Consider optimizing the allocation of a given input across multiple output opportunities when the input is obtained from a fixed endowment (that is given input is unpriced and the amount of input available is a given limited quantity). Consider the case where you have underlying algebraic representations of the respective production functions for each output.

(a) State the requirement for the optimal allocation of a limiting input resource across multiple potential outputs as an algebraic condition. Provide definitions as needed.

(b) Discuss how the algebraic relashionship you provided can be described and motivated entirely through characterization of a set of "oppotunitt costs". Describe the specific "opportunity cost(s) represent by specific elements of your algebraic condition in (a) and describe WHY this condition must hold for profit to be maximized.

(c) Discuss how a properly formulated Linear Programming model of an equivalent production problem could reflect anything relevant to your answers in (a) and (b) above noting any significant parallels and/or differences.