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Math for Economists - Lecture 15

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Figure sample final the
The aircraft in here onto a bunch of wealth
We'll see how much I do today but the TA do some of OK discover scanner can scroll through its CDC is antique comport up
Tablets are earphones to look at it more closely the basically what it's made of it's it's it a little mini midterm from your midterm registered some problems off the mid-term and a little many midterm here and then you have a mini midterm further steps since the term which is all the calculus that This is a long years of prisons this problem here I just put a bunch of stuff here and you can expect if you get this kind of problem on your finally maybe have 3 of 4 But more than you would need their help you get ready the determinant of a 4 by 4 solar system of equations test whether vectors orthogonal Material dependents quite ready All that stuff was The from the midterm and then warmly I want to get to number 8 years and it's all since the case a level sets Sloper level said that this section irradiated say that their that RTS marginal rate technical substitution that is a slope of the level said differential Maxmilian saddle points The same thing but with 3 variables here And had 2 variables but within a rectangle fined the maximum in inside a rectangle and then and the bronze multiplier 1 and that I had to 1 last 1 on Douglas Lagrange multiplier Probably point of you can to unite to gain yourself to know whether you're ready for the test and you can use sample mid-term the sample sample final and turned pretty much
What you see on the tests and tried at the Agassi a big surprise finally they maybe 1 problem the your expect for the most part of you can do all those tests the samples at a given you the actual midterm and you're ready so you might RDB there for some of it but I have the sample final practice Any questions for me A review some of the concepts and formulas and things in with whatever left over time I have all of the problems mostly start from the back angle for words that I get all the calculus stuff done in all the mid-term step this kind of you anyway But He focuses since the mid-air In doing multi variable calculus so let's go through all the little things that we've done a little recipes for each type of thing so the 1st thing we did was But if you look at it That single variable the situation in single variable It is you have some functions that are lower We go and we look at a point here And we find the Tianjin warrants and that Tianjin line his that Prime vex that's the slope of the Tianjin so me good Several variables really you can only see the case where you have to variables it 3 variables and you can't see the function anymore because the function itself as a graft in 4 dimensions of 3 variables plus the 4th dimension for the function but here we can we can see everything and it's also
They represents a good model for Multi variable a fee of 3 or 4 variables you could still use this picture Kind of guy G The situation is now Instead of a stream I now have a sheet and when I got was single point haricots gets get 2 variables X Y and now when I go up to a particular Point on on the surface now OK if you over here it was a Tianjin line now it's gonna be a Tianjin clean And each whom we wanna talk about slopes Slopes are always this important concept the words slope Is basically just to reach you have to thinks that the slope So now in this situation you only had 2 things that ratio the ratio of why the ratio The ratio of wide x that's the slow to nobody ever talks about it that way it's just the slow bread rise over that Riccio diffraction when you've already here you wanna talk about There's many more things that we can compare I can compare To the rate that that is changing compared the X the rate that is changing compared And then on top of that we can have the rate that Texas changing relative the There's 3 different ratios you get you get s and X and Y and then X and Y That X and Y part if you not sure entirely but there that will be the slope of the level of that that's the real that's the slope between X and Y Who are the 2 that the only talk about that issue but at the beginning only look back years they will you What's the slope Here would have to do is I have to focus on the slope and the direction and the slope of the wider picture of this little these little arrows their drive they represent the The number here but now I could have to look at 1 of the Ex-Im 101 Soviet at sub and that of what then what we do is see the visit to individual pieces of information and if you're in Yukon saying you'll stay here is that this is the marginal whatever rate if I had Did have sex was red Representing profit I take a derivative its marginal product if I have this represents cost than this is marginal cost use the word marginal For derivative product pretty so over here we have the marginal changes next election and the marginal changes y direction but those are 2 separate entities so high we put them together we put together and then ingredients We just store that information in a sector but that it also has a geometric interpretation and that is that so This point here exists on some level drove a level for Kurdish there's a procedure are at a particular point on the surface there's there's a level that it's that doesn't quite so every point on here gives you the same the same effort X Y and then that gives you a little Level set down here usually view the level set in the two-dimensional space it because when you set this constant only 2 variables Down here and you can think of the gradient Make sure my level set passes to point here There's little level set down here and it's got a point on it and and the gradient As always perpendicular to the level set that's the at X y And then we seem to radiant down the two-dimensional point it's always perpendicular to the levels it we use that what we wanted to find the maximum subject to a constraint strength was just came to the a to that a little later to but the basics said at a If you so the notation here where the FX and FY This is the dream with respect to next and means hold constant that language their brilliant if you wanna translated as something can understand the derivative with respect that doesn't usually need anything he stated Sony's hold why can't take derivative like you were in this situation and that this guy is a derivative with respect to why all ex-con class where you were really studying this for a while you would do lots and lots of these theories and practice of new products will coast and rule on all that stuff but for the most part discusses wanted basic wants We could focus on the techniques for finding Max's in men's and I wandered into complicated derivatives OK So I add 1 more here the partial derivatives and Barcelona include the notation here Dell at Dell X And Del At Delta why this is standard notation in now is the partial derivatives is not a Dietz a little The dealt almost a Delta that would be dealt So let's said Adel left LXU may come up against that a notation in the future so should be where we
OK Then The next thing that we started study was you want to get into these functions a little bit and work in order to study these functions because a complicated what we have to do is simple and even that doesn't mean that seem like we're simplifying it from your point of view when you look at a level said you doing bringing it back down into an environment that you're comfortable with the ex-wife so if we look at level sets out we find we have find out that had
Can't usually see take 3 different constants and don't try to be a hero and pick you know route high over he has you're constant 0 1 negative ones to so I you know it's not that complicated Arctic simple constants Anthony 2 or 3 Obama new graf Vermont on 1 set back down X points soon as you said the the sequel consonants like removing 1 of the variables now I go back down into the ex-wife point when you view several of them you can start to imagine a three-dimensional surface let's say this was At 1 and 2 and had 3 when you see that this constant put in there what you're supposed to imagine if you train yourself is you ramp going up this 3 years 3 years from the board this has to and this is 1 of the men may maybe have an extra 1 0 which is right on the board and maybe we have 1 here at people's negative 1 which is behind the board So if you train yourself and see this like a race just like if you get good at reading topographical maps you're a hiker something in your out the wilderness and you have a topographical map you can tell they tell you can see the elevation of the mountains and the hills whatever you're around by looking at the games chain yourself pursue three-dimensional OK So for once we learn about the concept of the level set the next question is here's a graph Figures a curve what's the slope of the slope here so this is similar to that situation it's just that this came from a more complex context so now we have to learn how
How we get the slope reverse slope is it's a ratio of 2 things so here this slope is the ratio of the function to this variable on this is the ratio that function that variable I give you a little slow the directions so this is what I was saying before there's another pair of variables that you find the slope between namely X and Y so what would be the slow periods DYP yet But since you notice the difference in notation This is a true derivatives like what you learn in half to wear This is just YDX is the only 2 variable sitting swatted we find those what will we do is we take these guys here and we do a ratio of of this turned out to be a minor setback at nite you can take IS taking take you through the derivation of that that is not relevant right now what you need is fine of this right here I want it if you wanna know what the languages that goes with this this is when you're asked a finder's slope level set so that's the that's the comparison of the variables involved in the problem and this is a comparison of the function with that a big source of confusion when you stating that there's a lotta derivatives floating around a lot and location OK so I the next concept that we looked at was called the differential the total could understand what this is
This is this is that it I wanna say any and more layman's terms if you have a function then variable if I let the variable change slightly than the question is how much of the function like the way you realize that you just estimated by using the tangible or so Here's This year's to typical picture years effort that looks at a single point year X and here's a point effort the next set of 6 murders of things and there's a change of in here so the question is If I let this change a little bit If I let the X the variable change a little bit and Woody call that when you talk about an infinitesimal change you call that EX The standard notation for little little wobble were little changed amid a little change years could cause a little change here in the way you determine that is you quizzically go on hearing you hop on the tangent line you see how much can candlelight changes and you just say Well that's an approximation for how much the function and the way you actually find that this little This little yellow here that the change if it's infinitesimal just like this since only ecology Y and so the formulas D Y is equal to that primer that's not TX it's at that point X focus of the only go to our new setting is just slightly more complicated but it's the same kind deal has a point clear x not why not just like that have there but now things can change in 2 directions I could wobble in the the x direction enacted wobble in the wider picture each 1 gives you gives you a little Little infinitesimal change in the variable 1 of those is He acts or put in because you will tell which would but this is the X axis so this is a little Y and that's a little DX good here this is attacks saying his is eyes that so that the little wobble here in the wider action that's little wobbled extraction and then it's gonna cause a little wobble here in the past so that DVDs or D why he likes look that's the name this function y equals that of Sadek someone like said so Casillas look back over here over here says about 1 another little change in the function and multiplied by the derivative times little change in the next year the derivatives more complicated that multiple derivatives that derivative NDX direction and I've got to do it in wider actions so should make sense that I Inc. both to find out what the change in the function as saying yes in your mind at least of the change in the function is it's just like that except 9 had pets setbacks minds But DAX little change in the extraction so he sees just looking at that that's exactly the same but I've got to woman I can't play favorites so had will not sure this is your total differential for to variables and I wanted add a 3rd Variable I just had At the New Times easy as many variables as you have you multiplied by the partial derivative in that direction and removal of the infinitesimal change yet I'm all out and that gives you a change in the the After bad here that this is that actually refuted the slope of that Level sets here don't forget that this is also the concept mrs mark it has targeted when I do a sample problem all it go through that again shorter remember remind you that concept is related to that term spoke
What OK the next thing we did was we started looking at Max's in the peso over here when you're in math to a this right here represented a maximum and this was a minimum and
That's the way you find did you find out where the tangent line source on all right that's that's where the men the maximum record so that we were at Prime that 0 and this will also be were and that was 0 But the incidence of Max Amir have a concave down at that point so I can also say that have doubled Bryant is less than 0 there and here I say this is a concave up You kind of that equal 0 for the location of the man and then to determine whether it was a manner not 5 You had to do the 2nd derivative and that said the injury was greater than 0 so that's the set up for that matter to a similar note are situation well I might have a maximum And I might have a minimum making maybe connect these so that they're part of 1 function So what happens at these points of this 8 years ago 1 of the Mac's phenomena have the Tianjin flanges horizontal rows the Tianjin line and the Tianjin plane horizontal really talk about it in those terms we said was this This Location near where the Tianjin Cuevas Orellano is where the Grady Siegel was 0 just like where the derivative was equal to 0 and then down here at this point have attended plane or allow all but that means the gradient is equal and that wasn't enough rated the green being 0 That was a put 1 over here finding the critical points so that the next step was we analyze the critical point by checking the 2nd derivative and so if you have a pointing down and this fiery here is come from the 2nd derivative the Hessian is that will this'll be negative definite
And then this guy here will be concave up so the scenario will be that the Hessian is But negative repository Said at this Guy weary covered the gradient gradient of That is why and that is the 2nd derivative so that we can show X X Y Y Y Y and don't forget that the mixed derivatives there always equal as long as everything's continuous his next functions This geezer equal by Young's their slow Show
OK woman the next level of this was to say what if we restrict it would we say that only allowed to look at certain points Fernandez's are of a global phenomenon you find any critical point and then tested that that was 1 scenario than the next scenario
Paid restricted domain so that scenario in index calculus it was have some function here But I'm only considering other region from say he so X has only made it be so I'm really only considering this piece of the grass it's a portion of the grappa that were interested in what we found out that the Macs and the man on a closed Interval all this was extreme value fear the the maximum on closed interval The 1st at the boundary The trip so all we have to do is go find maybe the critical point there and test the boundaries so it's 1 of those 3 points so that we go here situation is much more complicated than solving other problems takes a lot more work and now I have a rectangle that I'm allowed to check instead of just and honorable Mayo checked those points in what I get is The worked rectangle up there And I'm interested in finding all Lomax's amends along that ended here picture what I hear them acts and the end on a closed Rectangle that replaces the closed a over here it's still occurs As a boundary point so in this case may be the boundary point Ron Arad down as far as over here the boundary point where the edges of the the interval now it appears that Extreme boundaries but the word boundary really applies to the whole all the edges around there so we have 2 kinds of boundaries we can get Are we going to look at it as 2 kinds a critical point they at the normal panic critical points by finding it this way of letting the grating equals 0 or reading a critical points along the edges so it could that occurs at the boundary point half
For the critical point but We Kelso critical points edges So you go in you restrict yourself on the edge in you find the critical points yeah you get the overall critical point you might have a little critical point here which makes a little maximum there something she might get a critical point also might get critical points on the edges and you have to check all those Including the corners that I find your taxes and don't need to test the want you find the critical point you need to do that because here
Just like here all you do is collect the possible points plug-in the function and see what's the biggest value in the smallest value
They look Grimes multipliers was another former Max Min problem but it was slightly more complicated because here Rather than using a rectangle here we just have a little stream some functions that you're allowing you only get a dick part but picked points from that function and plug it and some now the scenarios guessing by the way Really have this is this type of thing in math to two-way this is truly new to this class and this is the thing that puts everything together We get nature matrices in this probably got partial derivatives we've got solving systems of equations area things in this problem so here's our scenario that level set Still There's ever X and then I've got some function down here We call the Jedi of X Y equals K and the reason we do that is to This implied that it comes from a larger function when I grab this little thing here is just 1 level set of some larger functioned UXY why when I'm only allowed the plug-in points from here then I I see something like this there and I'm interested in finding out who may be where the maximum and our along with little string of So what we do It is we joined the level
Of the vexed down here see this guy exist down here and the XY plane
So we draw level set this function here Jan Thanks the level steps Then where the level set suggests engine that reviewed relocations of the Max's isn't and in the doing see that as you see we have 2 sets of level sets here and what happens is the This yellow guy is the great of GE and the red Is the gradient to them when they're all lined up The level sets and and the the constraint are just in the gradients so that can be like a that land times operating in due to use all this time and location Can I give you a recipe for all this His giving overview it's helpful to you guys Prague in there in the trenches doing problems any kind of forget about the big picture what's going on and this is really what you need a have when you go on to We need a little bit of both in their ability to to solve at the you you have to remember when you take these classes in the future and you use this stuff you'll get practice to problems it's a concept that is The G So if you have a little overview of what this is all that said yes We It is not all show you that I do actually I would probably won't because she sample problem from the It is just like taunted the last lecture so slowly we tell you e-mail to the picture that I would expect you to drive is a two-dimensional version understand here so it could do that over here if I if I go into 2 dimensions that I have the constraint here might be something like this there so heated that supply quotes that constraint
Am What I would expect you use graphic constraint and then graf from levels sets that in particular the ones that are Tianjin so to see how however why there focused so if you take that picture you can turn it its bid for your brain practice that can take that Internet netted appeared but that's what I would expected draw the three-dimensional version
But I would expect to be able to dry certainly Graf the constraint might just a liner parabola Something and then the level sets of the functions that That's something that we practiced as well as general here said are demanding when you don't have an exact example but that will have that shortly look any questions revised the Thessaly 2 hours we'll have the whole class to to work on it looking at the sample meaning imagined that it would be 2 hours buckets start to go through some of the Is Check Got 3 levels sets of you like you meet you would draw more same time French teachers 2 1 at a time and it's up to you want to pick your constant but I recommend zeros always viewed as the best 1 2 equals 0 and that means that creates 1 6 2 equals 0 So you might be A little bit confused about the X 1 X to FedEx and why there you'd be fined that amount X 1 next to what to do I I would recommend just choose X to your wife but the larger ones as you're waters put him in the same order orderly with axes They so what I'm interested in here It is drafting this but I need to say what the axes and that's why I just want to play the next 1 attempts to I've committed that was the why wanna assault for that I get next to his equal to 3 x 1 And if you like the the charge you look at that rapid just changed to life goes reacts to and see what it So there's 3 x now what happens is The only thing that's going change when I grabbed 3 levels that is on a change the constant So in this particular case I'm just getting a number here and these are and be the same parallel lines is And have the same sort of 3 dealing a scanner changes the Weiner said so if I take a bath court to have 3 x 1 man 6 to equal to but this parts all the same the same slope excuse equal the 3 x 1 minus 2 so that's down here the parallel lines Neukom labeled the assesses that equals to this is was 0 And then making you 1 more it says it 2 3 Phillips said that he was negative to a 3 . 6 2 0 0 ext to equal to 3 nonplussed That is the line was slope 3 passes through that X to intercept usually called the white inertia Who are a Their standard warn you should villagers 1 in your sleep by now said that his company time this This Here a drive 3 dimensions that the travel and so the level set sail concentric circles including the 1 at the bottom which is just a single point so we go do this you can just grab as equals 1 that's X 1 square said chief squared we get a circle of radius 1 that was once and then that he 0 here And say said I got to a radius of 2 Don't just be be sloppy about this the radius of 2 0 occurs when Equals for because that's where we have 4 equals that square where the letter radius of to it doesn't really don't see that when you once you start to get in the habit of it 0 it is 0 others 1 1 but then you have to go that he calls for to get the radius
To get done I wanna life the hoof of practice at an act
It's a very few don't even know know what this means is that you gotta be able to say what this is not the say English but it isn't you guys early meters it would be helpful if you could say there is a marginal rate of technical substitution is card Douglas function which you guys will see a lot of you have already the marginal rate of tactical substitutions this is like a slope of the level set but slightly different so it's just write down what it is and hard T. S is equal to
It's pieces which variable that at 1st
But offers of help from the ratio of the derivative of each variable when you doing the slope I don't think I have it up there with the slope
Of the level said usually has a minus here and we don't include the Midas whom we do them just He so let's go find those things can said pal means a derivative of this with respect to L. get two-thirds out front In that case disconnect come along for the ride and when I do the derivative that put the one-third out front from multiplying at the Knesset tracked 1 nite belts of the negative too And we might as well as were interested in that point theory Myers would will get that Pl 27 meet next few feared OK so I have to do this algebra here when I put this is feature keep careful Ricardo Douglas Elfers and we think that appears so so Ellis 27 The parade in here so I had that each of the 2 periods And 27 to announce like that when you do these physical high school algebra lesson that on when I look at the highways say the 3 is going to make making things small because that cute route to do that would 1st City square 1st then 64 so what I do is I do the cube root Bursik is choose squares for 4 on top and we had a chance to do it again to Cuba root 27 is 3 and then 3 swear not The school teacher PKK means on multiply in front by two-thirds for 3rd and then I still have a L come along for the ride is a constant and then on the bottom The subtract 1 from the exponent like gained 3rd Now let's go Group P. That point 27 research 27 8 In the case of a have for 3rd 27 of one-third Over The force 3rd time's 3 your virtue Now I can ready and the values that I got he said Bell at that point was over 27 and A piece of cake was Jets the slope between the L able in the cave variable if you were to drop that it was hard drive In general level sexist Carl Douglas function look like this furious said the sequel The Constant solve this you'd get L equals something over K so whenever you have a reciprocal function whether you have roots are not as they they generally give these these shape so what we found there is the slope of 1 of those levels You don't have to know this commanding Indians since it's up here 3 questions about that
So the total differential problem I have to tell you how much the variables change I have to give you that infinite tests on these 2 were given and I want to do it at a particular point saying point before so deep he said is a matter what the variables are that I have that variable the parcel with respect to that variable times the amount of that variable changes phenomena have the derivative with respect to the other variable times the amount that variable changes in these 2 things together But those in Reddaway . 3 4 and P & L a piece of ability said K were found in the previous part who can write those in this was But he said Ellis 8 over 27 he said to 5 you like this is 3 over 10 20 17 plus This is for over 10 They elite tend toward that it are We reduce twist by by 3 bank So now by multiply this by 1972 Forget PE over 90 just
But To
OK so you're at the the Max Death will find a stationary points critical points to classify may so through this example review what it means to be a maximum or minimum herself when I talked about it a little bit before bed in mention saddle points before so it should be covered her focus we do we need to find a stationary point that that's just just this parang here is the Greens event he 0 sources could do that a F sub 1 is 3 x 1 squared minus 12 x 2 And have to is this part 0 negative 12 x 1 Last 24 X Q Square And setting a grain equal 0 means saying each of those equals 0 it's a system of equations but this is not only a system of equations like we did in the beginning of course because we have a squared You can use word linear for this but it's still unless system of equations is not really matrices to do it so what do you have to rely on is method of substitution Icahn sulfur X 2 here and get it in terms X 1 then take that result plug it in the other formula that orders of 1 variable assault from here this is the next 2 is equal to 1 quarter X 1 square If you believe that So now I take this version of X 2 and put it in I had negative 12 x 1 plus 24 times and won The x 1 squared square This is X to square this is thanks to you the square here said to put an square Yes I will become the next 1 to the 4th is all equal 0 that's clean it up a little bit of 12 x 1 24 times that this is square here so be careful It's 24 16 The minute Amin a this 24 over 16 additional reduce it yet and I get X 1 of the Equals 0 I use all that well I'm in a factory there's an X 1 each terms the X 1 out of this means that X 1 is equal to negative 12 plus this is 3 of her to come the common factors 8 That's 3 over 2 X 1 I just took gather next 1 8 8 8 6 1 times that factor at the Exxon X 1 Q Where were almost home free this means either at 1 0 That is something time something equal 0 means 1 of them has to be 0 or I have negative 12 plus 3 has 1 huge equals 0 over here That's also do this 1 that says the 3 had X 1 to do is equal to 12 with your X 1 Q It is equal to flip that multiply 24 over 3 matches 8 That could be know you know you did it right when you get to that step have something here because they fear that 17 year you know you did something wrong X 1 post to Oak so that 2 values x 1 when I try to solve this summer we want a pair of points will just 1 x 1 x 2 Easy access to legal one-quarter X 1 square So that lies in this case the next 2 0 0 So I have stationary point at a critical point 0 0 that's what I get from this analysis there and over here I have executed tuned people's one-quarter X 1 elsewhere so Next To is 1 The in the 2 here So I have a point to has meant a critical fallout analysis was to just get the stationary points to critical points now we analyze them are they be Max's millions saddle points neither that's are next year
In order to analyze the same year 2nd derivative detail about the Cannes cavity In this case this is bad 1 1 had to have 2 1 2 2 let's go get those things ethical already down have wanted 1 derivative with respect X 1 that's 6 That's what I want to is the derivative of the top 1 with respect X to Saigon negative 12 year remember these 2 were posting the same Celeste There's a check if you if you did this on our army that You don't you get something differed over here go back check supposed to be the same have to wonder if they were going under the 2nd 1 the derivative with respect to x 1 look we did get paid 12 good means and correct track and then effort to 2 I just get 48
X 2 of this guy is 6 said 1 negative 12 12 and 14 2
That's the Hessian but were interested in the Hessian at those 2 points so let's go check that at 0 0 then I get 0 NEGATIVE 12 negative 12 0 The ruin Analyze This now new voter tried a fine positive or negative definite The 1st thing you do you look at the upper left-hand corner so this is not a candidate 3 there it could never be positive definite because you need to have that positive and it could never be negative definite because an order be negative definitely death this negative so the maximum in out But we could have a saddle point with the condition for a saddle point we have a saddle point the only thing you have to check for the saddle point is if this guy determinant of that is lesson 0 that's continued assess what it's like to be the inflection point we see here at this point is a concave down its boat its alone the concave down from the essence of loaded concave So when you put a positive and negative together he Think that's the within tuition but I think that to put it back in and to a context that's why the saddle lessons 0 it's like the inflection Oak everywhere we have that situation the determinant of that matrix is 0 minus . 44 which is negative for 44 less So this is a saddle point OK we have 1 other point to check to 1 that would do that Book employee nearly 2 4 At that rate to 4 x once they get a 12 year And 1 very next to you At 48 and his fine any errors on here OK OK Soviet 6 For a comic 12 12 hours over this 1 reject the positive index positive and negative definite the only candidate is positive definite just yet 0 2 thanks I O carrying us OK Barrett so let's check the positive or negative definiteness of this matrix and the only candidate is positive definite so checked that you need the upper left-hand corner on It's positive 81 Look 12 and then 2 The determinant of all Thing that 12 times 48 minus 12 12 notes negatively canceled the negatives so that's equal to 12 times 48 minutes 12 The What positive about that so this is positive definite that means so when you when you get to that stage is to keep yourself From being confused positive definite there's concave up to think of it as As the ball 2 Positive definite concave up suppresses a minimum of
I don't have the money to get used to it for 40 years old houses
No we would be sending definitely have 0 in the upper left-hand corner that'll be sending you look in the book they talk about that but that's not where I could even kill you won't get them because I want you to get a saddle point of Maxtor so you're dying about none of those cases so I'm back and I'm back packages Shiites wanna you know the technique so you really get something that's gonna have maybe a saddle point maximum or minimum and a saddle point
2 maximum sir you just a couple of points to check and I'm not gonna traded Give me something Ahead of the you've never seen gap Yes There is beginning Don't have any of those 3 cases have been so because In deduced that I would give something Otherwise I would have done example like that in class to show you what happened to you can reverse engineer the kind of question I would ask based on what I've done in class but yet there are a lot of different possibilities when you talk about 2 variables in half you have none of those cases of there's all kinds of things it having its any definite sent positives and get negative new confined or those of the book but this was enough so we just stick to these 3 parts and honestly universe saddle saddle point I just throw that in there because it it comes up in these in these A types of questions and if if I don't tell you had analyzer saddle point you won't get it from the positive and negative death he kind you really disinterested in Max's saddle points with a 2nd before That's fear maximizing profit you want a saddle point I so if you look in the book he sees it as a really do the saddle point too much for that reason but because it keeps coming up and I will tell you about a book any other questions
Next OK 3 variables sustained processes that we have 3 variable so have 3 derivatives and that means that the Hessian is much more complicated Good work here but have warned that Siegel this is where here That's where had put it there so F-1 here is 6 x 1 minus 1 that 4 x 2 debts it is the only tournament as a next to that carried the derivative with respect it X 3 There are no extremes in those terms of 0 I get 2 x 3 plus 4 0 that's good because it is a simple equations here were setting these equal to 0 This is the gradient of this and I said it 0 0 0 of this guy Equals 0 we can say immediately next 1 has to be says that acts on his legal one-sixth of this guy here says that execute 0 and here the says X 3 2 So I have the only stationary point possible remember all 3 of these equations have to be 0 At the same time in order to get a stationary point critical point so we don't we found all that Possibility here's a stationary point could point is that one-sixth of 0 2 and now what I want is this the 2nd derivative major so I can see what happens at that particular point But No way around it when you have 3 variables that Haitians gave me a 3 by 3 matrix in 19 2nd derivatives luckily lot of emerged the same those 2 are the same At 2 1 is the same as at 1 2 and then these 2 really the same and then used to be so we don't have as many do but we still have OK That's so F 1 1 is going go with news that 1st equation into all the derivatives So if 1 wanted that 6 but at 1 2 that's 0 1 3 Now go on the 2nd line will do all the derivatives related to this effort to have chewed 1 that's 0 position is that 1 2 0 And then have to move to his derivative with respect fixed to his me for an end to 3 that's also 0 there's no Now will go down to the death 3 equation will do all derivatives for that 1
0 4 x 1 Jerry knew his history 1 that same as this 1 and then
At 3 2 with the same as at 2 3 0 and then F 3 3 . 1 or 2
OK so this guy here is 6 0 0 0 4 0 0
You can tell this it by a couple of met you communities as do our normal analysis of positive definiteness onto that they want the determinant of 81 determinant of 6 2 6 determinant of a few years 6 0 0 For 24 . 3 0 this was great 0 than the last 1 83 that's the determinant of Thing Pepsi-Cola 24 times choose 48 which is greater than 0 So this is positive definite has shed at the point of 1 6 0 2 is positive definite Positive definite figure that is been concave up votes and say there's another wages do this is if you look at the book there's a theory that says of all the agony values of a matrix of positive then it's positive definite Newton Look this is these are the idea that users is already a dagger matrix that in value eigenvalue agonize over all positive and I took a few wanna say that on the All I know is positive therefore positive or or liking value if these were all negative they say all agonizing negative therefore snake that's and you but he could always go back to the basics and you positive and negative definite analysis of for off The Hong Kong The figures for those restrictions along a rectangle where's the function of the rectangles worth To draw the rectangle of in this case it's a square for only talking about this region here are only allowed a point So by happening at a critical point that's outside reject that critical point because I'm only interested In points in that wrecked facility go work what were we need annuities edges are analyzes edges the scholars L 1 L 2 3 and 4 and easily the corners will take care those at some point but the 1st thing we do just find out that happens to be a regular role critical point that happens alive within that region of the want to that of 2 x 4 1 And then I have 2 is a negative 2 at 2 . 4 He said that he was 0 This was easy to deal with the top equally Asian here we get X We want equal 1 The bottom equation period X 2 equals negative to so when I do the critical point analysis just like before what I get his Critical point for stationary point that want a negative 0 0 1 negative too is down here that outside the region and interested in so I would not be used as is not inside the square rectangle scholar are Don't worry about that point that means the Max's either on this said on 1 of the edges were at the corners so I have to go to analyze all the crew
The I they methodically does go through each 1 don't try to be a hero and dual Manya headers just read all the No 1 it was the defining feature of L 1 it's that 2 variables won only 1 arms constant so final 1 the constant along L 1 x this variable X 1 is stuck it to the whole time once you have that then that takes that original function reduces it down to a function of 1 variable that that is now equal to survive plug into areas for minus 4 0 negative x 2 squared minus 4 x 2 plus 1
Now I only have 1 variable so I could see it on a partial But why would the zone 1 variable that Prime negative Culex to minus 4 Senate equal 0 you get 2 Equals negative It Notice that occur in a rectangle does not make doesn't occurred got to Negative This is not so we just not for about that didn't contribute to our List Potential potential point from the accident OK to the defining feature of their X 1 is moving back and forth along their next to his equal
So when x 2 equals to illustrated down families that have reduced to At 1 squared used to at once again minus 4 minus 12
Plus 1 minor but if you believe that
Valley of 1 variables analysis to the derivative to add 1 minus 2 equal 0 that means with a point will want to like using the location 1 common and this is in R and 1 comment to a world a a That is so that there want to a critical point that so far that's only 1 a L 3 at Sydney same as L 1 except Brunei have X 1 0 still get negative next in line for 2 plus 1 the same way that twice Acer saying he could see the the same thing that have to minus city people 0 executed negative too Same problem 0 negative to it is not at all so we'll have to worry about the afterward
Cause If here 1 senses patterns maybe you realize 4 will have a point as well for the defining feature years next 0
Along all 4 x 1 is changing the text to stuck at 0 quality next to equal 0 I get at holes had 1 squared minus 2 at all but 1 2 1 2 0 6 . 1 day so for this point get . 1 comma 0 Those 2 critical points focused on my final list that I'm checked to see would largest value relax in the smallest IV the men are these due critical points and out that in the quarters and now remain listed 6 points check at that those 6 points the largest ones will be
The Max And small store will be
From my potential maximum points are 0 0 the corners to 0 2 2 0 2 and then I have the critical points that I found that 1 to and That would want to get 1 to end a 1 0 So F 4 0 0 0 But she called 1 echoed to 0 to help me with this 1 that's also a global 1 another's yeah it's what they a have to pursue From that's going to be And negative for native is native let's negative 11 11 and then 0 2 But also negative 11 he judge check this for me here and want to Negative 1 That to be native The really not incidents negative 12 11 that's negative 12th You is doing this for me to make sure is negative and at 1 0 found little easier 1 minus 2 negative 1 this 1 0 OK so where's the maximum where's the here's the fears of men and then the Mac's history here these 2 locations
Do this stuff before I remind you because this is a good example of that there's nothing is no 0 no step in this problem is difficult but putting all packaged together that's enough that that takes a lot especially at a lot of other things so learned simultaneously this is the only thing I get the yeah put this together with all the other things so we need to do is this is always when every wanna learn something new and math
EDT a clean piece of paper and until you can reproduce it without looking at the solution you do not move on and you know my taking depends on the person but no about 20 minutes This may be a body done that he kind of might be That how learn In in it's also helpful billionaire Nazi this is what students to there's a mistake that and I did when I was younger too so that's why is it you think you gotta do a lot of different problem in order to get ready for really which needed be so much better sometimes it is is if he took this problem and it 5 times in a row you get so fast that it would be so fluent in so trivial to you that he applied had head because it is so bored that's what you wanted to you you want to do it once ago EADS guide you to have a bunch of times and you show good added It's you could do in your sleep then when you look at another problem you have that method completely burned into your mind and you just follow the recipe because you learned 1 problem really really well suited usually You feel like they can they can spend the time doing the same problem over and over again that is
Value and consider that move for here Look at number 12 irrigated lectures last lectures I'm not Reproduce city and get that from the note or you could watch the video that's that's available to do this last point here route
Things look a lot more intimidating when you have fractional Expo we agree on that if this look a little retained the pattern of that's intimidating adding this workers Peter weights can ugly Typical Cobb Douglas in fact it's more than typical it's the simplest possible case it's the 1 where you put these things So what we've done before uses a cart Douglas function defined the MRT yes in this case with dual Grant multiple multiplier problem somberly constraint till Ellen K. our labor and capital was a practice you this stuff you have to be there when you get into a year More and you're Yukon Major was below the level said the Michelle you why the levels sets are always do what they are with you well happen any kind of lonely level says it is Drawn thinking game Today has the shape of a C Y is a good example the sea what level said He saw you do for level said said equal to a constant will be the best choice for a constant here That is give you a single point 20 insists he's a 20 20 because it is what I want and that 20 of I said He for and I have 20 cost 1 EL one-half game want that And the 20 scans But I think it's a good choice will be another 1 maybe 40 60 OK So then what what is that we have 1 equals L 1 kicks and if you want Take a letter to be this is X and Y not a graph that is just the 1st one's always the x axis of the sector will vote the axis there's L K and I to grab this so what we do have a Texan why we usually self-reliant terms of excellent rapid To that here here might Why is it OK but at a quarter have people 21 or older adults and half that that's just so ugly red like this J. equals 1 over route L and that implies Nikkei case is equal to what So what's that gravity like it's wise Goes 1 acts that helped you see what it is and that this picture That's why I always get that she and it doesn't matter enough if I had different exponents than what'll happen joined up with something like a equals 1 over Overhill square 1 overall squared still has that shapes as when every had any kind a reciprocal function said that shaped This notice that I take a derivative period something negative its way down the 2nd derivative will be positive that's why concave OK so new that the level set of low-carb Douglas function you don't have a negative labor capital because you don't real-world OK so Mallette said
Let's get The rest of the problem so l labor and Have Labor costs
40 dollars per unit capital costs 120 dollars per unit And Oregon have is a budget constraints so we have 3 100 thousand dollars budgeted for Africa For the year-end the company now much you have to invest in the company naumachy Liu wherever it is you know how much that is my teachers and this is how much I'm a person working for me and this is the cost of renting the factory the machinery so I Hal sings known to have put it together well 40 of these this this implies this is mainly my budget constraints that 40 of the units of labor plus a 120 time units of capital I can't spend more than 300 Let my budget constraints
OK so what I wanna do that I wanna maximize production With this The strength of this constraint of beer It'll be something like this series budget This is what we're looking for Where the budget constraint This Tianjin to level said of the production from whatever that is as we want fight crime that's the picture you see that Picture so we Yukon class because this picture and very had That's also the same picture for utility function is just always that picture OK so we need really is the method of grants multipliers so what I need is let's call this the deed of L. A. budget Let us just say that this is
His Caribbean located sequel The 300 thousand that's the level said So will argue for furlough bronze multiplier take that function tickets Korean and said equal this Caribbean and put a constant in there and I find out what the relationship by derivatives
Said that's equal to see if you believe that success at 10 K
The derivative of of Alltel one-half is 1 of the 2 Elton one-half cancels the warning and he said OK they're completely symmetric they have the same exponents of the same switch letters lose my ears my partial derivatives of that the gradients that equals land at times a gradient of G that means I have 10 Group over Trudell 10 roof Elf over route J. people's slammed a the gradient of that only 40 and 120 It was at this point you have to do is have faith in me you have that faith that I've It's the numbers or perhaps even a little eclectic looks psyche it's a nasty algebra chose this That's not too bad to equations here and drew over rude L Equal slammed transporting them and the bottom 1 year I have 10 rude L. old K equals landed that the do that we always 40 landed 20 LANs Gallegos said you have that faith what do we do always at this step so let's say step here is just the algebras too messy it's you you just not confident which need to do is go back to 1 of the other examples of go state what would we do with this step would we do after we set these equal China solid what I've done in every single case if you look there's a sulfur landed here and then I get so I get Lander Divide by 48 every route over for Rubell is however that in every single case sulfur landed this debt and energy glamor and I plug it in there and what that gives me is a relationship between can l can I take that relationship plug it back in the budget constraints the constraint were not there yet but 1st do this so I have tendered Ruth L. over route K 1 20 Times that For health So I bring this route came over here to reality AEK they bring the route Eleva there Something ideas is 30 So The numbers did work I didn't look like it was going to be the next it would we do with that all we can hope for at this stage you look back at the other examples or at this stage all you could ever hope for was to get a relationship between the 2 variables that the last step was to take that relationship that we found and put in the budget constraint now actually give you the values that were interested
But Focus I have L equals 3 K and then the budget constraints says I have 300 thousand equals 40 120 day so that tells ago 3 Casler stake 3 came when in
In you really 0 here Like I have 30 thousand calls for times 3 day plus 12 OK Pockets of 13 thousand cost 24 K K is equal to Where is it He was 1 of 5 to where it Want us to OK and then Dallas 3 times that of that said This is the maximum production in order to really tell that you need the money back in the function in check another point on the streets so I didn't do all that you can do that arms program so if you plug another point here in just say Was You make this 1 0 and then you find at another point in check it out 1 will be larger than the production value this thing is with Carl Douglas You only get maximum production at is the minimum production 0 fat 0 labor and production 0 so I'm only that Dole is the minimum production 0 so anything else any other point you find is automatically get maximum and probably that's why we don't need to do that extra step OK so of as well how more questions for me hard leave that Abdullah calculus stuff you could look back at the midterm review I gave you the notes that post the post of Missouri A mail forget they still on their way to look at them a review for the germs but mostly like I said at the beginning if you knew the sample midterms the midterm a sample Final Four trip Yosemite questions for me talk Elsie on Wednesday
Offene Menge
Gerichteter Graph
Negative Zahl
Tangente <Mathematik>
Güte der Anpassung
Partielle Differentiation
Kritischer Punkt
Rechter Winkel
Sortierte Logik
Dimension 3
Ordnung <Mathematik>
Klasse <Mathematik>
Weg <Topologie>
Gleichgewichtspunkt <Spieltheorie>
Thermodynamisches System
Orthogonale Funktionen
Faktor <Algebra>
Turnier <Mathematik>
Abstimmung <Frequenz>
Prozess <Physik>
Natürliche Zahl
Familie <Mathematik>
Fortsetzung <Mathematik>
Kartesische Koordinaten
Konkave Funktion
Arbeit <Physik>
Einheit <Mathematik>
Regulärer Graph
Exakter Test
Wurzel <Mathematik>
Figurierte Zahl
Dimension 2
Lineares Funktional
Dimension 6
Arithmetisches Mittel
Gewicht <Mathematik>
Total <Mathematik>
Größter gemeinsamer Teiler
Positive Definitheit
Derivation <Algebra>
Physikalische Theorie
Ausdruck <Logik>
Matching <Graphentheorie>
Physikalisches System
Kette <Mathematik>
Parabel <Mathematik>
Numerisches Modell


Formale Metadaten

Titel Math for Economists - Lecture 15
Serientitel Math for Economists
Teil 15
Anzahl der Teile 15
Autor Kronewetter, Jason
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported:
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DOI 10.5446/12909
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik

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