Optimal Sales & Marketing Expenditures

by Tim Smith, PhD, 8 June 2005

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Is there an optimal level of expenditures in sales and marketing? A simple mathematical model of a business reveals that there is and what effects where it is.

Assumptions
In making any mathematical model, assumptions must be made regarding the business. Some assumptions are made to improve the simplicity of the model and others are made to reveal the underlying challenges that a business must face. These assumptions can be relaxed or changed to fit different business models, but the philosophical result will remain the same. There is an optimal level of spending for any sales and marketing department.

For our mathematical model of a growth oriented business, we will make the following assumptions:

1. The number of new customers increases with increasing expenditures in sales & marketing.
2. At a point, the law of diminishing returns lowers the rate at which new customers are created as expenditures of sales and marketing increase.
3. All new customers purchase one item at the same price.
4. The cost of producing the value offering has a fixed cost component and a variable cost component.
5. The strategic objective of the business is to maximize profits.

Mathematical Model
With the prior assumptions, we can model the financial position of the business with the following equations:

(Eq. 1) Profit = Revenue – (FC + SM + COGS x Volume)
(Eq. 2) Revenue = P x Volume
(Eq. 3) Volume = Number of Customers
(Eq. 4) Number of Customers = A x SM / [ 1 + (A x SM / B) ]

The following variables have been used to capture concepts in a quantitative form:

• FC is the fixed cost of running the business, not including Sales and Marketing expenditures
• SM is the Sales and Marketing expenditures of the business
• COGS is the marginal cost of goods sold, or the cost of producing one more item of the value offering. P is the price. Both are assumed to be constant.
• Alpha (A) and Beta (B) in Equation 4 are parameters that relate the sales and marketing expenditures to the rate at which new customers are created.

In Equation 1, we have distinguished Sales and Marketing expenses from other fixed and variable costs to higlight its effects on overall business profitability. Equation 3 relates volume to the number of customers as a one to one relationship. This assumption can be altered to accomodate other relationships and has been made to simplify the analysis.

Customers and Sales and Marketing Expenditures.

Equation 4 models the number of new customers created for a given level of sales and marketing expenditures. There are multiple means of capturing the relationship between customer acquisition and sales and marketing expenditures, but the equation that is used herein has the value of capturing the relevant phenomena. Specifically, the number of customers increases as sales and marketing expenditures increase, and that the law of dimensioning returns limits the ability to capture new customers.

For small levels of spending, Alpha is the effectiveness of sales and marketing. Specifically, Alpha is the number of new customers per dollar expended on sales and marketing. For instance, if Alpha = 0.01, then approximately 1 new customer will be created for every $100 expended in sales and marketing for low values of expenditures.

Beta defines the total potential market size. For instance, if Beta = 5000 and the company spends infinite amounts on sales and marketing to capture the entire market, the number of customers captured is 5000. (Not that we are suggesting an infinite sales and marketing budget.)

A graph of the number of new customers created for various levels of sales and marketing expenditures for the above parameters is depicted in Figure 1.

Figure 1. Number of customers as a function of sales and marketing expenditures for A = 0.01/$ and B = 5000.

Optimal Spending Level

From calculus, we know that the point of optimal sales and marketing expenditure with respect to maximal profits occurs where the first derivative of profit equation with respect to sales and marketing expenditures is zero. With the profit equation defined in equations 1 - 4, we find that the maximal profit of the business is achieved when the sales and marketing expenditures obeys the following criteria:

(Eq. 5) SM = B/A [ sqrt (A x (P-COGS)) – 1]

Equation 5 reveals that there is an optimal level of spending on sales and marketing for a business that desires to maximize profits. Furthermore, it reveals that the optimal level of expenditures is crucially dependent upon the size of the market, the marginal profit per deliverable, and the effectiveness of sales and marketing expenditures. (sqrt is used to define the square root function.)

As a function of sales and marketing expenditures, we can plot business wide profitability less fixed costs as a function of sales and marketing expenditures and visually see that there is an optimal level of spending for any given business. (It occurs where the business profitability no longer rises with increasing sales and marketing expenditures.)

Figure 2. Profitability less fixed costs as a function of sales and marketing expenditures with A = 0.01/$, B = 5000, and P – COGS = $10,000

Effects of the market potential, marginal profit per deliverable, and sales and marketing effectiveness

First, from Equation 5, we discover that the optimal level of sales and marketing expenditures increases linearly with the size of the market. Larger markets deserve larger sales and marketing expenditures and smaller markets deserve lesser sales and marketing expenditures.

Investment markets have long known this to be true as can be inferred from historical patterns of venture capital investments. Simply put, businesses with strong market opportunities are well positioned to receive capital from the investment markets in order to maximize their profits. The venture capital received that directly applied to sales and marketing should improve profitability for companies with large and untapped market potential. And, businesses with greater market potential deserve greater levels of venture funding. This principal is depicted in Figure 3.

Figure 3. Optimal Sales and Marketing Expenditures as a function of the market size, Beta, with A = 0.01/$ and P - COGS = $10,000.

Second, from Equation 5, we discover that the optimal level of sales and marketing expenditures increases with the square root of the marginal profit per deliverable. The marginal profit per deliverable is Price less the Marginal Cost of Goods Sold, the term P – COGS in Equation 5. This implies that companies with high marginal profit per deliverable should spend more on sales and marketing than companies with low gross margins. Furthermore, the relation between the optimal sales and marginal expenditure and the profit per deliverable is less than linear.

Figure 4. Optimal Sales and Marketing Expenditures as a function of Marginal Profit per Deliverable (P-COGS), with A = 0.01/$ and B = 5000.

Third, from equation 5, we see that marketing effectiveness influences the optimal level of spending on sales and marketing. The relationship is an inverse and more specifically less than linearly inverse. In other words, businesses that are more effective in capturing customers per dollar expended should have a smaller sales and marketing budget than those that are less effective. And, because the relationship is non-linear, a company that is twice as effective should not anticipate spending half as much, but rather somewhere between half as much and the same amount on the same market.

Figure 5. Optimal Sales and Marketing Expenditures as a function of Effectiveness in capturing customers (Alpha), with B= 5000and P - COGS = $10,000.

Conclusion

The mathematical model demonstrates the pertinent concept. For any given business, there is an optimal level of spending on sales and marketing. Spending less or more than that level decreases the overall profitability of the business.

The mathematical model also reveals that the optimal level of spending on sales and marketing depends upon the size of the potential market, the marginal profit per sale, and the effectiveness of the sales and marketing effort. Holding all else constant, executives should spend more on sales and marketing as the market potential increases. Likewise, holding all else constant, executives should spend more on sales and marketing as the marginal profit per sale increases. And, finally, executives should spend less on sales and marketing as its effectiveness improves.

An interesting aspect of the result is that the level of sales and marketing expenditures increases as the marginal profit per sale increases. This implies that if a company increases its price, it should also increase its sales and marketing expenditures. There are numerous consumer behavior reasons to explain this relationship. Still, it is interesting that an economic driven model based upon corporate finances reveals a similar result. Companies with higher prices should spend more on sales and marketing than those with lower prices if they wish to maximize profits.

For an astute strategist, the absolute value of the optimal level of sales and marketing expenditures is less important than the factors that affect where that optimal level should be. Different models and assumptions will produce slightly different results, but the general rules should hold regardless of the mathematical model used, (as long as it is within the realm of reasonability). Hence, while a specific business may have a different revenue or profitability model, and they will have widely varying potential markets sizes, marginal profits per sale, or required expenditures per customer captured, the rules of thumb will still apply.

• Spend more when the market size increases.
• Spend more, but not as much more, when margins increase.
• Spend less as the effectiveness of sales and marketing improves.

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Author
Tim Smith, PhD, Directorial Editor of The Wiglaf Journal and Adjunct Professor of Marketing at DePaul University.

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