Derivation of Marshallian Demand Functions from Utility Function
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- เผยแพร่เมื่อ 24 ก.ย. 2024
- Derivation of Marshallian Demand Functions from Utility Function
Learn how to derive a demand function form a consumer's utility function. In this problem, U = 2X(1)^0.5 + 4X(2)^0.5.
This video introduces the Cobb-Douglas utility function for two goods and demonstrates some of its properties.
I demonstrate how to compute marginal rate of substitution for Cobb Douglas utility. I use the resulting expression to show why Cobb Douglas indifference curves have constant slope along rays out of the origin.
I also demonstrate how to solve for consumer demand using the Cobb-Douglas utility function. After deriving Cobb-Douglas demand, I show that the alpha share parameter in Cobb-Douglas utility functions actually represents the consumer's expenditure share on that good.
This video uses calculus to compute MRS, but after that, the rest is just algebra. If you are a viewer who doesn't know calculus, you can take me at my word, skip the computation of MRS, and the rest of the video can still be worthwhile.
Economics Playlist
Derivation of Marshallian Demand Functions from Utility Functions: • Derivation of Marshall...
Derivation of Marshallian Demand Functions from Utility Functions (II): • Derivation of Marshall...
Derivation of Hicksian Demand Functions from Utility Functions:
• Derivation of Hicksian...
Deriving Marshallian and Hicksian Demand Functions (compensated and uncompensated demand):
• Deriving Marshallian a...
Derivation of a Generalised (n-good) Marshallian Demand Function from Utility Function.
• Derivation of a Genera...
Deriving Hicksian (compensated) Demand Using the Slutsky Equation:
• Deriving Hicksian (com...
Deriving Marshallian Demand From a Stone-Geary Utility Function:
• Deriving Marshallian D...
If you would like to see more of these more advanced microeconomics exam questions, let me know.
:)
As an addendum to this video please answer the question, "Does the following demand functions satisfy WARP?" by constructing the Slutsky matrix.
Solution: We know that WARP implies CLD, which in turn implies that S is negative semidefinite.
Thus, if we show that S is not n.s.d. then WARP is notsatisfied.
s11 = ∂x1/∂p1 + ∂x1/∂w x1 = p2/(p1 + p2)2 + p1/p2(p1 + p2) > 0,
which implies that S is not n.s.d. and WARP is not satisfied.
But, I want to see it worked out.
Please make a video on deriving slutsky equation through maximization of utility function subject to budget constraint
Most definitely, I'm in microeconomic theory right now at uni and this was extremely helpful. Also if you have anything for macroeconomic theory a few videos on that would be killer as well.
Plz let's have more lectures and questions as well. Thanks
wow, you made this way less complicated than what it seemed in my intermediate micro book. thanks a lot for taking the time to make this!!
one of the best micro videos ive ever watched. but i recommend to use different letters instead using x1 and x2.
Good call! It also gets pretty damn confusing and it’s a accident waiting to happen algebraically 😑
Explained better than a UC Berkeley professor - thanks!
+Grady haha, thanks. Just trying to help some folks out.
The professor that explained this to me was also hopeless. It took one of my mates from the year above to start from scratch and explain it without the jargon for the penny to drop with me.
I am glad that it helped :)
That’s exactly what I said. Kid needs more views.
God bless you sir. I'm trying to get through Hal Varians advanced microeconomics and it ain't pretty, but this helps a great deal.
No problems mate, glad the video helped 🤓
yep that book ./
I think there is an error in setting up the Lagrangian. U(x)+lambda(Y-P1X1-P2X2) or perhaps U(x)-lambda(P1X1+P2X2-Y).
Because the way it has been solved in the video, the FOCs are wrong.
Can you identify exactly where the mistake is, I don’t follow what you mean.
Bit of an update; the sign of the Lagrange multiplier would affect the value of the first order derivative however as long as you are consistent with your signs throughout the problem you shouldn't get a different answer.
I hope that helps :)
Here is a link to a great explanation on this exact topic:
math.stackexchange.com/questions/1099429/when-using-the-method-of-lagrangian-multipliers-does-it-matter-whether-i-subtra
Thank you very much Harold. I have just completed a similar question that I received as an assignment
Good to hear that my videos are helpful . Thanks for taking the time to leave a positive comment 🤓
ins't the Lagrangian equation + lambda, not - lambda?
in these cases I don't think it matters.
Bit of an update; the sign of the Lagrange multiplier would effect the value of the first order derivative however as long as you are consistant with your signs throughout the problem you shouldn't get a different answer.
I hope that helps :)
Here is a link to a great explanation on this exact topic:
math.stackexchange.com/questions/1099429/when-using-the-method-of-lagrangian-multipliers-does-it-matter-whether-i-subtra
Hello! What can we do if the partial derivative of x2 is equal to 1 ? How are we able to to plug x2 in the Y formula ?
I think the function should be : L = U(x1, x2) + λ (M - PxX - PyY), not - λ.
Is that correct?
It’s fine either way, the first order conditions will cancel to make the sign on the lambda irrelevant.
Good question though
🤓
I don't understand why in the original langrangian equation some choose to add the constraint and some choose to minus the constraint. Wouldn't this affect the first-order derivative?
Hi Dan, the sign of the Lagrange multiplier would effect the value of the first order derivative however as long as you are consistant with your signs throughout the problem you shouldn't get a different answer.
Here is a link to a great explanation on this exact topic: math.stackexchange.com/questions/1099429/when-using-the-method-of-lagrangian-multipliers-does-it-matter-whether-i-subtra
Hope it helps :)
clear, concise explanation and beautiful video. what software do u use?
I use a program called Sketchup and a Wacom Bamboo pen tablet and I use a program called Camtasia for my screen capture.
many thanks!
No worries 🤓
Hi! I know it's a bit late, but is there any reason why you subtract the Lagrange multiplier? I thought you added it, cheers!
It doesn’t matter, you can add or subtract it and the outcome will be the same.
Check if you like :)
Incredibly concise. Thank you.
Thanks for the compliment, I really appreciate it :)
you saved my micro mark thank you lol
That great to hear. I am happy that I could be of assistance :)
This is scary than a horror movie
Yep, that’s what I thought when my lecturer first went through it.
Why is the lambda have a negative in front of it, I thought the formula for the lagrangian was with a positive for maximizing utility.
Try it with either positive or negative, I think you’ll find it doesn’t make a difference 🤓
@@HaroldWalden Thank you!
No problem ❤️
Perfectly explain thanks.
+Gedleyihlekisa Dunah thanks, I am glad it helped. I always struggled to find a good way of explaining how these optimisation questions/problems work. I feel I am getting closer but I am going to keep trying.
why did you take a reciprocal formula when finding X and Y in the first part
Which bit exactly are you taking about?
after taking the FOC of x1 and x2 , you have to solve both to obtain x1 and x2 and in your solution you said take the reciprocal of the lamdar P =-2/x2^1/2 . so i'm asking why the reciprocal formula @@HaroldWalden
@@agnesphileoboateng6790 Oh ok, I understand. So to get LambdaP_1/LambdaP_2 we need to divide -1/(x_2)^(1/2) by -2/(x_2)^(1/2). When dealing with fractions, I find it is easier to use the fact that division is simply multiplication by the reciprocal
For example, 6 divided by 3 is exactly the same as 6 times by 1/3.
So taking the reciprocal has nothing to do with the intuition behind the problem, it is simply me trying to verbalize what I find is the easiest way to do the algebra.
I hope that helps, let me know if I need to clarify it further :)
Thank you so much!! Very clear and helpful :)
Thank's, I am glad that these videos are making life easier for people trying to wrap their heads around these kind of concepts, sometimes it feels like you are just banging your head against a wall. If there are any problems your having, I am always looking for new ideas for videos... Let me know.
:)
+Marilou Flandres glad I could help :)
I think I'll put up more of these slightly more complex economics videos, they seem to be getting a fair bit of love :)
Thanks so much i really enjoy the content can you please show how to get the Hicksian demand after getting the marshallian
Hey, already have put together a video answering exactly that.
th-cam.com/video/Zrs9uSMg6Sg/w-d-xo.html
Have a look and if it doesn’t help let me know :)
can't we reduce this problem to a single variable problem and solve it?
Good question, the answer is no. The reason being is because this demand function is based on two price levels that move independently of one another. We can find the demand for each of the goods as a function of the prices of both goods and a fixed income, Y but be cannot simultaneously solve it.
how can i solve if the function is a + b < 1. that is power does not add up to 1. I am stuck with algebra
well, in this case, a + b must be less than or equal to one. However, because we are trying to maximise our utility, the only way we can do that is if a + b equal one (if you're using a and b to represent proportions of consumers income spent on goods a and b. In the case you are describing, I would still maintain that assumption to be true and where you have a + b < 1 substitute in a + b = 1. In these type of problems, you should be spending ALL of your income on the two goods as the maximum utility can ONLY be achieved when you are devoting all the resources at your disposal to increase your current utility.
I hope that helps :)
Thanks that makes sense. I was thinking the same but didn't know i can change the utility function. I have powers of 2/3 and 1/5.
So i can make it 1/3 instead of 1/5
Aamir Khan usually the powers will add to one, it might be worthwhile to check with your lecturers to see whether or not there has been a mistake???
thank you so much!. It was very helpful.
No problem, glad I could help.
Thanks for taking the time to leave a positive comment 🤓
I hate math but you make me hate it a bit less but I still hate it a lot.
Please how do u solve for the marshallian demand function with U=xe with e raise to the power y
Have you stated the lagrangian correctly?
Yep
Do you have a video on how to do this but for 3 goods X1, X2 and X3 ?
No but I definitely should do one.
How do I derive the consumer's Marshallian demand function for min (x,y)
What is y greator and equal
Thank you!
No problem, I hope the video helped 🤓
Harold Walden it absolutely did, taking microeconomic theory and application. Feeling good for class tomorrow :)
Carson yeah it can all seem like gibberish is you haven’t got a handle on this nonsense.
So we don't need to find partial derivative of L here :/
Can you elaborate please?
@@HaroldWalden Just got it, thanks. Do you have a video for Stone-Geary utility function, I can't find proper material for this.
No I don’t but you have definitely given me a great idea for the next video.
Thanks :)
Sorry it's a bit late but here is my explanation on Stone-Geary utility functions: th-cam.com/video/qHZHAJozLc8/w-d-xo.html
Hope it helps :)
@@HaroldWalden Thank you! That is great!
Love the content, but moving the mouse around as much as you do is a bit distracting. Just some feedback!
Thanks, I am still getting used to the platform. Keeping the mouse still is something I will definitely work on.
Can you please solve for utility function u=x^1/2+y
Plz sir is this youtube channel still functional???
Yeah mate, I have just been a tad busy :-/
@@HaroldWalden thanks so much will always be at alert for lessons
Hello, I like to enquire if there is any software that can perform analysis relating to utility or any utility functions software packages. Your reply will be highly appreciated. Thanks
You subtracted the langragian instead of adding
It doesn’t matter if you add or subtract, check if you like.
I have something similar to what I emailed you. Please let me know if you are interested.
You complicate things to much. Useless?
How would you solve this problem?