Additive, Multiplicative and Exponential Systems

Whenever we think of the relation between efforts and results, we subconsciously expect a more or less direct relationship between efforts expended and results obtained. If I study for 3 hours everyday instead of 2, I would score, say, 80% marks instead of 60%. If I work a 60-hour workweek instead of a 40-hour one, I would generate an output which is approximately 40-50% higher. If I spend a few hours finding the best cake in town, I’d have the most memorable birthday celebration for my partner.

Most of us know deep down that the relationship between efforts and results is far more complicated. Broadly, there are 3 generic systems through which efforts translate into results. The systems that we operate in determine the relationship between efforts (inputs) and results (output).

I attempt to describe these systems through mathematical equations in the following sections. (For our purposes, variables such as x, y and z stand for inputs and R for results. Coefficients such as a, b and c stand for constants, that is, factors which cannot be directly or immediately influenced but play a role in determining the magnitude of output generated for a given value of input.)

Additive Systems

Additive Systems follow the generic rule:

This means there is a linear relation between efforts and results. More the effort, better is the result. Of course, the magnitude of the jump in output per incremental unit of an input is determined by the constant preceding it. When the constant is high, it would be a “high leverage” input, that is, it’d generate a higher result per unit of effort. Thus the trick to achieving superior results in an additive system is to expend efforts on factors that have a high leverage.

For example, drawing reference from my earlier post on improving team performance and output, managers who focus on high-leverage activities are able to deliver better results than managers that squander their time on low-leverage activities.

Multiplicative Systems

Multiplicative Systems follow the generic rule:

In multiplicative systems, increasing efforts in one area may not necessarily lead to improved output. For example, if x = 5, y= 0.01, z = 1000 and c = 1, the output is 50 units. Increasing z by 1 unit will take the output to 50.05. On the other hand, increasing y by 1 unit will take the output to 5,050! Thus, it makes far more sense to focus on y rather than z. An extreme case of a multiplicative system is multiplying by zero, in which case nothing you do on any of the non-zero variables has any impact on the end result.

This is often seen in production processes where releasing a bottleneck will lead to a quantum jump in overall throughput. On the other hand, spending efforts (or money) on process steps which already have surplus capacity will lead to no difference in the end result. Similarly, in life, we need to balance between work, personal life, relationships and other areas important to us if we are to be satisfied with our overall quality of life. Doubling efforts in one area while neglecting other important areas of life will substantially reduce the quality of life. Thus, in multiplicative systems, the best way to improve overall output is to balance efforts across multiple variables.

Exponential Systems

Exponential systems follow the generic rule:

Here, the critical thing to notice is that the variable x (the factor within our immediate or direct control) is in the index. In this case, if the base constant “a” is low, we are condemned to mediocre results, no matter how much we raise x. On the other hand, if a is high, a small change in x can have an exponential change in the outcome.

An example of exponential systems would be investing in markets. Even if the return is low (say, 6%), if we stay invested for a long period (say, x = 30), we will get a 5.7x output. On the other hand, if we have a 50% higher return (that is, 9%) but invest for 50% lesser duration (x = 15), then the output is 3.6x. The higher rate of return is the situation we find ourselves in. How patient we are determines the x. Thus the implication is to either move to areas that substantially increase the base or stick for longer duration to what you have. The latter will eventually trump the former unless we are able to combine the base advantage with the patience and discipline to raise x (remember the race between the hare and the tortoise).

In reality, the specific systems or circumstances we find ourselves in could be a combination of the generic systems I described above. I have used the mathematical framework merely to explain the conceptual differences among the various systems. Inputs and outputs may not be meaningfully quantifiable, that is, even if we find metrics to make them objective, those metrics may miss the essence of what we are trying to measure.

For example, if you want to create the perfect birthday for your partners, spending hours researching the best possible cake may be a low-leverage activity compared to compiling memories that are meaningful to your partner if your partner places more emphasis on experiences than on material things. On the other hand, if your partner is a food connoisseur, it might indeed make sense to spend a couple of extra hours finding the best cake in town. In this case, you are probably operating in an additive system.

If you are working to improve your overall satisfaction with life, it might be a multiplicative system between your personal development, career growth, financial situation and quality of relationships. If your result in any one of these areas is substantially low, it makes sense to address that area rather than doubling your efforts in other areas. For example, if your quality of relationships is poor, earning more money is not going to substantially increase your life satisfaction.

Thus, the underlying message is that whenever we find that increasing efforts in certain areas does not produce the desired result, it is worthwhile to take a step back and assess the kind of system we are operating in.