Consider the following utility function:
U = X_{1}X_{2}
Where X_{1} and X_{2} are quantities consumed of two goods. You are considering the actions of a consumer that maximizes utility. He or she has a fixed income m, and faces prices P_{1} and P_{2}. The consumer spends all of its income on the two commodities.
A. Assuming that the consumer maximizes utility, what would be the exact condition showing the ratio of each good purchased by the consumer?
B. Please derive the demand functions for X_{1} and X_{2}., providing an algebraic expression for each and a diagrammatic representation.
C. What is the price elasticity of demand for commodity 1? What is the price elasticity of demand for commodity 2?
For all of these questions, please provide an answer as well as a derivation or explanation for your result.