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Right this is the first video on comparative statics. I guess a good starting place place would be to answer the question. What is it what is comparative statics.
The way that i can define it would be that you find the optimal. I dont know consumption bundle or whatever is it whatever. It is that youre looking for you eat your first order conditions.
Then you say well what happens. If something was to change what happens if i all of a sudden get more income. Hows that gonna affect my desired quantity of good x good y etc.
So were gonna start by setting up. Our typical lagrange. Form eula finding our first order conditions and then moving from there for this example.
Were going to start with an expenditure minimization problem. So the first step will be set up your lagron and were going to do a very general example of this no functional form just very basic this is our typical lagrange thus is my letter for lagrange. This is going to be your price for goodwin your quantity for good one price for good two quantity of good two this is going to be our lagrange multiplier since this is an expenditure minimization problem.
This will be our desired level of utility that we want to keep in mind. And this will be our utility function. Which is a function of good 1 and good 2.
So you know just a quick reminder. This is going to be your total expenditure right here. And youre going to have the subject to achieving some desired level of utility.
And this is going to be your utility function so now that we have our basically rhyme setup. Were going to start this just like we would any other optimization problem so the first thing. Were going to do is were going to derive our first order conditions.
And i want to go ahead and write that out for you so just so you see whats going on here. I want to go ahead and just write it out like this first and then ill go ahead and clean it up so the first thing. Im doing is im taking the derivative of the lagrangian right here.
So you can see that weve got p1 and then i come over here up in here.
And so this is going to be a fixed value you could call this you know utility equals. 2. Or whatever you want so pay no attention to that its a constant and then were going to take the derivative of this new function here with respect to x1 and so you can see that this is going to be attached this is going to be a negative so to clean that up i just wanted you to see.
Where it was coming from we have l. 1. Equals p.
1. Minus beta u. 1.
And of course is the first order condition so we want to set that equal to 0. And in a similar fashion you see that we did the same thing here for the lagrant with respect to x2 and also for the grunge with respect to its lagrant multiplier and you can see that this just gives us back our constraint. Which says that for this given the desired level of utility.
You know subtract out its function you get back zero. Its important to note. Here that we can see this is a see this you one right here.
Its still gonna be a function. I think ive already drawn the arrow here of x 1. And x 2.
So its still a function of what youre consuming and you can see that these are actually going to be a function of your price and income. So thats thats something i think that a lot of people that tend to forget when theyre working on these problems. So dont forget about that.
And you know ill come back to that later. But just keep that in mind so our next. Important step is going to be we want to find a matrix of second partials.
So looking over here. We can see that this our first order conditions here. I guess we can just call those our our first partials so a little trick that i like to do is go ahead and just keep note of what youre going to be looking for so here.
Weve got l 1.
1. Then were going to have l. 2.
Just like before the second one 1l you could call this 3. But im gonna go ahead and actually make that 2 theta just like before 1 similarly. This is going to be l1.
L2 l. Theta 2 l. 1.
Theta l. 2. Theta l.
Theta theta. So basically. What we.
re. Going. To.
Do. Is. Have.
1 2. 3. 4.
5. 6. 7.
9. 9. Equations.
That we need to solve. Now. I cant remember what the name of the rule is but um since these are gonna be partials.
Theres basically a calculus rule that means that if you have a function. You take the derivative with one then two its going to be equivalent to two then one so youre gonna see that these two will be symmetric similarly this should be the same again this will line up with this so youre gonna see that you can really check to make sure you havent made any well any obvious errors as long as these guys line up. So.
Im gonna say recall that the l1 equals this and so were gonna want to take that the derivative of l. 1. With respect to this will be x 1 x.
2 and our multiplier. So our first will be 0. Minus theta u.
1 1. So we can go ahead and get rid of that and recall that i said u. 1.
Is still a function of x 1. And x 2. So.
Which you can really say is the derivative of u 1. X. 1 x.
2. With respect to x 1. And thats simply going to equal u 1 1.
But of course dont forget about your theta alright.
So using that same chain rule. We have l 1 2. Equals.
0. Minus. Again theta u 1 2.
So again this comes from the fact that x2 is still on that function. So thats how you do l1 and lastly. We have l 1 theta equals negative u 1.
So were working on our second line from our personal conditions. So this time well start with l2 similar to how we started with l1. So for l2.
Equals p. 2. Minus.
Theta u. 2. Again you just do with respect to x.
1. Which equals. This then with respect to x 2.
And with respect to theta so were again. Were really just working off with that first order condition okay. And so recall that this is just a constant so for the theta 1.
We just have negative you know and this is with respect to x 1. 2. Equals similarly with the with respect to 2.
And then for theta theta. We have no thetas so simply zero and of course. We want to go ahead and collect all of these terms.
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