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u(x, y)$ is a utility function. Sketch some of its indifference curves.$$u(x, y)=2 x+3 y$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Johns Hopkins University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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05:15

u(x, y)$ is a utility func…

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03:52

04:32

07:22

Graph a typical indifferen…

03:24

For each of the following …

So in order for us to sketch the's in different skirts for our utility function, remember, that's just kind of a fancy way of saying a level curve. So the first thing we would do, they just come over here and set this equal to Z, or you could do see on. Then we can solve for one of the variables than graph. But in this case, since this is just a line, I'll leave it like this because it would make it a little bit easier for us. Thio graph. Overall, I feel because then we could just graph the to zeros and then go from there. Yeah, but now if we were to come over here and plug in just some values for Z, um, so let's do so. One thing we're going to kind of assume is earliest normally. What we would assume is that these inputs X and Y are non negative values, uh, normally even more than there normally non zero values. So when we're looking over here at Z, this should also be like a non negative, non zero value or just a positive value. So let's just start with like Z is equal to one, and then we can go from there. So that would give us the equation. Two X plus three y Is he gonna one on? Then you could just grab this, however you one. But keep it in mind, we're more than likely just going to be staying in these first quadrant here. Eso Let's just quickly find what the intercepts they're going to be. And then we can plot those eso If why is zero? Um then that would give us two axes, Eagle toe, one which then says axes equal one half. So over here, if I were to kind of plots of these points down. So for this first curve, X is going to be at one half and then, um, we would need to also do the same thing for, um finding our y intercept. So we plug in X is equal to zero eso. Then that would give us three y Is he gonna one, or why is it to one third? So that would be a probably somewhere around here so that this little curve right here is going to be are indifference for utility of one. Now let's go ahead and do actually make the screen a little bigger. Let's go ahead and do the same thing. But let's just do it for like, a larger number. Now let's maybe do like or so that would give us two X plus three y Is it before? So now when? Why is he go to zero? That would give us two. X is equal to four or just X is equal to. And then when X is equal to zero, that would give us three. Why is it before or why is equal to four thirds? I'll actually graph this one in blue here, so we come over here and graph those so that would be, too. And then we have, like, four. There is probably around like here or something, and so that this is going to be. Our indifference for Z is equal to or thirds and then I don't know. Let's do just maybe one more and do like nine. So Z is equal to nine. Is going to give so two X plus three y is equal to um, Or better yet, let's do six because I think they will give us a nice number for each of these so six. So that would give when X is zero, you get three wise, you get sick. So why is going to be too? And then when y is equal to zero, we would get two X is equal to six. So then X is equal to three. So let's come over here and graft both those with why is equal to x 0 to 3 and then we just connect these here. And so then this is going to be our indifference when C is equal. E don't know why I wrote or thirds over here. This was Z is equal to four. And then over here this was Z is equal to six. Say I you could go ahead and make some other indifference curves, But I think at least doing these first three is good enough. So yeah, this here would be, um, just a couple of those sketches

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