# 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:

### Fixed Ratio Method

For the fixed ratio method, the optimal cost C’ is found by setting it equal to revenue times a ratio r (C’ = R×r). Graphically, this method finds the intersection between R(C) and the line R(C) = C/r. Using the revenue function gives: $C'=R(C')\cdot r=raC'^{p}$

Solve for C‘ and get $C'=(ar)^{\frac{1}{1-p}}$
Observe that the resulting cost C’ takes a very similar form from the profit maximizing C* and they are equal when p = r. That is, one only has to set the cost proportion equal to the exponent of the revenue function to achieve maximum profit. For example, if revenue is $4\sqrt{C}$, then the cost should be half as much as revenue. Further, when the revenue is nearly linear with respect to cost (i.e. p is close to 1), a large fraction of funds should be devoted to cost and profit margin will be thin. On the other hand, if the revenue seems to be stagnant when having higher cost (i.e. p is close to 0), one should be cautious of overspending on cost.

Given that the fixed ratio method is equally optimal once the best ratio is set, should managers continue to use this method? If the values of p and a are readily available, then C* is easily obtained. Empirically, finding the values of p, a, or r are all difficult. The value of r can’t be directly calculated; p and a are found simultaneously by fitting data to a curved line, not to mention it’s only valid when the revenue function takes the described form. The easier approach is to simply adjust the cost and see where ΔC and ΔR are roughly equal and forget about the fixed ratio method.

### Variations

If part of the cost is manufacture cost directly proportional to the quantity sold, the revenue function would take the form of $R(C) =\frac{aC^{p}}{1-d}$ where d is manufacture cost per item divided by price. The analysis would be identical after treating $\frac {a}{1-d}$ as a modified a.

If the revenue function has a linear component, it takes this form: $R(C) = aC^{p} + dC$ where d is the coefficient for the linear term. The conclusion is identical since C* and C’ become $(\frac {ap}{1-d})^{\frac{1}{1-p}}$ and $(\frac {ar}{1-d})^{\frac{1}{1-p}}$ respectively and only differ by the substitution of p with r.

When there is more than one product in the firm, each product can take on a unique revenue function, as shown below:

The differences in the shape of the revenue functions can be due to different values of a or p in the revenue functions. The same fixed ratio method can be used for all products if and only if p is the same across all products. However, it’s difficult to deduce the values of p and a based on sales data and therefore difficult to distinguish whether the functions differ because of different values of p or different values of a. It’s encouraged to use economic knowledge of the products in determining the revenue function.

### Usage

This article started by mentioning advertisement as the cost because the fixed ratio method is convenient in only products involving specific kinds of costs where the scale of the firm seems to be unbounded. Besides advertisement, research and development cost also fits this analysis. On the other hand, capital costs, for example, such as building a factory, is more conveniently analyzed based on productivity, quantity demanded, and interest rate.

Note that this analysis is not about setting prices for the products. When setting prices, if the production capacity can reach the quantity demanded, then the quantity sold is a function of price and higher price will ultimately result in reduction of revenue. Thus, the R(price) function takes a ∩ shape and so is the profit as a function of price. (If the production capacity, instead of price, is the limiting factor in quantity sold, then simply set the price as high as possible. Of course, the production capacity is rarely the constraint especially in the long run.) In the analysis, we have assumed that price is already set to be optimal for each cost chosen and therefore the revenue function is the highest possible compared to any other prices.