(M3E6) [Microeconomics] Utility Maximization Problem with Quasi-Linear Utility Functions

In this episode I study utility maximization problem with Quasi-linear utility functions.
It's crucial to watch lecture videos in the proper order to ensure effective learning. This is because the concepts in each video build upon those introduced in previous videos. To help you with this, I recommend visiting my website, www.selcukozyurt.com, for a recommended course outline.

[@julianrosas6197]

Hi profesor, and how I can find the optimal choice in a robinson crusoe economy?
If robinson have a utility = Log(c) - aH
F(c) = bH^u
0

[@dawmi3140]

x* looks like it could become negative
what values of Px would ensure that the quantity demand of good x is positive?

[@premeditz13]

Nice explanation
But what about the second order condtion?

[@TrymBraathen]

Double derivative = 0, to show concavity or convexity

[@tiffqnie]

I still can’t get it bruh got an exam in an hour I’m screwed

[@pablop.4465]

Hi Professor, don't you have to check for the feasibility of the optimal solution? I mean if it's in the bc? Thanks.

[@selcukozyurt]

Hi Pablo!
The solution of the Lagrangian, i.e., X and Y, always satisfies the budget constraint (i.e., XPx+YPy=I). But that doesn't mean that the Lagrangian solution is always the optimal (demand) solution. We need to consider if X (i.e., X=(I-Px)/Px) is positive. If X is negative, then the Lagrangian solution cannot be the optimal solution, simply because demands are not allowed to be negative. In this case, optimal demand for X should be 0, and so consumer must spend all his money on good Y.
The reason for this is simple: First, because Y enters into utility as lnY, it cannot be zero. Second, the optimal Lagrangian solution suggests that the consumer must spend his money primarily on good Y because the optimal Y is independent of I. Think it this way, if you don't have a place to live (apartment/room), then there is no point of buying more and more furniture. Your optimal Y is, say, a studio apartment. If you have enough budget, then you should rent a studio apartment, and then spend the rest of your income on furniture. However, if your budget isn't enough for a studio apartment, then buying furniture is pointless. You first need to fulfil Y as much as your budget allows (sharing apartment with somebody else may be) and then spend the rest on X (if there is any left).
I see now that providing this explanation would make things a bit better. Sorry for missing this out and thanks for the question.

[@pablop.4465]

@@selcukozyurt Thank you for the excellent explanation and for letting everybody access your videos. Best!