Math for Economists - Lecture 10

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Math for Economists - Lecture 10
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It will start up but what we looked at the derivatives a little bit last time partial derivatives even really talk about that name yet but I talked about the concept of when you when you have more variables you have to take a derivative for each variable so we saw some examples of that last time just using the power just extending the power rule from map to a mass for Dallas look at that the definition of a derivative of go back to match to a what was the definition of the derivatives there it's complicated and not just doing it at this level them only extended it'll be even more complicated so it's worth it to go back and can handle this this is step you've seen before
But may be your 1st time through calculus the Syrian mean a lot to you or you know you're just learning do derivatives really have the opportunity to look at that depth and and really get fluency with this so so this is another opportunity for you to just absorb some of the math theory from single variable calculus
OK So what was the definition of a crime of X look withstood by pictures
The slope of the tangent line right that was That's the idea behind it so I have The next year and then I have a point say X 9 We were just just say exercise had notation what we don't need to What we did was we said we won't by Goto . nearby Next plus Dr. acts of Delta taxis boasted to think of this as an infinitesimal amount to just adding a little bit But when I expanded like this they cannot exaggerated to make it a bigger distance so we can see what's going on here so I have 2 . 0 An X plus built text Making go plug them in Tampa Bay 6 And over here this is half of X plus Delta X Since I don't have a specific function I don't have to to specify any numbers I could just say what things are and what we do is we find the slope between those 2 2 points so what's the slope of the slope was rise over overrun the change in why over the change annexed what a change in y That's change in the white valued this little bit gray here and then we have to change in X on the bottom that X plus Delta X minus acts such as changing y ever-changing week and clean this up a little bit because on the bottom the Texas cancels the Norwegian Abbas there so I really have is back at just the slope persist as is what's that call the slope of the seeking 1 and just connecting these 2 points in finding the slope between so that's not the Tianjin line that's the slope of a line connecting those 2 points that's what it is calculated the battle I get the Canton Line what I do is I move this Delta X closer and closer to X and that brings that Tianjin line closer and closer And eventually you're the seeking line it brings them closer and closer union up with the Tianjin line wants at the limit when x don't X goes to 0 so the actual slope isn't written like this it's written with a limit if so at Prime of X that's equal to the limit As Delta Texaco's 0 of that thing Now let's just For practice looks due to an example would ever next Always go back there X equals X was not to that with 2 out 3 acts as I know the derivative the derivatives 3 bullets use the definition of a F crime of That's equal to half of X plus Delta Hexamita put Expos built Texan places the 3 acts Will plus Delta My At the next that 3 x over limit their limit as Celtic text goes to 0 The bottom line had dealt X and the reason that you can calculate this immediately is because you have dealt Mexico 0 on the bottom so that the problem points you have to do a little bit manipulation 1st usually what happens is the Delta X cancels and then you're able to all 1 of the Menendez del this is 3 x plus 3 Delta X minus 3 X Over Delta When you see those 3 axes cancel We get Arianespace clueless over here so than I have that crime of acts His equal to the limit as Delta that's close to 0 and all I have left is 3 Delta X over doubt that those canceled and make 3 so it's a long way of going about it I mean you learned you taking calculus at this point you know use glance at that you know the derivatives 3 but we just went through the definition from scratch and found out that the In fact the slope of the tangent line is 3 OK so how will it do is in see what this looks like with 2 variables
So enough for we now have
Not this situation we have 2 variables estate X and Y there's RFO that's why I but whether we had to derivatives because we have to to variables X that's military Dallas sex is this is it's just like this every year except it's the slope here's the pictures here have included draw something like this to give yourself some perspective here
And then the What is that deaths of X It is slope in the x direction that they would think it would have drawn here as a scholar sliced and I fixed y There's a fixed white here and so every point there's that is equal to that same y value but taxes varying and have now found the slope and the extra I could do the same thing for wider that's the general idea that's complicated won't get away from that We wanna have a little bit and tuition for what it's doing but popular just wanna get to where we can calculate OK so we have to derivatives here so what should be the derivative In the X direction while it's the slope and the way we find the slope is undertake a point here and then let extras very little bit just like they did their Delta X but why Isn't indicated change so let's write that down in the notation is Could it be the limit as Delta next goes 0 just like it is over there but now what the slope of the slope an agreed that the Plus Delta X Carmel why so I replaced the ex parte with X plus the exercises leave while on and then said voters over here I and that the variable changed a little bit minutes attracted awful functions on their here and around the bottom I have dealt so if you compare those you can see really the same thing the only thing that's different is I just had this little extra wide Fareham the wives the same formula for all were really doing is just finding that slope in that direction and then we may do the exact same thing in the white direction as well so we could just copy this down just say cable would have to be that guy here I'm doing with respect to acts of the Delta x goes 0 here I'm doing this with respect to why not dealt X going 0 here it's gonna be dealt 0 bullet that is now played by the rules but why so everything we see Annex over here you happened and why they now here Exs constant because if you want to run a little picture for that so now I'm and fixing X a little plane they're going wise changing that Texas fixed case so that gives me the little Slocum a wide direction of what So X stays fixed and then why changes a little And then subtract . global original functions Figure this is a change in y over the change in the function over the change in what just take a 2nd to compare those 3 things we have the formula from your math Put it their the Top from to air and it's really just a slight extension you have several variables and since we have this appear normal I wanna go through 3 variables and 10 variables but it's worth mentioning it if I had 3 variables here than I have 3 derivatives and I would just 2 Expos Delta excellent of wiped the wise men and others equals the disease due to the 3rd taking example just do want house That is why holds To Greek So we learned how to take the partial derivatives but I can tell you right now that X the derivative with respect that just remember what that is I'm holding Why can't since I'm thinking of it as a constant the derivative of the export of 3 the derivative of this part of 0 because women doing the derivative with respect to accent considering this constant 0 4 this is equal the 3 and the derivative with a that means I'm thinking of X is the constant Sonoma take derivative here that's 0 and the derivative with respect to y here not so there my answers Nellis Goosey if we can get that from just like we did here with the definition of tax that's going to be found using that definition of there's the limit as text was 2 0 ad
F O X plus Delta excellent that put Expos 6 here and leave while owner of the treaty next still my describe why minus and they will do for the 2nd guards minus if the whole function but over Delta So The same kind of thing should happen we should will assault down cancel the axes 1 up with answer we know they have 3 bullets to the algebra to see that that have a Harrogate 3 x 3 Delta Air Lines Letter and my aim is to be plucked And then minus 3 Minus 5 y over Delta and see they want you wade through all the notation earnings cancels out that That's what I need to talk about our 1st date is being a little mathematical maturity is nothing about this that's hard except the notation and want you can deal with the notation than than you start you get a different perspective on mathematics focus of canceled at the 3 X cancels that 3 x five-way cancels there and I just whittled the down exactly the same as this year a but that's 3 . that's over Delta the most canceled the that limits 3 If you're not really gonna have to do this but it is good for years practice and there's some examples of the book some homework problems that ask you to use the definition but the bottom line is for us or more in the middle of doing problems we wanna do it this way get derivatives and use that information assault the particular problems posed OK let's Generally the Theory behind the derivative just like it was before
We have slopes into directly over 3 direction would ever hollow many areas Those dealers the notation that you have to build the recognize the symbols in not to we had if I have but we could have us 7 example here OPEC's equals X squared
So What were the different notations we have for the jury there's Prime attacks but then there was also could users DCA remember that we get D E F G X and that was the equivalent noted this is this is Robert Bob Wright the same guy just a different name Now only good amount for it it's a little bit more complicated we don't have effort that we now that X Y or even if that's what will start to introduce a 3rd variable minute because they have this example here So now what We have that 2 act so we have ever of why that stupid why these 2 Here How however that crimes but then we have to we haven't notation like this for this situation as well and it's It's not quite a D partial D rats Any Cunningham said that the language used by the equivalently this is Robert Bob the same this is called the partial derivative with respect to exit here you'd say the derivative of that with respect The derivative of asked with respect to x here you would say the partial derivative of that with respect and you have the the partial derivatives of that with respect Why we're a little bit of dealt dealt tags this said It said a couple different ways 1 was is 1 of our he said Partial derivatives with respect to x it by adding a Y on the bottom I would say it's with respect to wide notice the language makes sense partial derivatives trader don't have a complete derivative because I have 1 1 variable on 1 for another variable So that languages appropriate part it's part of the jury not the complete state but then it's also because this is a little tedious I want to say this every time I write this notation on to say about the partial derivative with respect X we have a shorthand for this so we can say that all that Dell tactics Dell have Dell being being this is almost a Delta if it was a Delta a lower case Delta and in the Greek language is is that has a little tail on her like a musical note that looks like a musical but when we we don't include that And used cars considered dealt As part of its Del that's 1 way of thinking Justifying the language that as how we communicate it takes me a few minutes just to talk about it communicate with the stuff committee but that's a little bit I think you can appreciate even from having taken map to aid you prefer this this is this is more satisfies faster rate here we go this is more confusing to the fraction Everything that is D unit which is confusing for just like that will probably get a prefer S setbacks and of why and there's a little bit more notation to introduce you to him and they will get to that eventually 1 mission is the language of its use in economics in economics they might say partial derivatives I'm talking about it but usually not usually the words that they use marginal products that's the equivalent of that the Robert and Bob scenario so Hindi on are called partial derivatives are and and actually marginal products is it It is an abbreviation they're actually marginal product function But I'm just gonna that surrender room anyway a marginal products A If
But But we're at the point now where we want us put this together so I've got this situation is this I've got a a function of 2 variables which means 0 . here x y
Plug it in the me point up there at of X Y And then at that point we can say where the difference saltwater the wood the derivatives so we've got 1 that goes in the x direction at that time we got another slope that goes in the wider region so tolerated complicated I've got things gone in different directions the question is how we compilers information were wary put the food but that's how little example here said that the X Y equals X squared plus Y square standard example is wanna keep using the example because money ideas is is is have at least 1 function that you're good at doing everything the sums to keep using this example so you use as you see fit that part in of your head and you have some pictured a work Are a joke at but say the point Want negative ones that's Texas here next year's while I fell 1 negative ones back there I go there make it that way It's a bit too F X OK with the sequel 2 expert And this as to why we distributed a few times but now 101 is the slope the particular number associated to this point now I put that into get the actual value The notation I would use to say OK not just Partial derivative but at that point That would indicate that both the vertical line and put the point here that tells me I'm about to plug that it now I only had put that is 2 times 1 with just 2 and now I've got a number the slope is to with the extraction at that particular point but if I had a different point I get a different sort Now states numbers in here and then have why evaluated at 1 negative 1 that's negative to so What I was saying before is what we could do with that information where where we would restore the said that OK this Lopez two-on-one 1 direction is native to the other so we we want 1 thing that incorporates all that information and that's what leads us to the next concept which is called the Great But the answer to my question where he used to store information that at this point in the class that some idea about where you store information where we store numbers in a matrix as a matrix is a vessel of information you could put information into little store it therefore don't need a giant matrix because I've only in this case only at 2 pieces of information
2 really what I need is a vector it's it's nature experts a 1 column matrix But that now the idea is this guy This would in this example here I've got to do in this direction and get negative 2 in that direction so what that means is how to administer action when I go back native to know why direction and Enomoto I do with those 2 pieces of information that tells me how far gone Texas doesn't how far the And so that gives me a distraught here And it's getting in the way here that the gradient it'll be a vector that fits the diagonal of the parallelogram created by the deaths of X earlier similar to that of the we rate Set down for a year and that's why that 2 variables then making go get at next Connecting go that of why the 2 partial derivatives and store them in this vector called the gradients and that's the way stayed in English the gradient Of that I'm just a little which gradient diet The gradient is ditch the direction that a ball would roll futures dropped So in this room the slope of this term is coming down this wherever I take a bonnet put at the top of the Roman mosaic under my feet and if I put over their roles right down the buttered over their roles right by change the slope of the room that changes agreed to I think of the great is downhill that's the most downhill few you confined the most that whatever the most extreme letters in intuition for worthy dealing with mostly just how you get a clean and then using it and just to get a little preview of our ahead What we do in math to a We go get to derivative and then we ask the question What's the maximum aluminum frame and what we do a map to a when you ask that question you go you you take the derivative said equal 0 and then that gives you critical points in new test those critical points you find out where have a Maxtor remit the gradient is gonna play the role now have Prime of that we finally had 1 variable so we just 1 derivative now it to derivatives and so in the old setting you said the derivative eagled 0 that gives you critical points now organist at the gradient 2 0 and that's going to be a critical if you think about if I have a maximum here so would I just tell you that the gradient always points in the direction where it's like a maximum increase or decrease point downhill so if you're at a maximum there's no pointing down so it'll be 0 that's the equivalent of a horizontal tangent line will now become the horizontal candid plane and the horizontal Tianjin plane will just have a situation where there's no change in the text derivative and there's no change in the wife of the creator A top or bottom however you want to look at 1st is looking ahead showing you that there will really used alternately right now we just need to practice with the Moses Computers here is typical situations but you'll find yourself and I give you a function get to variables but you have no idea what that looks like nor should you nor is anyone born that way right now like there is no monk in Tibet was born you could shown that symbolism and are leaders knew exactly But like you have to practice and train yourself and you have to learn about these tools so that the situation we find ourselves in all the time when you just don't know what these look like a literary figure out what's going on you have to just practice these taking some derivatives are acts of the gradient of this OK but that just means that I'm take the 2 partial derivatives them and put them into a vector at subtext that means take a derivative with respect to x take it piece by piece what the derivative that Square that 2 X it was a derivative of minus X Y remember You think of wires a constant so If you follow me that the derivative of that piece rate there's minus 1 It's like having minus 2 acts the derivative of negative too And so the dreaded are another way to look at it a derivative of X is 1 of minus why comes along for the ride OK now and taking a derivative with respect to X of this case is a constant 0 rate city believe me that that's the partial derivative with respect to x 10 hours to So now for why this is thought of as a constant so it's 0 the derivative of this piece is negative X featured at your learning an unexplained should sing along in your head nature they try to get you didn't tell you that download the information to the derivative here when I'm doing with respect to negative X and the derivative of this with respect to allow 3 wives And then we could say What is this This is the generic derivative But the ex parte in the white part but then I wanna know what the Slope at that particular point of Inacomp plug-ins pond negative 1 there the notation leaders that just means a plug-in Mr. Exon and negative 1 for why those terms that those 2 times 1 is to minus native wanna 3 there and then negative 1 plus 3 years to the gradient of that function at that point pretty simple want somebody tell added to it and is the only place we really may be studied and get this right here to learn that take these jurors plugging in Why are they Our policy at Lake My life Yet years but you're taking a derivative of what's the derivative overcomes 0 rate so that can see when it's by itself drew 0 but when it's attached to an X you bring along just like if I had to exit And I do the derivative made to the FedEx plus and I take a a derivative I discovered star 0 yes OK that's in the important of IDS century not the only 1 that has this factor that fast enough sticking point whenever people learning let's get off OK Another owner had as if we didn't have enough notation owner had to the pile of in economics you don't usually use variables like X and Y So we had to get you ready for what you really and so if you think of a setting that year Iran in an economic model you might have a good 1 in good to break into your good said and a good 1 through good 17 label on the text of 1 x 2 orgies of 1 2 1 we label variables that way that we have a different slightly different notation for the derivatives
Give that a lot of the problems of the book a written in this form so to you have to know somebody has to tell you at some point put all this notation announced it X and Y X 1 next to a little tedious to deal with this example here Soap If I wanna right down the partial derivative I could do this they use the partial of f with respect to next 1 That was 1 way of note-taking taking it that Della Delaware well but even that's a little tedious like that then you my only used the F sub x 1 that's kind tedious to It was OK had just expert having the extra 1 that annoying so abbreviate that further Just call it at that's a lot nicer and doesn't lose any information and know exactly which variable and dealing with So F-1 physical want all businesses Robert is bonuses body all the same OK that's calculate that now so is a derivative with respect X 1 so if you want to think of that as a concept derivative of X 1 of 1 and then bring the constant with you and then same notation Bill F. Del Next to it that's the same as that next to like that was holiday 2 The derivative with respect to the 2nd variable hockey here I'm taking the derivative Ruth next 2 of the derivative of squared Exs 1 over to rejects the only to root X 2 that's a derivative that are only bring the constant along with I was thinking of it as it is a good example for you just make sure the you see that these are infected derivatives and be able to do them yourself make sure you get those with this is the only new part now we can just in a point half us so F X 1 X 2 0 sorry the gradient of the gradient of death at the point let's say a 3 9 make the water will this is sector during the top part I put S sub x Someone with fear that want to evaluated at those points us on plug-in four-year to in for here They form the bottom and 9 of the top That's the the gradient at that particular point that function again as hard as with this and that's why we have to learn how to do this without pictures off But So Of course Puts this is the next Concept level set of functions of the think about There's also an interchangeable word here call level currency is interchangeable terms level set some level So 7 example here The ex-wife and we want to do is to see this is a three-dimensional bullets it's it's a plane in three-dimensional lives in three-dimensional space at two-dimensional object but lives in 3 dimensions and the thing is that all your training was in 2 dimensions you're really good at BXY point so what we can do is we can reduce this down dimension and look at things in the XY plane and that's a valid thing to do and you will see some economics examples where we do that was to start off learning about water level said it's to see a new level said who wears a sword level come from what it is we're going to see Different levels of the function This is the level where down the Florida The level where the functions 0 But if I move up 1 then that's the level where were at 1 And then there's the level were it to go to Sealevel set next That's why Jews Effort to be a constant soft and graphs this Results Go as with most things and that when we describe it in English it doesn't help thinking if I tell you that isn't really help so tho have the Chadian discounted dissected aware the status of Jews Have to be a constant and we get to choose it's not like we're just as When a random but you get to choose so easy ones like 0 and 1 in 2 things like that so what do that but Jews have people 0 I just did this chose have to be a constant and then grabbed the results was at a monograph result after it chose see that doesn't do me highways at equal 0 busy all of this The same saying this sequel 0 so that's what it means by Graf the result so if equals 0 than that implies that 0 equals 2 x y Which then now stop that's why he was negative 2 X And I think at that a white equals native to X that's a slope of negative Tuan it passes through the origin So there's that there's the level set for while That is equal to 0 and the idea that is here we are passes through the origin of the job is done before so the level where an equals 0 is the Fourier that's where the value 0 so we're getting is getting a little lying there before but see that's hard to so we grow over here just Graf on original XY Also when you set this sequel to a constant forgetting where 1 of the variables so I don't have this side anymore now becomes a constant menace just a regular lying to graft that's what levels You wanna think of it like this you get this big functions and became you get on the ground floor you hop in the elevator and it's like a like in a parking structures in you you go up for sale the 1st level you get the elevator and you look at the ground you see what painted on the ground the graph At that level when you go up to the 2nd floor and get as military look around what painted on the floor on a graph that you can do that at every level here we just did at the ground floor so we'll do a few more
See another level set of effort excellent Tuesday to be another constant have about Jews that equals to case so then that means
Now I have 2 on the left side by the student peoples to experts y And that it was stated to choose ever to be a constant and Graf the result was grabbed the result is that the same as wide negative way to look at the same Lopez that will so
We should expect parallel lines that's if says the same slope is just as it did for a wider set out why intercepted 2 days a slope of negative to collect the level set for where the function is to be wanna think of that as we've gone out of the this the 2nd floor and a half that of the elevator announce it up there on the 2nd floor and then I see what painted on the floor it's this graph to lectures that Graf passes through
It passes through the wider 15th too That's that so I go up to the 2nd floor and I see this line painted on the ground that it is Graf Those 2 1 more but you have a false negative to that that I have negative equals negative that lost why the sales figures to be a consonant graf sorry to X was negative 2 X minus 2 The same slope but now the wider set down here Ringo gone on with this but the ideas that distance 3 and realizes that for this particular example each Level said is a parallel line now let's Graf that all on 1 set of taxis and then you start learn had a look at this in 3 dimensions of the goal here all really graphs on the same tax This is what enables to see what the three-dimensional graph looks like using these level sets The first one it was a slope of Jupiter pastor the origin and the other 1 passed through to other pastor negative to test so let's get these pictures here I prefer So if if you use your imagination you can think of those as parallel lines but we should see is that OK this is where you use the level said information this liner series We need your brain to do this this line is is on talk for this line 2 units out from the chalkboard This is this is this 1 near his ex equals to this 1 is people 0 And this is a negative to so train yourself properly you can see this as a rail concede this is behind the board this is at the board And this is 1 unit out of the book so you want you can actually see the tilting if you practice in train yourself to see that so that Current levels that means it's it's what distance out from the border behind the board adjourned levels set his years major graft so it's not 3 parallel lines all next each other's away appears this is further out from the board further in Threat on the board that takes practice but you can train yourself Easter to see that the next examples actually easier to see us take a look at the next It's our old friend square plus twice 1st not just for practiced you see the picture in your head now and maybe not yet but if I do this enough times you'll see that passes the paraboloid key So what about level set go through this process again a level See the level set of asked Tuesday to be a constant and then grabbed the result would take some easy constants but say have people who want their with back so when half because 1 which boasted thinker OK were at the level 1 I took the elevator up to the fore The 1st floor they got out and I look around Woods Gone on at that level Well if you if you can cut it with a plane here what he'll get is Gill circle here that's what happens at that level but but if we just follow that last example where we do we set the function equal constant and then grabbed the results of scholarly here As soon as they take the functionally a constant and take the rest of it and next to a plus Y squared equals 1 and was cool graph that that's a circle that's the unit circle circle of radius 1 So that's the picture that That's the level set for that function at the level of 1 now out with your mind This is more mostly Cheney added gets them Development of your intuition here don't take this plane here and move up and down So taken up to here What should you get there It's a circle also but wider If the different salutes the graph that matter to you that I mean by the but for the levels of a year appropriate procedures move up and down on all levels and changes the graph were interested in grabbing each 1 of those that may be putting a fuel together on 1 set of axes
We can see the three-dimensional picture using two-dimensional graph obstacle south
I kind of like a lofty goal that is focus another 1 will be now want pick something convenient TI equations circles the equation any circle has expiry plus Y squared equals square When I'm taking my values of F the Astaire picked to really Tata circle radius route to that's a little but rather have that's a lift to the next 1 or 2 or just to make the the numbers work out because Allionia circle agrees to pay the constant for the function then I for equals that square crosswise And that is a circle radius to and there's the Graph of the level at 4 4 Golway for their circle and 1 of the financier ones is actually equal 0 that that should be be the first one I error was 0 is always easiest number but 1 understated because it's a little bit different than you could see from the pictures what should we get it At 0 Just a single point right with just a single point counties that picture to guide you But do at 0 that means that I have 0 Eagles that's where plus Y squared Analysts think about this for a 2nd the values of X X and why they give me 0 Well OK let's say say why was something positive anything . 1 anything positive How could I ever cancel it would break could never had anything this other than 0 collating never be able to cancel it right so this this means that 0 and water So you just get 1 point level set for this 1 here it's a circle of radius 0 that's the way I look at it it's it's this circle here and at a radius of 0 Let's just a single point That point rate there is is the level set for sequels This is the single point of put on on the same set of axes and they will do that thing Richard Trainer break to see it in 3 dimensions Mr. saw I've got 1 of them is just a single point and then I've got At a radius of 1 gotta circle And that it really is stew I've also got a circle Woods labeled us this is where s is the circle agrees to it this was the levels that were at work for but that was a concept that we set it to and then this guy race here that's at 1 end and this 1 here is that equal 0 OK so the point here who were doing looking at this as were looking down the center of that parabola And when you look down the center you see a bunch of concentric circles that what you want to see this as pulled out from the board it's 1 of those that may be seen as hanging baskets where They're chain-link and when they came they three-dimensional thing arrest them on the ground they flattened Nanning EC about 2 rings ladies seemed so picture this This is here this is for units out from the board this year is only 1 unit up from the board and right on the border with you can visualize something three-dimensional from the picture of putting the 2 together There's another reason why we look at levels that will get to that but that The 1st step is just understand what they are It is we have got to stop and We have stopped for Yes did We so out when I see this because I'd labeled of LOST uranium from him this way and This 1 year I see this pulling out from the wall like a big cut Ford He through them Of course The really famous model in economics and this is this is the 1 mile you will see under final would use this function and goes through a lot of different examples different things that we steady with this 1 function So what is the Cobb Douglas and there is no there You want may be put to is that there is to it It's 1 of these people that they won the Nobel Prize anyway the difficult thing is a cart Douglas production function but added the word utility because if it works like utility to do whatever you learn when it comes to this you can apply a two-year utility functions as well and will all make those connections as we OK so wet what is this card Douglas production from this book has a particular notation that uses soldier stick to that but I'm not always has the standard notation future look this up on Wikipedia or something you'll find different mutations for a couple of things going on here that so I've so why is a function is a function 2 variables now the 2 variables er can l This is capital Couples overseas of its capital and maybe in America with the K that any of it Said The The capital labor and the idea is if you're gonna have any kind of productivity you need books right if you had a bunch labor By Justice years all hired let's go There's nothing the producer you need a factory somewhat lower creates sold labor by itself doesn't get you much production and if I had a big factory in anonymity to work in there that Allied Capital but I'm still not heavily productivity need to to pray so that's what economists notice that their dependent on each other you have to have some in productivity needs some capital needs some labor and the ideas let's say I wanted to buy
That a I want In corporate owner of a model that depicts what happens in the real world factors allows me to predict which cannot happen as Labor won a lot Comes into effect the Let's say a certain amount of pride productivity with a hundred leave a labor force of what I make it 201 I get double the productivity will assess so great the Moroney 400 wooden a quadruple the productivity is held works what what what plays a role The law of diminishing returns rate so far that depict what's happening in the real world what I do is I take these variables and if I want to incorporate the love diminishing returns I put a fractional On a day told loser fractions yet but that And then begin our constant from so you see it's kind of intimidating when you just look at it OK there's a model but then you have ask as it can't the question why it is that model will we see in the real world Meter They would often data are so we put that little stipulation on Alpha Beta 90 years a great example of math notation verses math in the English a god Take your math English dictionaries out and translate that in your head In your mind was he say that Is And then have say it is doing your head practice That How would you do if you reading this story time to your child I think will have you read that English or fewer saying it to your grandmother So a lot of you might be saying OK 0 less then Alpha Beta less than 1 summer like that procedures reading reviewing the symbols but that doesn't really give you an oppression of what it will be help you would advisers say how often bitter between 0 and 1 don't you know exactly what I'm talking about to see how math notation is intimidating You have to learn to read that if you if you If it was possible I'd like for you guys Just have a picture pop in Mindful that soon as I see that I see 0 and 1 9 0 Alpha and Beta and living in a picture of manga works much better with a picture but when you dealing with the within equality is message I wanna give you is used the word between the 2 Greek word when it comes to very few anybody say that in your mind when you were rehearsing it did you say out there between 0 own 1 signally desert but then when once they say is that all yet quarters and I know exactly what the fuck anyway but still does take 1 step further with the wizard meeting between 0 and 1 what kinds of numbers are between 0 and 1 Fractions break two-thirds one-third one-half rates so so if you want just give yourself some example and those numbers there between 0 1 1 and then As economists is betting economist you wanna save yourself Why would we choose those fractional X on and the answer is because on model the law of diminishing returns these types of These X Parliament you of the soldiers before but if I just go back to calculus single variable calculus this function gives you that picture sophomore Inc. That idea the law of diminishing returns on capital and labor and insists that TX Pontefract nuts with That's that's the matter is that that's introducing the Cobb Douglas production pollution from a math petitions point as I say why would use those kinds of function was was so great OK yet do model we see in the real world are saying about a there's not much to say it's just a constant so we allow his by just that there might be a real life example might actually have an example we take the data We However have a constant from so we allow That doesn't add much as just a constant to throw in the docket let's label it everything so why is the total production that you get out Based on a certain amount of capital labor So let's go get that does go do some partial derivatives And maybe some level so that's those only thing we know where we know do Parcel derivatives at levels that sold to those 2 things that carry of this is an example of of but this is a 400 in your book fee Of the course Which the KGL another name in practice they always put the 1st y The goddess Kansas who have a look at OK so they get the Alpha their less than 1 another thing they usually at this since might be your 1st time seeing this model by usually they add another thing and that Alpha plus peoples On matters makes pattern whether that models what happens in the real world I doubt that but it makes the computations a lot simpler on a lot of our examples have that features of the the numbers work out that look at what we do is we take care derivatives paths of K That's going to be a case of derivative with respect to cases figured this part right here that helps you could write this out like that are like this And they're all the same but I do the derivative with respect Acadia won over to root K I get 5 Ruth L. over group so I bring this along for the ride at the Rudolph and then I get worn over to Ruth K. changes tandem of 5 deaths of adults but that's going to be But The same thing every day No school look at some level sets could prompt Key
OK so where the level set out on the board any more means you take the function you City who constant bullets say will constitute about 10 of the yet many equals 10 route K L Rates attends cancel that's why I'm picking had that to on pick together equals attended and so so what would that mean that saying I'm really a production level of 10 so I could do that when 2 different ways very I could have a certain amount of capital and the Malabar the gives me a production of 10 Americans have more labor laws Labor and be less capital there's a bunch of different combinations they give me this and I wanna find out what they all are that the scrap so OF equals 10 that means 10 equals and Group K the tens canceled 1 equals Group K L and then it says Graf rate that We do grab the result after we set the function equal to a constant factor decide here which 1 is the X which is why I so let's let's just do it alphabetically with Rudolph is equal to 1 over Route K and 7 out vise that Ellis want By graft that used decay axis to L axes waters Over here just do the math X and Y axis that I have a light was over X that this graph but we don't have 1 over X we have l and k is still the same shit them this is what's called an indifferent were actually in a production function Colin Hisar called well into these vocabulary words that since the talk about a production function maybe you've heard Eastern Aris this so Willis that's analyzes term want I still means sane Kwan quantity of this Graph represents all the combinations of labor and capital to give you a production level of quality can so if you think about it from manufacturing point view do I care whether I had Not bunch labor a little bit of capital if I got an output of 10 horrified had a budget cap all over the labor either way it's an output of so there might be some little sweet spot on the curve that you might find and that's maybe you Indifference curves Over Isaac once before Usually in early trade make it Kantor the budget constraint or something like that eternal how much it is seen that anyway at typical Isa call that shake and so that's why this graph here 1 over X is important for economist enough that mention that before but that's see if you don't have those graphs at your disposal and it's really hard to do both Fractional exponents which square root the law of diminishing returns for you it's like this and then the other 1 that comes up all the time is this graph here which is a nice acquired or in indifferent to talk about utility functions therein There um indifference curves was probably the more common term it if So so far that the Tricks of the tools that we have tool bag are with their partial derivatives and together level set so eventually would combined with the degree of former put together the level set sail McGrady would put a ball on 1 graph So what I wanna communicate you about this is that the gradient is always perpendicular to the levels set in a graph 1 example and see that Burma write that down 1st the gradient it is always perpendicular orthogonal to up in the food see that Solis take a typical example of the X Y equals X squared plus Y squared you know what the level sets like Let's do I at the point Fed with say a 0 . 1 so 0 at 0 1 we've got that is equal to 0 plus 1 coaches 1 store on the level set of Of f equals 1 over at that point were on the level said of that was once scrap that It's a circle of radius 1 with crosswise squared equals 1 that particular level set in Dallas with the gradient the gradients there that's Partial with respect to annex the partial with respect to why this guy is to 2 y and then at that point events at 0 1 OK that means it put him 0 4 x 0 plug-in 1 4 too So if I go to the point where the . 0 1 hears the . 0 1 And at that point I Graf the gradient What is that that means it it is 0 2 0 0 and the extra actions and to the wider for the yellow The with that 0 recruit and you see that perpendicular Points straight What do another 1 the gradients at the point was to make here 1 0 means employees in here get a wall I get to hear his Exs once for to 1 0 tonight too 0 So let's go this point here 1 0 pending graft that did says go to the extra action and 0 and the wider Action or that's the gradient Of effort that point and you could see here That's perpendicular 40 million this topical this is what get that out there with any this fact later on this want it to be something you've seen before that the gradient is always perpendicular to the levels set if if if toss off
But OK I guess what's next 2nd 3 So it would about What about In math to where would we use 2nd derivatives for 4 was a tool I had a really Will it is the information that you got from the 2nd period can't cavity yet they still would have now if you look at the 1st with this guy Here is that Conte leprechaun came down
You don't always get life depended on it had to guess which is a great has agreed to have this is found it is going to be the same way as the gravel was concave up In calculus Even for concave down but also have an inflection point that that was usually found from the 2nd derivative take the 2nd release a legal 0 view another critical point or something might have happened namely inflection point always happen so that so we need to deleted start investigating 2nd derivatives of these guys Well that's a little bit complicated amid both notation simultaneously this is sort of a typical always have to variables X and Y depend on economics said it had difficulty have X 1 next to What we did our 1st earlier this year we got F X F Y and over here we have 1 2 OK then when it comes to the 2nd derivative don't we have to Derivatives we can take here so this is a function here sitting there was sexism wise and potentially a 180 during the day and I have 2 variables so I have to do to just on There were leader called the notation well I've got that the derivative with respect to X and then I do the derivative with respect X again so how about you F an X exact that tells me that I did I did with respect to x both times that I could come along here and I did the derivative with respect that they now wanted to the derivative with respect to why women indicated that way The 1st X-Men walk and that symmetric situation over here I do do with respect to then I could do X next alright dude why both Times I can't think about what you do and when you doing the derivatives here Here you're saying OK I get the slope and the extra action and I find out what the caddies Amelie directions and then Here I found the slope of the wider action find cavity and texture and teachers at this as a matter which ordered to do was get that out from Reddaway that's what's called a below more formal added a 2nd but since I got set there won't be clear these for your notes that geezer are always the same for our purposes that's called Young's those derivatives of the singer's recalled the Knicks partial The shorthand language that we focus Gloria This is a derivative with respect to x once and I've got to Lotus call this half 1 1 the derivative with respect to x 1 twice and then had had 1 too Over here that hash 1 and 2 and just like over here the mixed parcels be the same How
Actually wanted continually there's anybody need these here canaries as Bush continue on underneath that OK
OK So how where we do with this information we will get these derivatives we had what was the thing where we compiled all the 1st that was the gradient so for the 1st derivative we had agreed that that was just compiling consolidating information and 1 matrix And then we had the 2nd derivative what should that be We see about 4 pieces of information the where would put it However matrix rate is now we store information store the information in what's called was called the Hessian or the has seen the passion is saying that Both So this is the notation for this is guaranteed A matrix that 4 pieces in the upper left on how their X X and that's why X Y Y now it's a little bit when you had axes and why he may be stable here however member axes in the upper left-hand corner work Penagos alphabetical order soared over the 1st goes there but it's a lot easier to deal with this This notation because look we see this is that the 1 1 location that to be in the upper left-hand corner and this is 1 2 locations that should be in the upper right and 2 1 and 2 2 of those That notation is perfect for matrices because Darty got the subscripts that are appropriate for the Matrix location so here the gradients for this situation is 1 2 and then the Hessian Mrs. called the That's named after person Peter think it is the 2nd derivative matrix legislator gradient is the 1st derivative the the gradient plays the role that Prime from your math to a glass this is playing the role of that doubled from Er There's a little bit of weirdness here that you might have picked up on that he we use a lowercase that you use and appreciates that assists following the book as by the should be questioned it because is really that anything is getting all that stuff all the notation ballots
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Stuff This is the level of derivatives unit be happy to take slightly harder with some more expires but not much the only other 1 he might have to consider it We have the Things fractional expert but this is typical of the type of things that you can handle this and that year here where you want are at Solis go get these different things and Find the gradient of that and that owes the Computers to add another point From so the gradients and X While they're still at Texas 6 sexy there's no why they just data derivative like it was to a 6 and then derivative hero 2 Y The new derivative of negative for white square 1 doing it with respect Exs era that's considered a constant that that would reduce deaths of why this is 0 no why there Then a 2 x and made a lot cases there's a gradient if I wanted a plug in a particular point in this was that that My 1 negative 1 of the great hints at 190 . 1 years before Tehran if if she He knows we're 2nd derivative matrix the Hessian let's have X X X Y Y X Why why there was go get these things still have X X and means they take the derivative At the next with respect to x again so I'm taking 6 x plus wireless was put it here That is equal to 6 X 2 Y so now derivative with respect to x again in 6 and derivative with respect to why this and is key to the deaths of y It's 2 at minus a so they get me to derivatives year F Y X Now what I told you over here is that bad These children should be the same as with Scott Young's there but all called state formerly in the next anyway The derivative of this with respect to taxes too The notice you get the same year F X wiser singers FYX The system was saying demand FYY the derivative of this guy with respect to why his native vote now The Hessian matrix is 6 To Carson get so really do Now If We needed to buy 2 matrix now ask yourself whether all the different things and if you look at it too late to matrix could you see a picture in your mind Is there a picture to be seen Area over the determinants is the area that Is Tell you that time cavity area Well the determinant is Give us the kind cavity information so I've got a matrix 2 by 2 matrix we could take determinant in this case was negative 48 plus Florida's negative 44 Let's get it means something to it said that negative determination is It means something in this case it'll mean we have what color saddle point will get all that but that's our new uses information that our 1st derivative better 2nd annual dues Ravenna take that and said illegal 0 which is giving us a system of equations assault so that do what we did in linear algebra little that And then we will we set that equal to 0 will get a critical point and we take a critical point we put it in here and then that tells us whether we have a concave upper conquered down and I'll tell us what we have by now in Bangalore to Gaysville relief for 1 day Today is the day that it had been trying to get at a hearing and vision that I always run up against other start right now and will continue on with the next half and if you have questions about your midterm them