Economics with calculus: optimization problem?

manpreet Best Answer 2 years ago

The problem: I am currently working through a course on edX, Principles of Economics with Calculus. The course is structured such that the only prerequisite needed is univariate calculus; accordingly, I have completed the equivalent of Calc I and II, through Taylor series. I've come across the following economic optimization problem, and I've had quite a bit of difficulty reasoning through it. It is worded as follows:

"Imagine that you own a small business and that you have to decide how much of your product to sell every year, and which technology to use to produce it.

Your market research suggests that the price at which you will be able to sell each unit of your product is given by

where q denotes the total amount sold over the course of the year.

You have access to two different technologies, which you can use in any feasible combination. The technologies have different cost properties. Producing q units with the first technology costs

while producing q units with the second technology costs

How many units should be produced using the first technology at the profit-maximizing decision?"

My attempt at a solution and concerns: Initially, I made quite a gaffe and did not pay careful attention to the wording of the problem; at the outset, I interpreted it to be asking what the optimal production level would be if we were only using the first technology. I simply wrote the profit function as

expanded, differentiated, located the critical points, and verified using the Second Derivative Test. With that process, I arrived at q = 50. After having submitted that answer - which was marked as incorrect - I realized my first error: this question was asking how many particular units should be produced using the first technology if both are used in some configuration.

Knowing the real question being asked, I re-attempted the solution, but my concern is this: how do I - if this is at all possible in the first place - write a revenue function that accounts for the fact that a combination of these two technologies will be used? If q is our total amount being sold, then there will be a particular amount produced using technology 1 - let's call this x - and another amount produced using technology 2, which would be q−xq−x. Assigning these variables, I went through the analytic process of optimization once again, but the roadblock I continue to hit is not knowing how to actually differentiate with this involvement of both q and x; should I just assume that q is fixed, and treat it as a constant when differentiating?

Any assistance would be greatly appreciated. Thank you in advance.