maximizing utility with quasilinear demand

Instead of the usual cobb-douglas, here I illustrate maximizing utility with a quasi-linear utility function.

[@Lumchuck]

Thank you so much for this. Such a good explanation. Thanks for not skipping any of the maths either, it really helps to go through step by step.

[@BurkeyAcademy]

I am glad it helped you! Cheers!

[@akashkalyana8295]

Not many people shows how to solve numericals,thanks alot.

[@BurkeyAcademy]

You are welcome- if you have any special requests, let me know!

[@akashkalyana8295]

yes sir,if u can post a video about numericals on phillips curve and inflation and stuff,its just nowhere to be found,i'll really appreciate it.

[@akashkalyana8295]

numericals on unemployment,inflation and phillips curve is what i want.

[@BurkeyAcademy]

I don't do much Macroeconomics, but if you have a specific request for a topic maybe I'll do one. Here is an interesting Phillips curve one I made: kzbin.info/www/bejne/e36ThIyqeaypadk

[@akashkalyana8295]

Thanx sir,you are really doing it to help people,i really appreciate it.

[@mikegargaro]

That cleared everything up for me, thank you so much.

[@tausal1]

Thank you so much! Extremely helpful.

[@jamespatrick9191]

thank you so much, this helped me in my microeconomics course! btw has anyone told u that u sounded like Iron Man?

[@abbymelvin890]

Is the marginal utility of x increasing ? Is the marginal ulitity of y diminishing ? And the Marginal Rate of Substitution diminishing ?

[@BurkeyAcademy]

Perhaps this will help: kzbin.info/www/bejne/rKXSoGCfoLFgnZI

[@grup9499]

Thank you so much! This is a great video! :)

[@divyanshiraj4416]

Thanks a lot..

[@WheelhouseHockey]

Does the rule that slope of the indifference curve is steeper than the slope of the budget line mean that this is a corner solution for other types of common utility functions as well?

[@BurkeyAcademy]

Hmmm. Just to make sure we are on the same page, exactly what rule are you talking about? I generally thing of the rule being that if the solution is not feasible, e.g. involves a negative amount of x or y, that indicates a corner solution where the solution is to spend all money on the good with the positive value. It has been a while since I made this video, but I don't think I know of a "rule" about the relative slopes. Do you mean steeper at the x intercept? If the corner solution were on the y axis, the budget line would be steeper. But in general, any utility function that has indifference curves touching the x or y axis could have a corner solution- bot a Cobb-Douglas cannot.

[@david_felipe]

When we talk about MRSx,y we use UMy/UMx and the other way MRSy,x is equal to UMx/UMy. It´s that correct? Thanks for the response

[@BurkeyAcademy]

You have it backwards. Though I think the notation is a bit confusing and backwards myself (don't blame me! ☺). MRSx,y =MUx/MUy, and means "how much y you would be willing to give up to get one more x".

[@elferi]

great lecture, sir. may i know what are the book you refers to for the lecture? thanks

[@BurkeyAcademy]

None really... but when I recorded this I was teaching out of Besanko and Braeutigam's Microeconomics. So, this probably relates to one of their end of chapter exercises.

[@TheArkus2002]

It doesn't seem like there's any solution where you will actually have a corner solution where you consume only y and no x. Mathematically, it seems like there is only one type of corner solution in this problem, where you consume only x and no y. With any positive prices and income, setting MRS = px/py results in a positive value of x. Am i missing something? As you keep increasing px in your example, you continue to get a positive x value, albeit closer and closer to 0. So there is now way we actually get to the corner. I can see it graphically, but not mathematically. When I graph this quasilinear function on desmos.com the indifference curve becomes nearly parallel with the y-axis, up to 3 decimal places, an then reaches 10. But presumably you could take this value to a closer and closer value to 10- and never actually reach it. This would mean that x is always positive. The way this was taught to me was that you will always consume some of X, but your decision to consume some of Y depends on the amount of income. i.e. MRS = 1/2*SQRT(x) = px/py, and therefore X* = px^2/4py^2 (a positive number, provided px and py are positive). I = px.X + py.Y, so I = px.(py^2/4px^2) + py.Y, and I = py^2/4px + py.Y, Y* = I/py - py/4px (a negative or positive number) So we need for Y > 0 to be am interior solution, otherwise we will just spend all of our money on X, and none on Y, i.e. [m/px,0] Y>0 I/py - py/4px > 0 I > py^2/4px So the problem is solved in the fashion X*,Y* = [px^2/4py^2, I/py - py/4px], if I > py^2/4px and [m/px, 0], otherwise.

[@BurkeyAcademy]

You are right, thanks for the comment! I hadn't thought about the math very carefully. but since the MRS is 1/(2sqrt(x)), as z goes toward zero the slope goes to infinity. So, no matter how steep the budget line is, you will always buy some x.

[@gurnoorsingh4103]

is under root x+y is cobb douglas function

[@BurkeyAcademy]

Nope- think about it: You can always replace 1 x for 1 y and get the same utility. root(8+1) = root(7+2)=3. So, these are perfect complements with MRS=1.

[@gurnoorsingh4103]

What about root X × Y

[@BurkeyAcademy]

Root(x*Y) = y^.5*y^.5, so ...

[@gurnoorsingh4103]

Is it cobb Douglas or not

[@BurkeyAcademy]

Do you know what a Cobb-Douglas looks like? If you don't know, I have a video or two for that... I am a professor, not a do your homework for you service. ☺ You can look elsewhere for that, if you want to.

[@namratamishra6909]

But the optimal bundle doesn't satisfy the budget line If we put optimal bundle in budget line it turns out to be 5.125 which is greater than 4 Can you please explain this part

[@LeandroRobert1]

Good video. But I don't understand something: why didn't you use the lagrangian method? Thanks.

[@BurkeyAcademy]

There are two common ways it is taught, this is one of them. Here is a video where I explain how the two methods are really the same thing: kzbin.info/www/bejne/hWSwd4uKbKaXhsk

[@LeandroRobert1]

Ok, so langrangian method is suitable for quasilinear demands aswell?

[@BurkeyAcademy]

leandro8894 Absolutely. Do the problem in this video using Lagrangian for practice, and see what happens.

[@LeandroRobert1]

Please, correct me if I am wrong, but I think I can't actually "solve" it, since I have a function of X expressed in general terms, and I can't obtain a plain 'X' from it. That is because when I equalize the lagragians Multipliers, after setting the first orden conditions, I get f'(x)=px/py. The following step would be obtaining X=*something* and replace that in ∂L/∂ λ. But only by knowing what f(x) stands for (for instance Ln(x)), I would be able to solve for x. But still, I think I get the general idea. My only question is: You say that in quasilinear Functions, we obtain parallel indifference curves, with the same slope if X=Xo. Now, Given a Cobb-Douglas function, and a fixed value of X; If I move along the Y axis, do I get a different Slope value for each Indifference curve?

[@BurkeyAcademy]

For a cobb-douglas, you will get a different slope if you hold x constant, but change y. Easy way to see this: Let U=sqrt(xy). Then the MRS (slope of indifference curves)=y/x. Hold X=5, and increase Y, the slope goes 1/5, 2/5, 3/5,...

[@rhea_khurana]

Will quasilinear preferences always give corner solutions, or interior solutions are possible?

[@BurkeyAcademy]

In my first example there was an interior solution, so yes, it is possible. Just remember, that a corner solution is impossible with a Cobb-Douglas.

[@rhea_khurana]

BurkeyAcademy Another question: How do we know when we'll get a corner/interior solution?

[@BurkeyAcademy]

Good question! 1) If you get a negative solution for X or Y. 2) Another way we might be able to tell is if the slope of the budget line (Px/Py) is either greater than the MRS when Y=0, or less than the MRS when X=0. See if that makes sense to you- if not, let me know, and it might make an interesting video...

[@rhea_khurana]

BurkeyAcademy This makes complete sense. Thank you so much!

[@emreyilmaz7641]

I dont get how you made the MRS shouldn`t it be 1/( 2*square root (x) ) ?

[@BurkeyAcademy]

Yes, what you have is just another way to write what I have. .5=1/2, and x^.5 =sqrt(x).

[@emreyilmaz7641]

Thank you , i`m from germany and we don`t use that type of writing so i was confused

[@anais5811]

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[@anais5811]

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[@anais5811]

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[@anais5811]

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