-9scR's Proposed Solutions (Work-in-Progress)

I'm currently testing the "TeX the World" extension to see whether it has the packages I need to properly format my solutions.

Problem 1

Since the problem specifies that the function to be optimized is a utility function as opposed to (e.g.) a production function, I'm going to take a different approach. I first take a monotone transformation:

[;\alpha\log x_{1} + \beta\log x_{2} \Longrightarrow \dfrac{\alpha}{\alpha+\beta}\log x_{1} + \dfrac{\beta}{\alpha+\beta}\log x_{2};]

This is a Cobb-Douglas utility function with scalar coefficients summing to one. The following is a useful fact: for such a function, the maximization solution under parameters [;(p{1},p{2},m);] is given by

[; x_{1}(p_{1},p_{2},m)=\dfrac{\alpha}{\alpha+\beta}\Bigg(\dfrac{m}{p_{1}}\Bigg);]

[;x_{1}(p_{1},p_{2},m)=\dfrac{\beta}{\alpha+\beta}\Bigg(\dfrac{m}{p_{2}}\Bigg);]

This fact can be obtained by the method of Lagrange multipliers suggested by the problem text. This is also the solution to the initial problem because (1) every consumption bundle is totally ordered by the original utility function, (2) the transformed utility function has an identical total order albeit with different utility values, and (3) consumers are modeled as choosing between consumption bundles only based on that total order.

Using this result, we can easily determine the income shares of goods 1 and 2, respectively:

[;\Bigg(\dfrac{p_{1}}{m}\Bigg)x_{1}=\Bigg(\dfrac{p_{1}}{m}\Bigg)\dfrac{\alpha}{\alpha+\beta}\Bigg(\dfrac{m}{p_{1}}\Bigg)=\dfrac{\alpha}{\alpha+\beta};]

[;\Bigg(\dfrac{p_{2}}{m}\Bigg)x_{2}=\Bigg(\dfrac{p_{2}}{m}\Bigg)\dfrac{\beta}{\alpha+\beta}\Bigg(\dfrac{m}{p_{2}}\Bigg)=\dfrac{\beta}{\alpha+\beta};]