Question

The manager of the XYZ has determined that when monthly revenue is modeled as a function of the amount of money spent on advertising in thousands of dollars per month, R(x) = ?4x^3+36x^2+30000 (for example, the fact that R(5) = 30400 means that if the company spends $5,000 on advertising in a month, the company’s revenue for that month would be $30400). Find the point of diminishing returns to determine how much should be spent on advertising per month.

Answer 1

Given the firm's revenue function R(x) = ?4x^3+36x^2+30000, the revenue-maximizing level of advertisements is given by taking the first order condition and putting it equal to zero.

I.e. Differentiating R(x) with respect to x, we get:

-12x^2 + 72x = 0 which gives x = 0 or x = 6.

Taking the second order condition in order to find maxima/minima, double-differentiating w.r.t. x, we get:

-24 x + 72 = 0. At x = 6, this is negative. Thus, the revenue is maximized.

Diminishing returns to scale would occur when R(x) starts falling with increase in x. This happens at x >6.

Thus, the firm should spend $6000 per month on advertising.