Now develop the counter-case for your response to Question 1 (posted below). That is for the situation where the condition(s) you identify does not hold. For example, if you wrote that something is equal to something else, now take the case where that is not true. Illustate through discussion how profit cannot be maximized if this counter-case is true. Thus, you are providing evidence to support your claim to Question 1.

Question 1:

The input decisions for a profit maximizing producer can be solved using marginal relashionships thag are derived from the production functions and input and output prices. Consider the circumstance where you face multiple production options, each driven by a single purchased input used in the production of a single salable output rach with profuction function Yi = fi(X) , e.g., where you could thinkof X as fertilizer or irrigation water, Y1 as corn, Y2 as wheat, Y3 as hay, etc. And you are a price-taker in both the input and output markets.

State the general mathematical relashionship(s) that must hold in your input utilization decision(s) if you are to maximize profit in this situation. Be sure to decompose all elements of the relashionship(s) into their most basic terms relating back to the underlying production function(s) and relevant prices. Be clear and specific in your notation. (Note: do not derive the gondition but be sure to define any symbols/abbreviations used).