## Why It’s Usually Suboptimal for Businesses to Set the Cost Equal to a Fixed Fraction of Revenue

A firm is considering how much to advertise its products to maximize profit, which is the difference between revenue and cost. In this article, we will investigate the profit maximizing conditions and compare it to the convenient method where the advertisement cost is set as a fixed fraction of revenue (fixed ratio method from now on). We can model the revenue (R) as a function of cost (C):

$R(C) = aC^{p}$

Where a is the scale parameter and p is the shape parameter. When 0 < p < 1, the function is concave and diminishing marginal returns is observed. (Other values of p are uninteresting or uncommon: when p = 0, revenue is independent of cost; when p = 1, cost is directly proportional to revenue; when p > 1, marginal returns increase with cost and expansion is surely needed.)

### Analytic Solution

The firm makes positive profit when the function lands above the line R(C) = C. Maximizing profit is equivalent to finding a C* such that the vertical distance between the two curves is greatest. This happens when the derivative of R(C) is equal to that of the R = C line, which is 1:

$\frac{\text{d}R}{\text{d}C }=1$ when $C=C^{*}$
$paC^{*a-1}=1$

Solve for C* and get $C^{*}=(ap)^{\frac{1}{1-p}}$
Graphically this means finding a 45 degree tangent line on the R(C) curve:

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