Extended utility function example. Monotone transform. Quasiconcave and quasiconvex functions. Characterization in terms of convex upper and lower contour sets.
Thank you so much for the class. It helped me a lot. I have just one question. Is U(x1,x2)=x1x2 strictly convex? How could we verify if a function is "strictly" convex/concave?
Actually, it's not even convex (or concave). It's strictly quasiconcave. The first answer to your question "How can we tell?" is to try to apply the definition directly. It's easy to do that to verify that this particular function is not convex: just apply the definition to the two points (1,3) and (3,1) and their convex combination (2,2): you'll see that the condition defining a convex function is violated. You can do the same with (1,1), (3,3), and (2,2) to see that the function is not concave. The second answer is that you can use calculus. See Lecture 33(B).
Spend some time thinking about the example f(x) = x_1*x_2 in the video. It's quasiconcave but neither concave nor convex. Try to understand the explanation why that's so, given in the video.
professor, nowadays, do we have methods to find the global solution about the qusi-convex functions? If yes, can you give me some references? thanks a lot.
That was just a delight to watch, couldn't wrap my head around it until I saw this video. Thanks a lot!
Glad you found it helpful, and many thanks for the positive feedback!
This is the only place where I could finally understand it CLEARLY. Thank you professor!
OK, glad you found it helpful!
This was absolutely incredible!! Thank you so much :)
Thanks! I'm really glad it was helpful.
Another great video! I believe at @15:08 you meant strictly quasi concave function, not strictly concave. Thanks!
Yes, thanks -- that correction is actually in the video, at the bottom of the screen.
Thank you so much for the class. It helped me a lot. I have just one question. Is U(x1,x2)=x1x2 strictly convex? How could we verify if a function is "strictly" convex/concave?
Actually, it's not even convex (or concave). It's strictly quasiconcave. The first answer to your question "How can we tell?" is to try to apply the definition directly. It's easy to do that to verify that this particular function is not convex: just apply the definition to the two points (1,3) and (3,1) and their convex combination (2,2): you'll see that the condition defining a convex function is violated. You can do the same with (1,1), (3,3), and (2,2) to see that the function is not concave.
The second answer is that you can use calculus. See Lecture 33(B).
@@ArizonaMathCamp Thank you so much!!
thank you so much Sir...
I saw the model of convex function and quasiconcave function are the same? How can we distinguish them?
Spend some time thinking about the example f(x) = x_1*x_2 in the video. It's quasiconcave but neither concave nor convex. Try to understand the explanation why that's so, given in the video.
Superb
Thanks for the great one-word review!
Thanks a lot for uploading all your classes on youtube. But will you be making videos on dynamic analysis like Difference and Differential equations?
No, I won't be doing any lectures on dynamics, except for a single lecture on a simple growth model.
professor, nowadays, do we have methods to find the global solution about the qusi-convex functions? If yes, can you give me some references? thanks a lot.
Sorry, no references.