I have the question. And the question asks
A)Show (without solving the problem explicitly) that a solution exists to the above problem.
B) Show (without solving the problem explicitly) that there is a unique solution to the above problem.
C) show (without solving the problem explicitly) if $x_1$ and $x_2$ in $R_{++}^2$ solves the problem it must be that $x_1P_1+ x_2P_2=I$ for $x_1>0$ and $x_2>0$
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I can solve for this question with Lagrangian method explicitly.
Please share your knowledge with me and inform me. Thanks a lot
For part A)
Since the utility function is continuous and the constraint set is bounded and closed, there exist at least one solution by Weitress theorem.
For part B)
I check for whether the function u(.) is quasiconcave or not. For that I use hessian matrix as follows:
Since $B_1\le 0$ and $B_2>0$, this utility function is quasiconcave. But since the quasiconcavity is not strictly, there is no unique solution.
I cannot show the part C.
I am not sure about the part B and I could not do the part C.
Sorry for hand-writing, but cannot write matrix in latex. Thank you.