Mets Whittle's a little bit I just take the the amount that X with multiplied by the derivative that tells me how much y And the intuition that is basically hop on the Tandem line indices are much at changes use that information as an approximation Sosa's could example of that in end a single variable setting will make a lot more sense we do it in multi
Talk preamble would be that I have wide those eggs where and I wanted add eggs equals 1 so that
When I say addicts equals 1 of picking this location here with saw how much does why exchange backs changes is that were wiggle raises a little in economics the little with little might come about by inaccuracy in the data you collect you collect others day-to-day you know is a off But a little bit so you wanna calculate how much is your function off a little bit if all your data was off a little bit that's the role that this might play an economics said OK but some asking How much as white changes exchanges bank I tell you it it is it's set point with a 1 . 1 . 1 day so if I look over at the formula How much does why changed this part right here What that's the question that Formula I making go get that kind of not then it got the exits given me the only thing we need to do this problem is wide time Plug-in why Prime ad A X equals 1 that was to conclude D Y like this is a prime of banks and this is how bags as well as its so what's DY that's that kind of elects not I'm DX this that we just computer to this was given to us when 1 of the amount that white changes . 2 The picture that goes with this year we get a complete picture because we know everything we know a parabola looks like we know where x equals 1 nice Picture here there is at the VAXes there's X equals 1 and then what I've said is DX is . 1 says just as little wobble their that's that was equal to . 1 And then what that cause if I go appear to have y equals 1 the function value that little wobble here cause they Wavell of . 2 up there that's what happens that's what we were that over China discovered this meant that here this week questions about the differential about to move at a variable yes yes it is the same thing as the approach to become the same section of the book and cut and that's were using the blue line the Tianjin line to approximate that change when you do linear approximation of what you do is you you You would sell them like this you have why equals why not plus That crime that's not X 9 aspects not the approximate the function using this to you bring Missouri other side that's a linear approximation But I will pit where the father of a son really emphasize in calculus it's easy to forget that topic probably repo getting trade now related the buyer So Rueda in math for Alice although the trailer and this is what we really have to do so Now that 2 variables so I've got to things that could walk the same concept that How much it variable wobbles Lula how much is the function change tonight that 2 variables will say I keep the notation and say Why is the function of function 2 variables but Sabol picture the goes with this got some floating sheet there That said That's my functions to make that died We like I am so the setting yeah but say this is X 1 at this is X 2 OK Over here had a single point will have a single point here but here at the Collett But that's not why not that's it said it's got to value sex not wine at how like that notation the mysteries X 1 next to because I have a big otherwise there and that implies that the function of this huge exporters next OK so now what's gonna happen they seem situation year said that you variable so This guy can wobble And this guy can wobble in each direction and that creates instead of creating a little segment of DX segment creates a little square within that Square I go here and that caused the function to wobble a little bit So uninterested in here just like I was over there and D. Waggoner found it's going to be found by using the differential from these variables so in this direction I have X 1 and in this direction Ranieri that's the other direction too You put those together tells you the change in the function listed duties that same day I had an extra variable that I don't see itself isoleucine equation of tangent but now that 2 variables kickoff and go here and I got a steam engine plane there and so will reduce their hop on the engine plane and then plug in these these wobbles from these variables see how much the function changed on that plane this is not really part of the class calculating equations of Tianjin plans although it could be as we can skip over it to do that but Show you where it is Directly analogous to that If on the left side wine might well I'm not just just like this over here look at what I had I have this slope times X minus X not so now when I have to variables I might have to slow some in 1 of these little gizmos you're free to variable rate that so where are to derivatives There's F-1 Andrews 2 that did you the slope in the X 1 direction limited U X minus 1 plus 9 of the slope in the white next you directions X but that's a little bit that's slowed We're there and that's I I don't really need the engine plane show you the saying OK So so now what's gonna happen is this right here just like over there that the change in what This is going to be pretty I have F-1 what's this when these 2 things are closed That DX 1 and when these 2 things and close just low did Infinitesimal So that's not how you find the differential In 2 variables it's exactly the same as 1 variable assist since we had to do is do the same thing for each variable and they had the results and I give you the change in the function but intuitively what's happening here hopping on the cans plain using that to approximate Then there would be a linear proxy This is a linear approximation to variable if this is not a line it's
Ruth Eisenhower what new an example we had a formula If this If you get over the intimidating nature of the occasional that calculations for these problems but it's it's the notation in the terminology that's a little intimidating pockets of 1st example we have Hawaii also really X 1 minus Q
That's to say what this ad . 1 2 I tell you what the differentials are each very DX 1 is . 1 I'm back . 2 It's really just computation everyone 1 of finest Dewalt D Y is equal to be that the formula for it Variants That 1 c x 1 DX to and The DX 1 of the DX to given the only thing you have to go get that the partial derivatives and then plug in that point at 1 . F 1 is a sequel The Three Here visually nothing the players is a number 1 this constant 3 and negative to close GUY that 1 D X 1 That pursued DX that's just 3 TX 1 . 1 that to is native to DXQ . 20 so that gives is me a point 3 reminders for Negatives . 1 They're so that says that for this particular functions This is a linear function as the plane for this particular functions but if I led the very at this point I let the variables wiggle by that much
It causes the functional down by by . 1 at that location and back Do But I had another example they do so 2 1 more example Take that example here and just where those guys get that 1 where you plug in the valley sold Take that Function was extend that 1 and will stay at the same point at 1 2 the same preferential so let's just go The process is the only thing I needed to do in that case to get the result was defined the partial derivatives so here that 1 is 6 x 1 ahead to it negative for for 2 and then I want at that point so that 1 evaluated at 1 2 That's 6 and had to evaluate it wants to that I 8 that 1 GXY That is not DX I am the only 6 Times . 1 plus native gave her point or 2 . 6 minus point once Things I went out whack a lucid 1 . 6 per se What the proved negative case it's a simple computation that that um no understanding intuition behind it that could apply showed it to use maybe add a little better that now that's easily something that you don't come a class with something that you can forget lady had tuition for Pierce a different is how much is the function changes that the variables change that's what's gone E
Back but us Plea Is there have it thinking caps on this year's if you're just gonna half watching this next part will be something that little challah G little salt water and do here
If we can use this concept of the total differential I find the slow of level said so maybe there's terminology that is you not totally comfortable with Sosa's posture 2nd 1st just a maybe have a picture in your mind it where level said is is a couple examples so I let let let's just have level sets here Let's take up paraboloid example is to remind you The level sets errors as they move up in levels I just look and see what what Graf I had at each level at the level set so as I go up as galleys concentric circles and we grab them in 2 dimensions so when I got last time was a budget concentric circles geezer the level sets Of that Function years that the X Y that's where most squares and those levels said where said these equal the constants and OK those go back and look the language would use this differential who knows how face escape that fire The find the slopes Of levels that Casey that we might be interested in The slope of that function once we go get the level said that's the slope It is just a two-dimensional instead to variable so we not a fine slows from calculus but on The little bit of a job here OK so that our goal now we have to discover how can we use the differential to find slow planning to see how they relate it I was just talk about all D Exs and wobbling in everything right now all the sudden entirely slopes of levels that they seem completely disconnected until you realize that had At on a single level set pace and I write that language there is wanted a picture of a year you're stuck on this level right here what's that Wet Where's the differential here What's how it how much the function change when you're on that level members The d why tell you how much it moves up and down right is the change in the function so if you're stuck on 1 level It doesn't change it all so on a single level set the function and will What we do a final levels that we say will concentrate so that means that function is constant on a single level set in order that meet that needs the differential 0 It's on 1 level said if not the wise not changing so no matter how much I wiggle the values as long as they stay there The the function doesn't OK so that's how renews the differential a final assault boats that see that owner But I don't But Arrives over DEY that we know the formula for that I will not if it finds said eagled 0 so on a level said DYE 0 so I could replace D. wireless 0 I could see what happens will at this stage is still We don't even know overhead a where leaders slope that think about that would slow is a slow slope is the Grazia but to variables Please comparing 1 changes a little bit Dillard changes some I compare that change that gives me the slope And what Where similar to the notations that we have Ursula away at that time that we have DUY The what were wine asks that might have D next to DAX that's a slow that's the slope that's that Little change in next to over the change in tax 1 it's just that you not used to using X 1 next to you used using wine expert if I replace y with X X by X 1 and that's the slow Still it's all 4 years This implies that a pursued DXG was tabled negative 1 DAX someone A man implies that EX 2 over DAX wanted well OK I bring the DAX 1 or there that Disney DX 2 idiots 1 and they really have to order the other said and get that and this slow out of the Lopez said It's mobile missiles delay they'll all we can write out pictures and everything see what's going on here saw it in general it
When you deal with X 1 next to it helps to replace sex to why you use the higher number as that Higher variable so usually exigent 1st Variable so that X 1 usually wiser 2nd variables LEX to 200 easily the 3rd once about exterior that helps you can switch it back so that you can deal with that Then a beachhead go back and do it together notation again this is just a computational
If if I gave you the command find the slope of the level said leaders of the religious use this formula and what does that entail basically this entails getting the 2 derivatives that's always better than going get derivatives and he makes it in this particular way And that's what you know the formula that I'd do it understanding tuition is a little bit more will try to get a little bit of both book at solicitor still seeking can visualize what that my look like that's all hard that With Danone enough of this maybe in the utilization so that hard for four-year that's OK just take away the 3 in the 6 were aware if it didn't have the 3 in the 6 banker David Villa will yet it's it's this picture keep seeing it over and over so I want you to see over the question is what was the 3 in the 6 do with Maybe they just become a war that thing little bit instead of getting circles as my level said some it ellipses so it's it's an elliptical parabola that's what the kind that at the conference in front of a did that passes it appear paraboloid if the counters with the same like born for that still in India parabola When they're slightly different I wore pit 1 in 1 direction awarded slightly different another treats ellipses still basically the same picture but you make college the I am parabola I I I I I saw each 1 of these It's level says Ellipse again user command find slow blood above the level set up at 1 1 What Book OK there's a lot to do here so 1st of all let level said early on I told you that we can find out buckle units and little tell us what levels that were on so at warned 1 that I want 1 is equal to 3 . 6 9 so we're on the level set Where S equals sows go graft that means it would do if I told you go Grethel the level set People's 9 put a 19 year new draft that equation of 9 golds 3 x 1 square plus 6 6 2 square Legislator totally accurate picture but that what I want is to realize that this is the way you like the 1 equals at When X 1 square over 3 years plus QXQ squared over 3 that's an ellipse You know how much you remembered that loose OK Cell but yet so lips there's might Level said that he now I'm interested in the slope at 1 1 0 1 1 Israeli here What Is the What is the slope of that that's basically what we're looking for and the formula for his given by that So I need my partial derivatives have warned that a 6 x 1 2 is equal to 12 x 2 and then I wanna do that at that particular point stood at 1 . 1 1 and that 6 2 at 1 1 6 12 tonight have numbers that I need How the crawl on the globe
Still the slope above the level said that that people's minds it's that particular 1 . 1 1 years X to it over DX 1 legislator DYO 30 X formula for that Israeli it's minus F 1 or F 2 negative and then a half 1 was found to be 6
It s to was found to be 12 I made a hat and that's the the slow that negative I thought I Good house That is not as bad as you think it's worse right now and then you will think take with you from this class when you take economics courses in this can issue comes at McGill you all that terminology From economics that goes with all this terminology so we're learning the math but then it has very different vocabulary when you get into economics Wanna make the connection between the map that were doing and causing deal breakers this here so we stopped residual vocabulary from economics and put this in an economic setting makes connections for quick end when I do this with 3 variable of more attacks You need this a little bit do the problems in the book because in this particular section of the book only has economics examples Emily had any pure all the terminology in the book is from economics when you try practice this since I have to make sure that you have a little bit of that without having to spend too much time reading it looking it up on Wikipedia OK So in economic theory there's 2 types of things they come up there very common for this context 1 AR production functions and production functions The EU there really interchangeable with utility function as production functions where my inputs Think of them as good my goods are had that labor and capital and my utility increases when my production increases so high I want maximum production or I want maximum utility will you decide what you wanna call utility production so they are kind interchangeable but but slightly different because In an utility model if you have substitutes like Coke and Pepsi those urges completely interchangeable wears laboring capitalism necessarily perfectly interchangeable so they are slightly different models but when you're learning it shed think of them as the same That's OK so we have a production function example of that might be something like this is to keep something in mind there's a there's a card Douglas production function and by the way they have Cobb Douglas utility which is exactly the same thing is that these fractional exponents of that really is interchangeable talk at so ballots gets That the map here so here's a function P of 2 variables if you like it here on your mind you could just think of it as that of X Y It's a function at 2 variables but when you have a function to variables will you do you go when you look at the level set so the function but you don't level sets when you're in a cup Douglas situation an economics class they will use this term level level Kyrgyz they might introduce it but the term the years I want I hope that last time Isaac wanting same quantity of the level said is where the function is constant so think of it like this if I picked this to be set and the level of production gives me a graph that's called an icicle because every point on Iberia graft is The same level of production talk it so there's the level said and then we have the slope model set which is what we studied here An editor for that too that's called the marginal sing-along in your head Marginal rate of new wanted cellular ears Technical substitute then ask yourself why it technical substitution wise substitution widening that language somebody somebody made a conscious effort to grab that word from the English dictionary and put there by the technical substitution You're substituting labor for capital is that a true substitution nite is technically substitute I hate it if they don't are not interchangeable like Coke and Pepsi that's a true substitution when you're substituting laboring capital yet you're so the team could you end up getting the same production out but it's not a truce of technical such a of course that's abbreviated and
Some us Get the other situation is facility Barry
If they so For a utility funds not used key of labor and capital others have here and that good wine and good actually utility functions are more consistent with the math notation as easily just good 1 good too good said whatever Howard The you Avenue get certain utility at enough Using the mixture of all those goods by OK Zollar typical 194 X 1 X 2 is given each says Lord diminishing returns by putting Ysquierdo around is is also called the geometric mean I'm OK so that the local utility EU to indicate utility think substitution in offset competitors OK anyway Let's go through the same thing level said Work Over here they're called Isik wants when year Cup Douglas production functions situation over here they're not call a quality they are Isik wants the same level of utility but here they're called different so did you ever think a widening called indifference But he is awaiting difference quite different But make this connection between a level here's the match the level said says you take the function you keep a constant and you have a graft and every point on deputy the same value so what that means is If I said utility constant but means I've got the same level of happiness a utility from wherever I am on the left and different aware monograph I don't care The same level utility faggot at 3 coke someone Pepsi that is me in that The utility of war but I have 1 Coke 3 Pepsi's agony utility for an indifferent to the situation that's why they're called different F because you always get the same level on a particular 1 and then the slope of the indifference curve but it's really a spilled level set by OK so that's not called slope of The difference Kurth knuckles local level set in a time of economic setting for it's called is the marginal rate substitution And that slope they're just tell you how
How he'll change from 1 product to the other because they have a certain relationships with their perfect substitutes Then now have a certain slope if imperfect not different slope That's the connection 1 more outlets mrs them RTS fur
Production functions and my arrest for utility function which is cut codify that 1 little diagram care head A production that we we had not had utility for Play here We have things look at next 1 had And then we do is we take this and we said illegal book constant we call that a level set and that the slope of the level set off saw it This deal X owed DX 1 Which was minus That 1 over at That's a week discoveries in the differential by over here we have a view of X 1 X and then we go down weekly said people took on their way call that difference there and here's this year is the whole point is doing all this when Economists look at this year's we get this negative when studies the study and that he had a native here but for economist point you with the difference between switching from 1 good together such as the rate of substitution that's all I care but they don't care whether positive or negative so they eliminate the negative to the Raiders substitution is slightly different for this eggs That Taiwan right the slow this is gonna be adjusted anyone over you know mind get real good answer to why they did it by what I ask that you can't Professor my understanding is You just want the rate itself a matter whether it's a negative greater positive is 1 0 with the Raiders and always indifference curves They're always that shapes and has always had a negative slope anyway so I is convention leaders say the Raiders substitution is positive because you go this way giggle It's positive review doubts depends on what good you looking at so that it is always make positive that's important distinction actually I looked it up on the Internet searched for this idea and I found book I found in our book they did they take me away the miners
But I'd also found it down Your elected Usually experience for our class it needs to be without my book every year we have a production function and then we said illegal to a constant have Isaac wants and then the slope is going be threat that out the slope of the ice
Concur They were doing the math that should be P sub but a L over peace that with a minus but we get rid of the mind
Shit
So I had I know you put them on top of the Camelot But the serious said in the 1st Variable goes on top of the 2nd variable goes on the bottom so that was my 1st Variable put on top of that the blood bottom bad KNL switch that would have this watched as well It's because there was for a high-level Moss just no mind here I couldn't go Apple What I want Sir but we don't know all the bubble open But what want Yeah winner for my next trick when a new 3 variables is fairly complicated the 1st thing I wanna communicate he about slopes Is that it's always between 2 variables the slope is a ratio of 2 variables so when I have 3 variables president hounding different ways Their apparent variables there's sex but then to this year's mud l had less to keep it on head gets Ohio has 3 variables then that Different slopes of the level sets harmony different slopes somebody said 6 illustrate this summer we don't account it so I have DX 2 over DX 1 that's always has dealt with but then you say Well what about DX 1 DX to They're the same Meciar we were still comparing the same variable so they don't really deal with that distillate 1 and general always put the 1 with the bigger subscript on top so there is 1 of the next 1 I have is DX 3 over D 1 that's Another racial I I had DX 1 of RDX 3 but this same focus on that 1 and then that D. The other 1 I have the include DX 2 in the X 3 Solana put the data 1 on top here I could add DXQ over DX This fall will deal with the as just a reciprocal and say Here's my 3 slopes that I won computer Your salad example and that will work it out here grew out of hand It would open up because there's so that 3 sublet a budget different slopes I'm an array Downer write down 1 general formula by 1 D X I over DD X J on Friday mall down at the same time year-ago So that Q over 1 that's where ideas to and James 1 of I tell you with this is that is told you what all the slopes I can tell you that Closes guests could get us go back to the original bad X to Annex 1 I get a minus F 1 over to basically the slope of a level said is the reissue of these partial derivatives If I follow that example I should get a minus for what should be bad To hear in 1 day in the switching again S. J overhead that's the general for for all that's even if he had 5 variables away that 5 variables USA earlier what's said DX 5 over X 3 0 KDX 5 over DX 3 will be at 3 over Please go get us to partial Make a fraction out and make it you the slope Placebo again tackle 1 of these this'll be real push on near the frontier of your knowledge I heard
For all of us are At back back back a back at of bomb at a fast fast fast fast
OK go there's there's a function to barrels yes Dad No comes when you look in the book of these exercises there always utility functions of production there are Teddy yet on the final I had warned a production from it The yet because Well what'll happen is if I don't want to use the Minus I'll ask you for the MRT at and so when I say mrs that means you not to include the mightiest that Iger said find the slope of the of the level said then you would think that but only a tours of fine Ollie pretty clear about which ones are OK to get a function as 3 barrels to a level set looks like With a level such as failure minded imagine what the levels It's gonna be What's the process for level said will you do she said this sequel to a constant and you see what the graph giddy pistol grip decide that X 1 X 2 X 3 I give you a number Vs. this to say OK basis equal to a constant so what is this kind of of thing look like what Gloria if I just had X 1 squared plus sex to swear plus tax 3 squared that's a spear is going to say 1 that's a sphere The dislike with the circle Bester putting numbers and others variables scanner stretch in 1 direction a little bit more than the other and see don't end up with this year because at 1 year to hear a three-year that's going to start the war below that and what I end up with is ellipsoid 1 level said It is the work of lower than until year with go at a particular point here so let's say I want negative 1 negative 1 Let's not back here somewhere her down here there's 3 slopes of many yet there's in India A slope in this direction The slope in that direction and slow this way we calculate the 3 slopes pretty complicated things this is just 1 level said it's already some big complicated ellipsoid mirror that word him now but you don't really have to know all that you know And intuitively Set to go calculate so defined Slopes level said adds 1 man 1 negative 1 if I were a plug-in that 1 native 1 when I find out that death that 1 negative 1 negative 1 What's that that's 1 plus 2 Peers I 3 minus that Plus 3 6 so that the level set at equal beleaguered is out there's 3 of the yet I need DD X 2 D X 1 But the school get there during its 1st half on need my F 1 that's just 2 x then anymore Q. that's for access to remedy that 3 the 6 x 3 But let's see Edward put This is for sale a year and a half villagers follow this now thanks to the IDF is 1 of of the relationship between those 2 variables that F-1 1 over at 2 And they had 1 donned negative ones this here is to what sets negative and then added to line up and then head to you at this point is negative for that equal to 1 D X 2 X DD Adams by 3 DD X 1 The relationship between the 3rd variable on the 1st variable that's can be mined F 1 over at 380 minus 2 a S 3 at that point their native won his name said 1 of the last 1 the only other pair that I haven't gotten his along with 2 and 3 D X 3 over DXT that's mine set to over over 3 he sees switch on race That's 3 into jewelry that's my S 2 wood is negative for and F 3 is negative Sam's that's it and made it to the those of the 3 slopes of the of the level set at that particular point of is computationally is not that bad I mean you have to memorize a bunch of stuff that formula in particular and beneath you if you were
Interested in writing it the other way wanted DX Tuareg EX 310 and EX to over 3 . 3 of her too But the convention is you put the larger 1 on top to those of the three-day have to 3 variables Just just to help you out little notation if you had them that this was now a utility function instead of just a regular math functions and I wanted you M R S I can just tell you calculate MRS because I have to tell you which 2 variables I'm interested so I might say MRS 5 June 3 that tells me I'm doing the 2nd variable 3rd very book that's going to be Eagleton negative soon 1 one-way again is it
Doesn't have the negative or the multiple comes does the opposite of this year D X 3 over deemed too which is negative negative attitude MRS to comma 3 that means you go to calculate the other 1
Multiplied by negative To switch the sign and 1 more if I had and Todd said 1 that would be negative X to Do Which is of key negative negative The 1 place in the book would have a combination of math problems and come from Connor forces media coverage counts but it's good anywhere think that's a good review of the step you'll see a lot of opportunity he can't and the thing about those classes is that I usually Instructors don't have time to teach you the man have money can't teach you And then I really used to teaching the Makes it hard for flat at this club 8 questions
So it's uh uh might Flood ahead here a negative negative thank you but yes I know of the slump missiles that would vote You just think if that's what negative of this So this 1 had a natives of I don't wanna Nader multiplied by native
So Alleges 8 years the original 1 that once part bath and modified by natives annihilated the MRT S & M & RS always the opposite side of the regulars So if I get a half for the 1st 4 mill owners
Yes is made They always is opposite side that silly How so Ah Louie a T Back off a little bit here after 2nd was on lot We'll go back and do little linear algebra At about 4 . 4 4 But he did tell you about how we identify these matrices they tell us about can't cavity that Nasser and telling that advance Manchaca see that a little bit
Who going talk about matrices in general and the special nature Of all the positive definite matrices boy audited think of positive definite is that skinny concave up We discuss spilled beyond that away So positive is going to be like Can't make the 2nd derivative is positive and negative definite will be the 2nd derivative is negative picture something concave down something conquered but the 1st thing we have to do is reacted 1st C 1 and we have to learn had identified matrix is being positive or negative that the best way to resist taking examples gets pretty tedious to read all out as theory In usually can just pick it up from seeing 1 example Kaesong on telling you in advance this is a positive step but you can't identify from this by saying that everything's positive There's something negative in it so it's not like a the definite matrix always only has positive entries That has a negative entry but it's still a positive that nature so you can't look at it immediately know unless ITT similar tools Solana dealers on a break this matrix up until 2 or 3 different nature seek a 1 by 1 matrix at 2 by 2 matrix and 3 by 3 matrix and Called the league team principal miners minors from this it's it's Iker matrix within a matrix the Russian doll situation so a 1 that's going be the 1 by 1 matrix will 1 it's the upper-left corner that's a 1 and then I to you but that's get indeed 2 by 2 matrix starting in the upper left-hand corner the 1 1 0 2 and then 3 years is going to be the whole matrix it's that 3 by 3 major expected in this case a title at the Lorena is annotated determinants of the terminal today 1 that's equal to 1 and terminal at 8 to that article that To and turned so they 380 that's equal the 6th this is an upper triangular matrices just multiply straight down to get the terms and the key is is that it All of those their positive then that's a positive definite matrix fish so if all the determinants of of the principal leading miners are positive Then I saw Thanh a clears positive dead and We say It As a positive using your intuition say negative definitely switches which should make definitely got off Carlos has positive was make with knowledge to think that to show it the as if it was just so simple as Although their negative that I could just say that we be done but it's not quite Show you a negative death matrix and show you what happens with the negative definitely tricks and then Maybe explain a little but why that happens Jim He if was negative but she is Wooters of negative diffidently jerks some us definitely say negative depth But again it's got some positive terms and it's not like I just look at it sale of scattering negative that means naked at infinity related to this definition sows go get those leading principle miners everyone in this case is committed to the 1 by 1 matrix negative to a To matrix again and starting the upper left-hand corner and continue on to the 2 by 2 is the upper left to right field and A 3 is again the whole matrix that's a 3 by 3 at a cup OK so those principal miners again was go the determinants This what I have to Be real careful with the determinant today 1 OK that's the determinants of This 1 by 1 matrix which if I were a ragged that looks A lot like absolute value C. be tend to direct to there but it's not it's the determinant of this nature of such as negative to this is not a bad back they entered the determinant of 8 This 1 is just as is rapper lower triangular so I just multiply the diagonal sets to so already a sort of violating the tuition you expected negative definite matrix that have all these miners be negative the same whether they were all positive for a positive definite that sorry violated so something different 83 years that 2 native one-time negative 5 so that So what's gone on here from 1 of the more stood negative definite matrix where happens is beat determinants of the principal miners they Alternate between positive and in the same way that powers of negative 1 alternate between positive and if I take 1 negative that's negative as I take to natives that's positive That's why it's skips as wet switches back and forth between negative and positive because this is a one-dimensional that 1 negative times itself but the negative You're 2 negatives times each other that giving me a positive that I have 3 negatives Here and what I'll get to an amended is it really comes down the Aggies values Matrix of all 0 the Eigen values if all the Aggies valued negative that's a negative definite matrix and if so when you take to alone together that you're positive that light switches write down with that with that is there so negative that had a determinants
But scanner right miners mill in the context or talking about here survived alternate signs starting with a naked that is as it may it gets us matrix Spoke It had been here the alternate signs starting with letters that public example of you leaders even do it to buy 2 it's easier to explain when Abbott 3 by 3 political back and look at a couple to bake to a is a good time See here As June 1 1 more this is positive definite at the upper-left use glance at it not to write at all the stuff and I'll believe you if you features rated posited that that's not even be the biggest part of these problems but anyway Here the upper left is too I want to and at 2 that's the whole thing so that's For minus 8 miners want to set that's greater than 0 this is greater than 0 so this is posited that E. E. yes But I also have cost If up up up up up up up up up up up based But but but This was the negative death in the upper left-hand corner the upper corner has Negative for it to start up a candidate for day-to-day here and 1 that's equal to the tournament that is equal negative to And the tournament today through May AS to switch that's the definition has to start with the native switch sides of the secular tested positive period 8 minus 3 to 5 sets Graham 0 that's less than 0 so this is later Horak half so now we have everything we need done on almost done for the day this 1 I gave you little that where were where were headed for now already started out we did a little calculus with the differential calculus by learning partial derivatives but now what calculus entail entails buying Max's now we need to learn how to to do that OK math to aid to a Amax problem this when you there's all know added Y equals X squared I know every thing there is to know about know whether Max men are there is no Max the minimum is there's 0 was just review how we find that So what's the what's really move over here said they support so that the X equals X. Switzerland will quickly derivatives 2 X and you do it too derivative he said there Kennedy sulfur give you the critical point so that does x equals 0 we call that a critical Meant to determine at critical point something might happen mania Max medium and maybe an inflection point the something's gonna happen at this critical point and the way we determine whether it was 1 of those things a Maxtor minaret critical point is we take the 2nd derivative
In this case at 2 in the 2nd derivative but he tells me that this is can't that's all we care about being care was to readers wanted of those positive or negative so that meant concave up And so that means that we have a man adds of 1 0 0 A standard calculus Problems not ours purposely not that 1 illustrate the Met good amassed for we have a function that were a little bit comfortable with that but that's why text would oppose script the paraboloid but I know where the minimum it like aluminum minimum was there So let's see how we find that gay that's going to be our goal is to deter discover that that's the critical point and then it happens also water we do they were here we took the derivative but over here I get to derivatives to take so let's get em in fact I don't want to see the derivative I want the gradient right of the gradient that death that's gonna play the role of that crime of X agreement is to x 2 What I would do it did during the day here was a Eagle 0 So let's do that here gradient of that equals 0 that's can say there is no change desk in a put where that that tangent plane is horizontal just like this is a hint of online news horizontal here will get Tianjin waning as horizontal that means if the Tanzim plans horizontal that means it is no slope in India the direction has to beat is completely flat So this means just like over here we take that 0 here we have more means that equals 0 0 there's this single 0 and that equals a little extra would but then be amended x equals 0 NY so there's a critical point critical point is located where the gradient is 0 That occurs where x 0 wiser which alerting you could see it Now we discovered it so we have But 0 0 0 is a critical for just like over there What we do now at a critical point on discovery is that Max's is at a man is an inflection point is what we do we take the 2nd derivative here we took the 2nd derivative we get a two-year we thought it was greater 0 That meant concave up so that the minimum focus allows could get the 2nd jury The 2nd derivative is not a lot of them we put on a matrix that can be well what's the Early added that X X X Y Y X Y Y them we compute that what's at XAX that's too that that's 0 FY X that should be the same by Young's 2 0 in the half is also too so here's my 2nd year religious like this to This this is my matrix for this day and Over here we just said to you is greater than 0 metals is concave up here We say this is positive definite And that's the equivalent of the 2 being greater than 0 Insulin that implies concave up I can't hear myself into a corner here Space Look up a bit and so the last absences can't up Let's get that we have a minimum 0 0 yes Hope it that Part deal be Dozens more complicated will get that a lot can happen in that that inflection point category a lot of things can happen when you're in more dimension but yes the saddle point is in play the role of the inflection point But there's is more cases still So you say It is not positive definite negative definite if you look in the book lawyer by the way is
There's a page numbers go back to page 386 years of chaos go back with a 386 due problems from there there's that maybe they give you a few matrices the test but anyway when you go back there
You'll see in that section they have all kinds of things their positive definite they have positive Sammy definite they have negative definite negatives any definite they have all these different things so there's a lot 3 categories of matrices borrowing a deal positive and negative debt and 1 other case the saddle were not deal with any that's any definite In all that is that's that's like when 1 of those derivatives is 0 that they're not both are all positive not paltry minutes is too much this class Get plenty of SO that's the that's a renewed our we're taking bad example and were now than I apply the functions of several variables in the strategy is that it's a derivative and said Eagle 0 Take the greedy instead of taking the 2nd derivative and knowing whether it's positive or negative you take the passion and you find its positive definite so that Will start to do next and you can go ahead
Subtraktion
Kalkül
Punkt
Hausdorff-Dimension
Klasse <Mathematik>
Gruppenoperation
Zahlenbereich
Derivation <Algebra>
Physikalische Theorie
Übergang
Negative Zahl
Zahlensystem
Variable
Vorlesung/Konferenz
Indexberechnung
Schnitt <Graphentheorie>
Gerade
Lineares Funktional
Mathematik
Güte der Anpassung
Vektorraum
Frequenz
Quadratzahl
Flächeninhalt
Menge
Dimension 3
Ordnung <Mathematik>
Numerisches Modell
Punkt
Exponent
Zehn
Güte der Anpassung
Ruhmasse
Derivation <Algebra>
Term
Eins
Gradient
Konzentrizität
Negative Zahl
Variable
Weg <Topologie>
Zahlensystem
Quadratzahl
Flächeninhalt
Forcing
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Mereologie
Vorlesung/Konferenz
Zusammenhängender Graph
Fünf
Turnier <Mathematik>
Lineares Funktional
Abstimmung <Frequenz>
Quader
Matching <Graphentheorie>
Mereologie
Vorlesung/Konferenz
Derivation <Algebra>
Gerade
Matrizenrechnung
Lineares Funktional
Differential
Kritischer Punkt
Punkt
Kalkül
Abgeschlossene Menge
Vorlesung/Konferenz
Derivation <Algebra>
Indexberechnung
Gerade
Analysis
Subtraktion
Differential
Kalkül
Physikalismus
Vorlesung/Konferenz
Abstand
Mechanismus-Design-Theorie
Lineares Funktional
Subtraktion
Punkt
Approximation
Mathematik
Gleichungssystem
Derivation <Algebra>
Ausdruck <Logik>
Variable
Zahlensystem
Multiplikation
Menge
Vorlesung/Konferenz
Abstand
Indexberechnung
Grundraum
Gerade
Ebene
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Kalkül
Punkt
Sterbeziffer
Klasse <Mathematik>
Gleichungssystem
Derivation <Algebra>
Ausdruck <Logik>
Richtung
Zahlensystem
Variable
Differential
Vorlesung/Konferenz
Tangente <Mathematik>
Schnitt <Graphentheorie>
Gerade
Lineares Funktional
Oval
Approximation
Mathematik
Physikalischer Effekt
Primideal
Quadratzahl
Menge
Rechter Winkel
Mereologie
Garbentheorie
Parabel <Mathematik>
Ebene
Lineares Funktional
Punkt
Natürliche Zahl
Zahlenbereich
Fortsetzung <Mathematik>
Partielle Differentiation
Rechnen
Ausdruck <Logik>
Variable
Zahlensystem
Negative Zahl
Differential
Vorlesung/Konferenz
Resultante
Lineares Funktional
Variable
Punkt
Prozess <Physik>
Mathematik
Klasse <Mathematik>
Vorlesung/Konferenz
Partielle Differentiation
Lineares Funktional
Prozess <Physik>
Mathematik
Wasserdampftafel
Hausdorff-Dimension
Stichprobenfehler
Kugelkappe
Übergang
Konzentrizität
Variable
Zahlensystem
Differential
Quadratzahl
Menge
Rechter Winkel
Mereologie
Vorlesung/Konferenz
Ordnung <Mathematik>
Punkt
Kreisfläche
Zahlenbereich
Gleichungssystem
Derivation <Algebra>
Partielle Differentiation
Übergang
Richtung
Ausdruck <Logik>
Knicken
Variable
Zahlensystem
Einheit <Mathematik>
Quadratzahl
Ellipse
Vorlesung/Konferenz
Parabel <Mathematik>
Lipschitz-Bedingung
Randverteilung
Einfach zusammenhängender Raum
Lineares Funktional
Punkt
Exponent
Graph
Sterbeziffer
Mathematik
Klasse <Mathematik>
Güte der Anpassung
Biprodukt
Term
Physikalische Theorie
Ausdruck <Logik>
Übergang
Variable
Arbeit <Physik>
Menge
Residuum
Vorlesung/Konferenz
Garbentheorie
Substitution
Numerisches Modell
Einfach zusammenhängender Raum
Randverteilung
Lineares Funktional
Subtraktion
Punkt
Matching <Graphentheorie>
Sterbeziffer
Kurve
Mathematik
Güte der Anpassung
Stellenring
Biprodukt
Übergang
Konstante
Zusammengesetzte Verteilung
Arbeit <Physik>
Zahlensystem
Menge
Geometrisches Mittel
Vorlesung/Konferenz
Indexberechnung
Substitution
Beobachtungsstudie
Einfach zusammenhängender Raum
Lineares Funktional
Subtraktion
Punkt
Oval
Mathematik
Ortsoperator
Sterbeziffer
Biprodukt
Übergang
Negative Zahl
Diagramm
Menge
Vorlesung/Konferenz
Substitution
Lineares Funktional
Subtraktion
Variable
Mathematik
Offene Menge
Reelle Zahl
Minimum
Klasse <Mathematik>
Vorlesung/Konferenz
Partielle Differentiation
Biprodukt
Fokalpunkt
Übergang
Lineares Funktional
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Kreisfläche
Punkt
Graph
Zahlenbereich
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Biprodukt
Ausdruck <Logik>
Übergang
Richtung
Eins
Variable
Kugel
Menge
Basisvektor
Vorlesung/Konferenz
Tonnelierter Raum
Lineares Funktional
Mathematik
Klasse <Mathematik>
Kombinator
Euler-Winkel
Zahlensystem
Variable
Negative Zahl
Multiplikation
Regulärer Graph
Forcing
Vorzeichen <Mathematik>
Vorlesung/Konferenz
Negative Zahl
Matrizenring
Mereologie
Vorlesung/Konferenz
Matrizenrechnung
Matrizenring
Determinante
Ortsoperator
Natürliche Zahl
Ähnlichkeitsgeometrie
Derivation <Algebra>
Dreiecksmatrix
Term
Physikalische Theorie
Unendlichkeit
Sechs
Negative Zahl
Hauptideal
Menge
Betrag <Mathematik>
Sortierte Logik
Radikal <Mathematik>
Körper <Physik>
Vorlesung/Konferenz
Analytische Fortsetzung
Leistung <Physik>
Turnier <Mathematik>
Matrizenrechnung
Kalkül
Existenzaussage
Extrempunkt
Differentiation <Mathematik>
Wasserdampftafel
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Tangentialraum
Derivation <Algebra>
Äquivalenzklasse
Raum-Zeit
Richtung
Gradient
Vorzeichen <Mathematik>
Gleichgewichtspunkt <Spieltheorie>
Vorlesung/Konferenz
Lineares Funktional
Mathematik
Kategorie <Mathematik>
Wendepunkt
Partielle Differentiation
Frequenz
Fokalpunkt
Kritischer Punkt
Menge
Rechter Winkel
Mereologie
Standardabweichung
Lineares Funktional
Matrizenring
Ortsoperator
Kategorie <Mathematik>
Klasse <Mathematik>
Zahlenbereich
Derivation <Algebra>
Gleitendes Mittel
Negative Zahl
Variable
Exakter Test
Gleichgewichtspunkt <Spieltheorie>
Unordnung
Strategisches Spiel
Ablöseblase
Vorlesung/Konferenz
Garbentheorie

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Titel Math for Economists - Lecture 11
Serientitel Math for Economists
Teil 11
Anzahl der Teile 15
Autor Kronewetter, Jason
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported:
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DOI 10.5446/12905
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2013
Sprache Englisch

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Fachgebiet Mathematik

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