On Gurudev’s Happiness Equation

Recent discussions in the faculty meetings on Happiness triggered a memory from some time ago – 2012 to be precise. I had just started attending my first study group. The text was Gurudev Chinmayananda’s all-time most widely read book, Self-Unfoldment. Happiness is the title of the 2^nd Chapter in this book. Apt to the saying, ‘If all you have is a hammer, then everything you see is a nail’, I wanted to see everything in a mathematical way. Thus, I was enticed by Gurudev’s definition of ‘Happiness Equation’.  I could not help but do some Mathematics with it. I wrote it all up in the form of an article for the CMWRC (Chinmaya Mission Washington Regional Center) newsletter called Smrithi.

The article was an outgrowth of discussions during the study group which met at the house of Dr. Sitaram and Smt. Radha Kowtha. I am grateful to the stimulating discussions of fellow enquirers and Sitaram’s suggestion to write this article. The Upadesha Sara verses and meaning are taken from Swami Tejomayanandaji’s commentary on the same. Insightful remarks by Sri Raju Chidambaram and his suggestion of including graphs helped me improve my own understanding. The article also benefitted from a brief review by my economist friend Sri Praveen Kumar.  Finally, I want to acknowledge Chetana, my partner in ‘crime and punishment’, for her thorough reading and her insistence that what I write should be meaningful.  So, if what is written is meaningful, Chetana should get the credit. If something here does not make any sense, that means I must have won that argument.

I have attached the article in its entirety. It is unimportant to follow the Mathematics if you are willing to accept my claims of certain quantities being either positive or negative. The same facts are also illustrated using the figures and examples involving laddus. In order to understand this article, any one of the three things (mathematics, figures, laddus) is sufficient, but none of them is individually necessary. Well, on second thought, I would say laddus are nearly necessary.

It seemed unfair to exclude the note Dr. Sitaram Kowtha kindly included in his capacity as the Editor of Smrithi at that time. It gives a good summary of the article and also provides a reference to a book which contains many more analyses of this type involving spiritual concepts using mathematical tools, to whet your appetite.  The current version including this brief forward, was reviewed by our Managing Editor Rukma Sadekar in a very short notice. Thank you, Rukma. I also want to thank Lal Ji and Shefali for indulging in my insistence to use Tex commands (via MathJax) for the equations.

[Smrithi] Editor’s note: This article explores the connections between the Happiness Equation given by Gurudev with the law of diminishing marginal utility. Using some basic mathematics, the Happiness Equation can explain the apparent paradox of humans’ relentless pursuit of desires while being aware of the futility of such a pursuit.  Readers of Smrithi, who are of this bend of mind, are invited to contribute articles involving mathematical analysis of vedantic concepts. Law of Love: The Mathematics of Spirituality (www.ceasis.com), by Dr. Raju Chidambaram is an excellent point of reference on this topic.

****Start of the Article*****

In his book Self-Unfoldment, Gurudev gives the following Happiness Equation:

$$ {Happiness} = \frac{Number \, of \, Desires \, Fulfilled}{Number \, of \, Desires \, Entertained} $$

This article refers to the value of the fraction in the Happiness Equation, as the Happiness Quotient (HQ). The HQ can be increased by either increasing the numerator (number of desires fulfilled) or by decreasing the denominator (number of desires entertained). In Self-Unfoldment, Gurudev explains that desires create agitations. One can attain greater happiness by either fulfilling desires and quietening the agitations, or by simply not entertaining desires. Of course, Gurudev is quick to warn that fulfilling desires often leads to more desires. For example, eating one laddu will result in a yearning for more laddus, eventually resulting in a pressing desire for medical relief from indigestion. 

Gurudev has embedded significant life-lessons in this innocent looking HQ. These lessons are revealed by taking a closer look at the Happiness Equation. Let us assume that a sadhaka entertains one desire at a time in a sequence 1, 2, 3… so that at any point in time, a total of n consecutive desires have been entertained and a total of $k( \leq n)$ desires have been fulfilled.  The value of HQ at any point in time is given by

$$ HQ(k,n) = \frac{k}{n}. $$

Note that HQ is the proportion of fulfilled desires. If all the desires ever entertained are fulfilled, HQ is 100%, and in the unfortunate situation of none of the desires being fulfilled, the value of HQ is 0%.  Let us now consider the change in the value of HQ as additional desires are entertained. At this point, if the sadhaka were to entertain an additional desire, and by God’s grace it is fulfilled, HQ will become

$$ HQ(k+1,n+1) = \frac{k+1}{n+1}. $$

Thus, the marginal change in HQ is given by

$$\eqalign{ \Delta HQ(k,n) & = HQ(k+1,n+1)-HQ(k,n) \cr & = \frac{k+1}{n+1} – \frac{k}{n} \cr & = \frac{n-k}{n(n+1)} } $$

Note that the marginal change in HQ given above is always non-negative, i.e., fulfilling an additional desire always gives us some happiness.  An immediate question then is, ‘If our goal is to maximise HQ, why should we not entertain as many desires as possible?’  Of course, if the additional desire entertained is not satisfied then the marginal change in HQ is negative. This can be seen as follows:

$$\eqalign{ \Delta HQ(k,n) & = HQ(k,n+1)-HQ(k,n) \cr & = \frac{k}{n+1} – \frac{k}{n} \cr & = -\frac{k}{n(n+1)} } $$

The preceding two equations make perfect sense in practice. We try to fulfil an additional desire. If we are successful we are happier than before. If we are unsuccessful, we are less happy. Of course, the sadhaka only controls whether a desire is entertained or not, and how hard the desire is pursued. It is not entirely up to us whether or not the desire is fulfilled, as eloquently declared by Bhagavan Sri Ramana:

कर्त्रुराज्ञ्यया  प्राप्यते फलं |
कर्म किं परम् कर्म तज्जडं ||

By the command of the Creator, the fruits of the action are gained. Then how is action the All-Supreme? It is not, since action is inert.

Even under the optimistic scenario of a fulfilled additional desire, it then follows that the marginal increase in HQ decreases as you entertain more and more desires. That is,

$$\eqalign{ \Delta HQ(k,n) & = \frac{n-k}{n(n+1)} &= \frac{1}{n+1} \left(1-\frac{k}{n}\right) \leq \frac{1}{n+1} } $$

From the inequality given above, we can conclude that regardless of the proportion of desires fulfilled (that is, regardless of the value of the fraction k/n), the marginal change in HQ is bounded above by 1/(n+1). The marginal change decreases to zero as the number of desires entertained increases without bound. Hence, the well-known law of diminishing utility applies to each additional desire fulfilled. Almost all of us relate to this experience. The 11^th laddu never tastes as good as the 10^th, and none of the later laddus can match the taste of the 1^st laddu. In fact, this is true regardless of the proportion of desires being fulfilled.  Figure 1 below illustrates the diminishing marginal utility obtained from continuing to fulfil additional desires. The increasing curve is the plot of the Happiness Quotient, whereas the decreasing curve (in dashed line) is the marginal increase gained from the additional fulfilled desire. [For illustrative purposes, k is taken to be an increasing function in such a way that k/n approaches unity as n tends to infinity.] 

In fact, further examination of the inequality reveals that larger the proportion k/n, smaller the marginal increase in HQ! In other words, the additional satisfaction obtained from the 2^nd laddu, relative to the 1st laddu, is greater than the additional satisfaction one would obtain from the 11^th laddu relative to the 10^th!  While an additional fulfilled desire has taken the HQ closer to unity, we are left with an empty feeling and a craving for more due to the diminishing impact of this fulfillment. In fact, this ‘diminishing impact’ of the additional fulfilled desires is the cause of unhappiness. The 11^th laddu, which came from the same batch as the 10^th laddu and every bit as good, simply cannot not stand up to the performance of the 10^th laddu relative to its predecessor. The 11^th laddu fails to meet the expectation created by the performance of the 10^th laddu. This disappointment is the cause of unhappiness and unfortunately drives humans to entertain more and more desires. Figure 2 illustrates that someone with 90% of all their desires fulfilled seems to gain less by fulfilling an additional desire, than someone who has only 80% or 70% of all their desires fulfilled. With each new fulfilled desire, the HQ itself increases. However, in practical terms each new fulfilled desire seems to bring more unhappiness, and the impact of this unhappiness is greater on someone who has more fulfilled desires!

As Gurudev asserts, each fulfilled desire invariably leads to more desires.  Thus, his Happiness Equation reveals the secret of this apparent paradox surrounding humankind’s continuing quest for happiness, driven by the disappointment resulting from the diminishing marginal returns obtained from each additional fulfilled desire! This observation rings true in the second verse of Bhagawan Sri Ramana’s Upadesa Sara:

क्रुतिमहोदधौ  पतनकारणं  |
फलमशाश्वतम्  गतिनिरोधकं ||

In the vast ocean of actions, impermanent result is the cause for fall (of man), and is a barrier to progress.

This insight into the workings of HQ, and its behaviour as the number of desires increase without bounds, proves that entertaining more and more desires and fulfilling as many of them as possible, is NOT the appropriate way to maximise happiness. Instead, one should focus on minimising the denominator in order to maximise HQ. In the extreme case, if there are no desires to be entertained at all, the denominator is zero and thus HQ is infinity. For a true seeker (mumukshu), there is only one desire to be entertained, namely, the desire to attain self-realisation.  Thus, HQ for a mumukshu will remain at the value of zero until self-realisation; will become unity at the time of self-realisation and reset to infinity from that point onwards because there are no more desires to be entertained!

REFERENCES

Self-Unfoldment by Swami Chinmayananda. Chinmaya Mission West Publications. 

The Essential Teachings: Upadesha Sara by Bhagavan Ramana Maharshi. Commentary by Swami Tejomayananda. Chinmaya Prakashan.

Law of Love: The Mathematics of Spirituality (www.ceasis.com), by Dr. Raju Chidambaram.