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Groupon’s Business Model Profits-Revenues Data Analysis Using the Work Function

Summary

Groupon is a relatively new web-based company, founded in October 2008. It

became a public company in November 2011. However, the company has reported

a profit for only 4 out of 14 quarters for which data is available. It reported a small

profit for 1Q2010. Since then, although revenues have increased steadily, the

company has consistently reported a loss. Nonetheless, things have been actually

improving, with the losses decreasing with increasing revenues. Groupon has

finally reported a small profit again for 2Q2012.

The profits and revenues data is shown to follow a simple linear law y = hx + c =

h(x – x0) where x is revenues and y is profits. This is a consequence of the classical

breakeven analysis for the profitability of a company. Depending on the numerical

values of h and c (which can be either positive or negative), we have at least three

possibilities: Type I behavior (h > 0, c < 0) increasing profits and profit margins

with increasing revenues, Type II (h > 0, c > 0), increasing profits but decreasing

profit margins with increasing revenues, and Type III (h < 0, c > 0), decreasing

profits and profit margins with increasing revenues.

Early in 2009, Groupon was operating in the Type I mode. Revenues were

increasing but the company reported a loss in 1Q2009 followed by profits in

2Q2009 and 3Q2009. Then profits suddenly took a “plunge”, although revenues

continued to increase. Groupon reported a loss in 4Q2009 and then a profit in

1Q2010. This “fluctuation” in the profits can be explained on the basis of the

“fluctuations” in the nonzero intercept c, or the cut-off revenue x0 = -c/h, and

actually represents a “fluctuation” in the costs. Thus began a prolonged period of

Type III behavior with losses increasing (profits decreasing) with increasing

revenues. Following the IPO, Groupon has slowly made a transition back to Type I

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mode, which began in the quarter ending Dec 2011 and has now reported a profit

again for 2Q2012. A brief description of this transition was provided earlier (click

here). The purpose here is to discuss the significance of the nonzero intercept c,

which can be shown to be exactly analogous to the work function W conceived by

Einstein, in 1905, when he enunciated his famous photoelectric law, K = E – W =

hf – W = h(f – f0). The analysis of the quarterly data, using the “work

function” model also shows that Groupon could potentially convert 80% of

the revenues (in excess of cut-off value x0) into profits.

The analysis of the limited amount of annual profits-revenues data (2009-2011),

using these ideas, reveals that the “breakeven” revenue is about $5 billion,

annually, based on the current cost structure. The cumulative revenues for the six

month ended June 30, 2012 was $1.13 billion. Hence, it is unlikely that a profit

will be reported for the full year ending Dec 31, 2012, or even in 2013.

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1. Introduction

The main purpose here is to analyze the profits and revenues data for Groupon

Inc., a US-based web company which was started in October 2008. It became a

public company in November 2011 with its IPO being the biggest since Google’s

IPO in 2004. Groupon has just filed its second quarter financial results on August

14, 2012 (for quarter ending 6/30/12) and has reported a profit of $28.39 million

with revenues of $568.34 million, for a profit margin of 4.99%.

Based on the available profits-revenues data, the company last reported a profit for

the quarter ending March 31, 2010. The profits were $8.551 million with much

smaller revenues of $20.272 million. The profit margin was a whopping 42.2%.

However, as seen from Table 1, the company was not able to sustain this high rate

of conversion of revenues into profits in the subsequent quarters. Although the

revenues increased steadily, losses were reported, consistently, starting with 2Q10.

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Table 1: Profits and Revenues Data for Groupon

Quarter

ending

Revenues,

x

$, millions

Profits, y

$,

millions

Profit

margin,

y/x %

Change,

∆x

Change,

∆y

Slope, h

= ∆y/∆x

31-Mar-10 20.27 8.551 42.2

30-Jun-10 38.67 -35.929

30-Sep-10 81.78 -49.032

31-Dec-10 172.22 -313.23

31-Mar-11 295.52 -113.89

30-Jun-11 392.58 -101.24

30-Sep-11 430.16 -10.57

31-Dec-11 492.16 -64.95

31-Mar-12 559.28 -3.59 264 110 0.418

30-Jun-12 568.34 32.33 5.7 Additional entries to this table (with slight differences) may be found in Appendix I.

The change in revenues ∆x and profits ∆y between March 2011 and 2012, yield the

slope h = ∆y/∆x = 0.418. The above figures were obtained from the company’s

SEC filings and differ from those reported here. Some losses/profits here are those

values attributed to Groupon (in the financial statements) and those attributed to

common shareholders. Data sources: http://files.shareholder.com/downloads/AMDA-

E2NTR/2021598481x0xS1445305-12-1731/1490281/filing.pdf and

http://files.shareholder.com/downloads/AMDA-E2NTR/2021598481x0xS1490281-12-

15/1490281/filing.pdf and http://files.shareholder.com/downloads/AMDA-

E2NTR/2021598481x0xS1445305-12-922/1490281/filing.pdf

******************************************************************

A turn around, which has resulted in the first profitable quarter since March 2010,

began with the quarter ending Dec 31, 2011. Losses have been decreasing since

then, which is equivalent to an increase in profits. Thus, it appears that Groupon

may have achieved its “breakeven” revenues, see Figure 1, and is now ready to

report profits consistently.

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Figure 1: The quarterly profits and revenues data since March 2011 indicate a

general decline in the losses with increasing revenues. The only exception to the

general upward trend is the data for quarter ending Sep 30, 2011. Has Groupon

revenues exceeded the “breakeven” revenue needed for sustained profitability?

See also an alternative view of this same data provided later in Figure 7.

2. Linear Profits-Revenues Law and Breakeven Analysis

As noted in the SEC filings, Groupon, Inc. is a local commerce marketplace that

connects merchants to consumers by offering goods and services at a discount.

Traditionally, small businesses try to reach consumers and generate sales by a

number of well-tested methods: listings in the yellow pages, direct mail and

newspaper ad campaigns, radio, TV and online advertisements, special in-store

promotions and occasionally even using someone dancing at the street corner in a

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gorilla suit. Groupon offers a new way for local merchants to attract customers and

sell their goods and services.

Essentially, Groupons sells “coupons” online, offering say a 50% discount, to

customers who will then redeem them at local stores which have entered into a

contract with Groupon. The Groupon business model has also been the subject of

some scathing criticism (click to see the recent article by Farhad Manjoo in Slate

magazine) as being more like a loan sharking business.

Groupon was started in October 2008 and has experienced a significant growth in

revenues (see also the revised Table 1 in Appendix I). Also, according to its SEC

filings, the number of active customers, defined as individuals who have purchased

Groupons during the past twelve months, has increased from 8.9 million as of

December 31, 2010 to 33.7 million as of December 31, 2011. Groupon launched its

IPO and became a public company in November 2011.

As noted already, the purpose here is mainly to analyze the profits-revenues data

for Groupon using a new methodology which has been described in several articles

available on this website (written since the Facebook IPO on May 18, 2012). A

bibliography list is provided at the end of this article.

Very briefly, it has not been appreciated that all companies follow a simple and

universal linear law, y = hx + c, which relates revenues x and profits y. The

constants h and c in this law can be deduced using the available financial data,

such as that compiled in Table 1.

Indeed, this linear law can be shown to be the consequence of the classical

“breakeven” model for the profitability of a company. Consider a company making

and selling N units of some product (e.g., internet coupons offering a discount, in

the case of Groupon). The total cost C associated with the operation is the sum of

the fixed costs “a” and the total variable costs “bN” where “b” is the unit variable

cost. Thus C = a + bN. If p is the unit price, the total revenues generated R = pN, or

equivalently, units N = p/R.

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Hence, the profits P = Revenues – Costs = R – C = pN – a – bN = (p – b)N – a.

Eliminating N using we get, P = [(p – b)/p] R – a which is a linear relation between

profits P and revenues R and can be rewritten as y = hx + c, where

Slope h = (p – b)/p = 1 – (b/p)

- Depends on unit variable cost b and unit price p

Intercept c = - a

- Depends on the fixed cost a

Although deduced here by considering a single product, the linear law can be

shown to hold for a number of companies, large and small. Some examples are

Microsoft, Apple, Google, IBM, Southwest Airlines, Exxon Mobil. Depending on

the numerical values of h and c (which can be either positive or negative), we have

at least three possibilities.

1. Type I company (h > 0, c < 0): Profits and profit margins increase with

increasing revenues. (The reverse is also observed, i.e., profits decreasing

with decreasing revenues and is called INVERSE Type I behavior.)

2. Type II company (h > 0, c > 0): Profits increase with increasing revenues,

but profit margins decrease. (Companies are usually observed to make a

transition from Type I behavior to Type II behavior with increasing

revenues, as seen, for example, with Microsoft).

3. Type III company (h < 0, c > 0): Both profits and profit margins decrease

with increasing revenues. (Sometimes, even the opposite is observed, i.e.,

profit and profit margins INCREASE with DECREASING revenues).

Each of the linear laws described here applies over a limited range of revenues

(and profits). Transitions from one type of linear behavior to another (Type I to

Type II, or Type I to Type III, or Type I to Type II to Type III) are observed and

the general profits-revenues law is a smooth curve with a maximum point. We thus

observe small linear segments of a more general non-linear law, which can be

written as follows. Further details about the implications of this general nonlinear

profits-revenues law may be found in the articles cited.

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y = mxn

[e-ax

/(1 + be-ax

) ] + c …………(1)

and, dy/dx = [ (y – c)/x ] [ n – ax - axg(g - 1) ] …………(2)

where g = 1/(1 + be-ax

) note g = 1 for b = 0 …………(3)

As discussed elsewhere, equation 1 can be derived by generalizing the statistical

arguments invoked by Max Planck to develop quantum physics, in December

1900. Instead of the total energy UN associated with N particles (called oscillators

by Planck), we now consider the total revenues (and profits) associated with N

different products each having its own triplet (a, b, p) which defines the costs and

revenues. Thus, energy in physics is equivalent to money in economics (or the

business and financial world). The power-exponential law, given as equation 1,

thus follows if we systematically reinterpret and extend the meaning of each of the

mathematical symbols in Planck’s famous paper.

The derivative dy/dx of this general function (the slope of the tangent to the curve)

relating profits and revenues is given by equation 2 where the function g, defined

in equation 3, is introduced for convenience to perform the operation of

differentiation. (The rule for differentiation of a product is all that is required.)

The linear law is deduced as a special case for n = 1, a = 0, and b = 0.

The power law, y = mxn + c, is deduced as a special case for a = 0 and b = 0.

Profits increase with increasing revenues at an accelerating rate (n > 1) or a

decelerating rate (n < 1). For n = 1, we get the linear law.

The power-exponential law, y = mxne

-ax + c, is deduced as a special case for

b = 0. The x-y graph now reveals a maximum point at x = n/a.

The case of a, b ≠ 0 is the most general and again reveals a maximum point.

The profits-revenues data for several real world companies can be shown to reveal

such a maximum point. Some examples of companies operating past their

maximum point are Ford Motor Company, General Motors (which went into

bankruptcy in June 2009), Air Tran (which entered into a merger with Southwest

Airlines), Southwest Airlines (even prior to Air Tran merger), Yahoo, Verizon

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Communications, and Kroger. The implications of the appearance of the

maximum point and prolonged operation in Type III mode (profits decreasing with

increasing revenues) have been discussed in detail and needed not be repeated

here.

The various transitions outlined (including Type III behavior and operation past the

maximum point), very simply put, are due to increasing costs as the company

grows and its revenues increase. As we will see shortly, Groupon has been

operating in the Type III mode and seems to be making a transition to Type I

mode. This transition also reflects the increasing costs of operation as revenues

have increased.

3. Analysis of Profits-Revenues data: Breakeven revenue

Although a nonlinear trend is evident in the profits-revenues data in Figure 1,

caution must be exercised in applying nonlinear models, such as the power-law

model, y = mxn + c, to describe this data since extremely “bullish” or “bearish”

predictions will be obtained, which are eventually proven to be inconsistent with

long term observations (see, for example, the discussion of the data for Google and

Facebook, click links here; see also links given in the bibliography list).

True nonlinearity, such as power-law behavior, is often merely due to a transition

from one linear mode to another with increasing values of x. Essentially, the slope

h and the intercept c change as x increases and the system is operating along two

linear segments for different ranges of x and y values. A smooth curve can be used

instead of two line segments. Engel’s law relating income x and food expenditure y

is a good example of such a power law (n < 1). Another example is the frictional

drag, or aerodynamic resistance, experienced by a moving object (such as a

spaceship, an aircraft, an automobile, or even an Olympic runner) as its speed (or,

more strictly, its velocity v) increases. Such nonlinearity should, however, be

carefully distinguished from what we see here in the financial data.

The Ockham’s razor principle is especially applicable in the analysis of financial

data where one is often interested in the short term predictions (see Jeffreys, W. H.,

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and Berger, J. O. (1992), “Ockham's Razor and Bayesian Analysis,” American

Scientist, 80, 64-72 Erratum, p. 116). This principle states that, of all possible

models, the simplest model is the one that is to be preferred. “Keep it simple”, said

Einstein, “but not too simple”. (The latter quote can also be used in a detrimental

way to promote complicated theories. It is NOT a polemic against simplicity.

Rather, what Einstein is emphasizing is that we understand the “essence” of the

problem that we are dealing with.)

Thus, after careful consideration, it is suggested that the nonlinearity embodied in

the power-law, or the power-exponential law (equation 1) is NOT a good model to

analyze financial data, where one is essentially looking at the short term

projections. What we often witness in the financial world are “line segments” of

the more general nonlinear curve represented by equation 1. What we observe is

really what amounts to “small fluctuations” (with occasionally large fluctuations)

in profits along a Type I, Type II, or Type III line, with increasing revenues. Good

companies, such as Microsoft, which has certainly set the standard for excellence

in the business world (at least as far as financial performance is concerned), seem

to operate along a set of parallels in the profits-revenues diagram with profits and

revenues jumping back and forth (“fluctuating”) from year to year, or quarter to

quarter, between these parallels. The reader is encouraged to review, in particular,

the analysis of Microsoft profits-revenues data.

Kia Motor Company also reveals an exactly similar behavior, with the profits and

revenues data following two parallel lines.

The variation in the numerical value of the nonzero intercept c is thus the primary

reason for this movement between parallels on the profits-revenues diagram. Thus,

we must understand the significance of the nonzero intercept c. The breakeven

model teaches us that this is related to the company’s “fixed” costs “a”.

The quarterly data for Groupon plotted in Figure 2 reveals significant scatter.

Nonetheless, a careful study of the data shows an unmistakable confirmation of the

linear law. Consider, for example, the two recent quarters ending March 2012

when Groupon reported a small loss ($3.59 million) and June 2012 where a profit

($32.33 million) was reported for the first time since March 2010. Revenues

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increased between these two quarters and one would therefore expect an increase

in the profits (or reduced losses). Since Groupon seems to have been close to the

“breakeven” revenue, a profit was indeed reported for June 2012.

The “breakeven” revenue represents the minimum revenue x = x0 = - c/h which

must be exceeded to report a profit. This can be deduced from the linear law, y =

hx + c = h(x – x0). The constant h and c can be fixed by considering any two (x, y)

pairs in the data set. The equation of the straight line joining the March 2011 and

March 2012 data points is y = 0.418x – 237.47 = 0.418(x – 567.9). This is a Type I

equation with slope h = 0.418 > 0 and intercept c = - 237.47 < 0. Thus, based on its

current operations, we estimate the cut-off revenue x0 = $567.9 million. Groupon is

therefore expected to report a profit is revenues exceed this minimum value.

Figure 2: Quarterly profits-revenues data for Groupon Inc., confirming the linear

law and the classical breakeven model for profitability of a company.

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y = 0.42x – 237.5 = 0.42 (x – 568)

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The revenues for the quarter ending June 30, 2012 were $568.34 million

confirming both the linear law and the breakeven model. The profits for 2Q2012

were actually higher than predicted by the Type I equation. This can be interpreted

as a slight “fluctuation” in the value of the nonzero intercept c from the value

deduced using just two data points. Notice also the general upward linear trend in

the data if we overlook the “outliers”, close to the origin (small revenues and

profits/losses) and the (x, y) pair for Dec 2010. Indeed, the following equations can

be deduced if we consider the four data points, at smaller revenues, that lie just

above and below the Type I line indicated in Figure 2.

y = 0.36x – 244.3 = 0.36(x – 670.4) June 2011 and Dec 2011

y = 0.31x – 144.1 = 0.31(x – 464.2) Sep 2011 and June 2012

Best-fit line, y = 0.47x – 259 = 0.47(x - 549) with r2 = 0.714

The slopes h = 0.36 and 0.31 deduced here roughly equal suggesting a movement

along parallels. The slope h = 0.42 deduced earlier for the March 2011 and March

2012 is also roughly comparable with an “average” slope of 0.364 for these three

sets of (x, y) pairs. Furthermore, a linear regression analysis (excluding the

outliers, standard practice in any such statistical analysis) yields the “best-fit”

equation with a higher slope h = 0.47, with a very high positive regression

coefficient r2 = 0.71. (Recall that r

2 = + 1.00 for a PERFECT positive correlation

and when all points lie on a perfect straight line.)

The best-fit line has a higher slope because it must pass through the point (xm, ym)

= (456.34, -43.65). These are the “mean” values of x and y for the six data points

mentioned above. The best-fit line pivots to a higher slope about this mean point to

minimize the sum of the squares of the deviations (y – yb). Here yb is the predicted

value for profits on the best-fit line. The mathematical formula for the slope h in

the linear regression analysis is derived by minimizing the sum ∑ (y – yb)2. The

alternative viewpoint is that of finding an “operating” line such as the line joining

the March 2011 and March 2012 data and examining the data carefully for

movement along a set of parallels with varying values of the nonzero intercept c.

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Figure 3: The profits-revenues data for Groupon, for the most recent quarter, can

be explained by invoking a simple linear law, y = hx + c. The constants h and c

were determined using classical linear regression analysis. Using the analogy

energy in physics = money in economics, the nonzero intercept c is seen to be

exactly analogous to the work function W in Einstein’s photoelectric law K = E –

W = hf – W = h(f – f0). The “outliers” here thus represent “fluctuations” in the

work function, or the “costs” associated with converting revenues into profits.

4. The nonzero intercept c and the work function W

Groupon is an interesting company for two reasons. First, it is an infant company

with only three full years of operation. Second, unlike Microsoft, which reported

its first ever quarterly loss in 26 years, for 4Q2012 (for the full fiscal year

Microsoft did report a profit), Groupon has been reporting quarterly losses rather

regularly since its inception in October 2008.

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The quarterly data for 2009 has been included in the revised Table 1 which may be

found in Appendix I. Prior to the quarter ending June 30, 2012, Groupon has

reported a profit for three other quarters.

Let us consider the evolution of revenues and profits in 2009. Groupon reported a

loss in 1Q2009. In 2Q2009, revenues increased and Groupon reported a profit. The

profit margin y/x was rather small, only 0.6% (see revised Table 1). Notice that ∆x

= $3.049 million and ∆y = $0.33 million yielding a positive slope h = ∆y/∆x =

0.108. In the following quarter, 3Q2009, revenues increased again (∆x = $6.697

million) and Groupon reported a profit once again (∆y = $0.829 million). The slope

h = ∆y/∆x = 0.124. The profit margin y/x, however, increased dramatically to

8.5%. If we consider the overall change between 1Q2009 and 3Q2009, ∆x =

$9.746 million and ∆y = $1.159 million and the slope h = 0.119.

In fact, the three slopes are seen to be roughly equal to each other. On a x-y graph,

see Figure 4, the three data points can be seen to lie on or very close to the straight

line, with the equation y = 0.12x – 0.34 = 0.12 (x – 2.85). This equation is deduced

by considering the two extreme points and is a Type I line (h > 0, c < 0). This

means there is a cut-off revenue x0 = $2.85 million, above which profits will

appear. This is the situation for 2Q2009. Profits and revenues continued to increase

along the same Type I line through 3Q2009. Also, it is now easy to see why the

profit margin y/x increased from 1Q2009 to 3Q2009. The linear law means the

ratio y/x = 0.12 – (0.34/x) will increase as x increases along the Type I line.

A straight line can always be drawn between any two points in (x, y) space.

Here, however, we find three points lining up nicely on a straight line. The

fact that the 3Q2009 data lies close to the extension of the line joining the

1Q2009 and 2Q2009 is indeed remarkable. (Alternatively, 2Q2009 data point

lies close to the line joining 1Q2009 and 3Q2009, which permits the slope h to

be fixed by considering a larger range of x and y values.)

Also, why do the three data points fall on a nice straight line instead of being

scattered? Why is it a Type I line, rather than a Type II or Type III line?

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This has to do with the organizational structure of the company being studied. This

“structure” dictates how costs increase with increasing revenues. The simple

“breakeven” model teaches us that profits will appear only if revenues exceed a

minimum amount, the fixed cost “a”. As revenues increase further, not all of the

revenues will be converted into profits because of the variable cost. Only a portion

(pN – bN) will appear as profits. The rate of conversion of revenues into profits is

given by the slope h = 1 – (b/p). The lower the unit variable cost b the higher the

slope h. Also, the higher the unit price p, the higher the slope h.

Figure 4a: The profits-revenues graph for the period 1Q2009 to 3Q2009.

Revenues increased with each succeeding quarter and profits also increased. The

three points are seen to lie on a nearly perfect straight line.

It is this subtle interaction of costs and revenues, in other words, the organizational

structure, or the “business model” of the company, which dictates whether the

three (or more) data points that we have just considered fall on a Type I line or a

Type II line, or even a Type III line.

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y = 0.12x – 0.34 = 0.12 (x – 2.85)

y/x = 0.12 – (0.34/x)

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As we from the profits-revenues data, Groupon was not able sustain this profit

generation rate beyond 3Q2009. For 4Q2009, Groupon reported a loss, which

implies a “sudden jump” in the costs, i.e., a change in the nonzero intercept c. The

(x, y) pair for 4Q2009l, see Figure 4b, falls well below the Type I line.

With increase in revenues (to $16.920 million), an extrapolation along the Type I

line would have yielded a profit of $1.673 million, i.e., the projected Costs =

Revenues – Profits = $15.247 million. Instead, Groupon reported a loss of $1.903

million, i.e., Costs = $18.823 million. The difference in the projected profits and

the actual loss 1.673 –(-1.903) = 3.576 is exactly equal to the difference in the

costs (18.823) – 15.247 = 3.576. The higher costs meant a loss instead of profits.

Figure 4b: The 4Q2009 data falls off the Type I straight line followed by the data

for the period 1Q2009 to 3Q2009. This can be explained by a “sudden jump” from

one value of the nonzero intercept c to another, i.e., a change in the fixed cost.

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A “sudden” change in the nonzero intercept c, or the “costs”, between 3Q2009 and

4Q2009 thus provides an explanation for the sudden plunge in the profits.

Indeed, as also discussed in earlier articles, the nonzero intercept c in the linear law

y = hx + c = h(x – x0) is exactly analogous to the work function W introduced into

physics by Einstein, in 1905. Einstein uses Planck’s radiation law (the nonlinear

law generalized as equation 1 above) to arrive at the simple linear law K = E – W =

hf – W = h(f – f0). As discussed briefly in what follows here, this analogy is indeed

compelling and cannot be overlooked.

In Einstein’s law, E = hf is the energy of a photon and K is the maximum (kinetic)

energy of the electron, h is the Planck constant and f is the frequency (of the light

wave). When light (which can be thought of as being a stream of photons each

having the energy E = hf) shines on the surface of a metal, it ejects electrons from

within the surface of the metal to produce a photocurrent. However, some of the

energy E must be given up to produce the electron. Einstein called the energy that

must be given up as W, the work function of the metal. This must be determined

experimentally. Thus, K < E. Also, the nonzero work function W means that

electrons will be produced only if the frequency f > f0 = W/h, the cut-off

frequency. Furthermore, the K-f graph will be a series of parallel, if we perform

experiments with different metals, having different work functions W.

We see an exactly similar behavior in the profits-revenues data for various

companies. Energy in physics is just like money in economics. Profits and

revenues are exactly analogous to the energy of the electron K and the energy of

the photon E = hf. Just as some energy W must be given up to produce the

electron, some of the revenues must be given up to produce the profits. The

difference is known as the “costs”. And, just as the frequency f must exceed the

minimum value f0 = W/h before electrons appear (in the external circuit), the

revenues must also exceed a minimum value x0 = -c/h before profits appear. Also,

just as the K-f graph is a series of parallels, we see clear evidence for a

movement along parallels on the profits-revenue diagram.

Although, Microsoft and Kia provide the best example of such a movement along

parallels, the data for Groupon reveals a similar pattern. Of course, it would seem

Page 17 of 33

that we have ignored the “outliers”. We have not. The “outliers” simply represent

“extreme fluctuations” from the general behavior that has been postulated here

(much larger variations in the fixed costs, embodied in the constant c).

Notice also that the data for March 2010 (y = 8.551) and Sep 2011 (y = -10.59) and

March 2012 (y = -3.59) all represent points where Groupon was very close to its

“breakeven” revenue. However, as the company has grown and its revenues have

increased, both the fixed cost (the parameter “a” in the breakeven model) and the

variable cost (parameter “b” in the breakeven model) obviously changed as

revenues increased. Thus, Groupon was unable to report a profit. It appears that a

“stable” operational mode is now established.

Perhaps, new insights can be gained, as noted also in earlier articles, with a wider

appreciation of the significance of the linear law being discussed here and also the

work function (nonzero intercept c) well beyond physics. Both nascent companies

like Groupon, and mature companies like Microsoft and IBM can be analyzed

using this model and the idea of a work function. The nonzero intercept c also

means that the profit margin y/x = m = h + (c/x) is not a constant and will keep on

increasing or decreasing as revenues x increase. The maximum profit margin y/x =

m = h the slope of the profits-revenues graph.

5. The Line of Excellence

Finally, let us re-examine the quarterly profits-revenues data to determine the

maximum profit potential for Groupon. This is exactly similar to finding the

maximum kinetic energy of the electron in the photoelectricity experiments. (An

accurate value of the fundamental Planck constant h can only be obtained if we

determine the maximum value of K as a function of the frequency f.) The quarterly

data for 2009, along with the data reported in the registration statement filed with

the United States Securities Exchange Commission has been complied in the

revised Table 1 in Appendix I. The data is plotted in Figure 5.

The following the Type I equation, y = 0.798x – 450.6 = 0.798 (x – 564.89),

“envelopes” all of the data in that all the points lie above and to the left of this line.

Page 18 of 33

It also has the highest conceivable slope h = 0.798. This line may therefore be

thought of as the Line of Excellence (click here) the significance of which also

been discussed earlier (see Ref. [13] in bibliography) within the context of

Google’s profits-revenues data.

Figure 5: The highest profit potential for Groupon is revealed here by considering

all of the available quarterly data going back to 2009. The Type I line joining the

(x, y) pairs for Dec 2010 and Mar 2012 data, with the equation y = 0.798x – 450.6

= 0.798 (x – 564.9) has the highest slope and “envelopes” the data. It also passes

close to the Dec 2011 data point.

The two points that fall below and to the right of the line were excluded since they

represent the data with the restated revenues (with no change in the profits)

between the S-1 registration statement (filed prior to the IPO) and the first 10-K

filing for the full year 2011, following the IPO.

-500

-400

-300

-200

-100

0

100

200

300

400

0 100 200 300 400 500 600 700 800 900 1000

Quarterly Revenues, x [$, millions]

Qu

art

erl

y P

rofi

ts,

y [

$,

millio

ns]

Dec 11

Mar 12

Dec 10

Page 19 of 33

The graphical representation of the profits-revenues data here suggests that

Groupon could potentially convert nearly 80% of its revenues, in excess of the

breakeven or cut-off value x0, into profits.

Indeed, the short line segment joining Dec 2011 and March 2012 data points can

be shown to have a slope h = 0.901. This means that Groupon was able to convert

90% of the additional revenues ($67.12 million) generated between the quarters

ending Dec 31, 2011 and March 31, 2012, into profits (∆y = $60.5 million).

Furthermore, Groupon seems to be poised to make the transition from a

sustained period of Type III (h < 0, c > 0, negative slope and positive intercept,

increasing revenues and decreasing profits, or equivalently, increasing losses

with increasing revenues, Figure 6) behavior to the more desirable Type I

behavior which was SEEN EARLIER from 1Q2009 to 3Q2009.

A higher slope is conceivable if we join the June 2011 and Sep 2011 data points.

However, in this case ∆x = 37.58 and ∆y = 90.67 and h = ∆y/∆x = 2.41 > 1. Profits

would have to increase at a very high rate, increase more than revenues do, for the

system to operate along this line. This is clearly not a sustainable proposition and

must be rejected. Likewise, other potential paths, with slope h > 1, must also be

rejected. For example, the line segment joining the two recent quarters Mar 2012

and June 2012, has a slope h = 4.1 >> 1. The revenues increased slightly by ∆x =

$9.052 million (from $559.283 million to $568.335 million) but the profits

increased by an astonishing ∆y = $36.8 million, which is clearly unsustainable.

Hence, we conclude that the highest rate of conversion of revenues into profits

is about 80% for Groupon. This profit potential could be achieved if efforts are

directed to understanding all of the factors in the Groupon business model that

influence the fundamental (a, b, p) triplet which determines costs and revenues,

and hence also the slope h and the cut-off revenues x0.

Page 20 of 33

Figure 6: Groupon went through a transition from the highly desirable Type I

behavior (increasing profits and profit margins with increasing revenues, seen

between 1Q2009 to 3Q2009, seen in Figure 4) to a prolonged period Type III

behavior, shown here using an expanded scale (for negative profits) compared to

that used in Figure 5. The Type III line, with the equation y = - 0.258x, joins the

origin (0, 0) to the June 2011 data point. This is added here only to reveal the

general Type III trend of increasing losses (mathematically equivalent to reduced

profits) with increasing revenues. The Type III lines also passes through the Sep

2010 data point. (Slope h = -0.251 for Sep ’10 and June ’11.) A linear regression

equation could be deduced but this is not deemed necessary at this stage.

A more viable and sustainable alternative is presented in Figure 7 where we

reconsider the data for the most recent quarters. A Type I line with a slope h =

0.372 joins the data for March 2011 and March 2012. This is essentially an

alternative view of exactly the same data presented earlier in Figure 1. Instead of a

nonlinear or seemingly erratic variation, we now envision a movement in the

profits-revenues space along roughly parallel lines. The June 2011 and Dec 2011

-350

-300

-250

-200

-150

-100

-50

0

50

100

0 100 200 300 400 500 600 700 800

Quarterly Revenues, x [$, millions]

Qu

art

erl

y P

rofi

ts,

y [

$,

millio

ns]

Type III trend line y = - 0.258x

Joins June ’11 to origin

Type I line y = 0.372x – 212.7 Mar ’11 to Mar ‘12

Page 21 of 33

data can be seen to fall on a roughly parallel line with a slope h = 0.364. The Sep

2011 and June 2012 data fall on another roughly parallel line with a somewhat

smaller slope h = 0.31. This viewpoint here is also consistent with the discussion of

the data presented earlier using the representations in Figures 2 and 3.

Figure 7: An alternative viewpoint of the profits-revenues data presented earlier in

Figure 1. The same six quarters, starting with March 2011 are considered here.

The data for the six quarters can be seen to represent a movement along roughly

parallel lines with a slope h = 0.372. The “fluctuations” in the quarterly profits

are thus entirely due to the changes in the work function, represented by the

nonzero intercept c. If the profits-revenue trend here is sustained, overall,

Groupon may be considered to have made a transition from the Type I line in

Figure 4 with a slope h = 0.12 to another Type I line, with a slope h = 0.37, with

an intervening “adjustment” period with the Type III behavior (Figure 6).

In summary, Groupon may have achieved its “breakeven” revenue level in the

quarter ended June 30, 2012 and so seems to be poised to report profits on a more

consistent basis. However, efforts must be made to control both the fixed costs and

-200

-150

-100

-50

0

50

100

0 100 200 300 400 500 600 700 800

Quarterly Revenues, x [$, millions] Qu

art

erl

y P

rofi

ts,

y [

$,

millio

ns]

Type I line y = 0.372x – 212.7 = 0.372 (x – 571)

Page 22 of 33

the variable costs which (as shown here) have increased since March 2010. The

application of the generalized Planck power-exponential law (equation 1 here) and

the extension of the idea of a work function to many problems outside physics

have also been discussed in several articles cited in the bibliography.

******************************************************************

Appendix I: Annual Profits-Revenues Data

The annualized profits and revenues data for the three full years of operation is

given in Table 2 and was again obtained from the SEC filings. This data is plotted

in Figure 8. Although quarterly profits have been reported on four occasions

(quarters ending June 2009, Sep 2009, Mar 2010 and the quarter ending June 30,

2012), Groupon has still not reported a profit on an annual basis.

The loss reported in 2009 was quite small and it appeared that the company was

close to its “breakeven” point. Since Profits = Revenues – Costs = P = R – C, for

2009 the costs C = 15.881 and Revenues R = 14.540. In 2010 and 2011, when

revenues increased, costs also seem to have increased and the company did not

report a profit, see Table 2. However, the reported losses decreased (which is

mathematically equivalent to increasing profits) between 2010 and 2011, as the

revenues increased.

The equation of the line joining the (x, y) pairs for 2010 and 2011 can be shown to

be y = 0.089x - 441.273 = 0.089 (x – x0) = 0.089 (x – 4952). This implies that the

“breakeven” or cut-off revenue x0 = $4952 million which is significantly greater

than the $1610 million reported for the year ending Dec 31, 2011. Hence, although

a small profit was reported for the quarter ending June 30, 2012, it appears highly

unlikely that a profit will be reported for the year ending 2012, or even in 2013.

Page 23 of 33

Figure 8: Annualized profits-revenues data for Groupon reveals a Type I behavior

with the equation y = 0.089x – 441 = 0.089 (x – x0) = 0.089 (x – 4952). This

implies a “breakeven” or cut-off revenue (akin to the cut-off frequency in

Einstein’s law) of x0 = $4952 million or about $5 billion in annual revenues given

the current cost structure of Groupon. For the six months ending June 30, 2012,

the cumulative revenues were only $1,127.62 million. Hence, it appears unlikely

that a profit will be reported for the year ending Dec 31, 2012.

Table 2: Annual Profits and Revenues Data for Groupon

Year

ending

Revenues, x

$, millions

Profits, y

$, millions

Change,

∆x

Change,

∆y

Slope, h =

∆y/∆x

31-Dec-11 1,610.430 -297.762 1,297.5 115.6 0.089

31-Dec-10 312.941 -413.386

31-Dec-09 14.540 -1.341

31-Dec-08 0.005 -1.542

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-400

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-100

0

100

200

0 1000 2000 3000 4000 5000 6000 7000 8000

Annualized Revenues, x [$, millions]

An

nu

aliz

ed

Pro

fits

, y [

$, m

illio

ns]

Type I line y = 0.089x – 441

= 0.089 (x – 4952)

2009

x0

2011

2010

Page 24 of 33

Figure 9: Revenues growth, as a function of time, for Groupon since December

2008 (taken to be month zero). The highest rate of increase of revenues is $1,297.5

million per year. This rate is used to predict future revenues for 2012 and 2013.

(The problem is exactly similar to predicting the future position of a moving

vehicle based on its current speed, i.e., the measurements of position x and time t.)

From Table 2, the largest increase in the revenues was $1,297.5 million between

Dec 2010 and Dec 2011. Hence, the highest rate of increase of revenue, ∆x/∆t =

$1,297.5 per year. If this rate of revenues growth can be sustained, the predicted

revenues are: for Dec 2012, $2908 million and for Dec 2013, $4205 million. The

projected revenues are therefore less than the cut-off revenue x0 = $4952 million

needed to report a profit, based on the current cost structure.

A significant acceleration in the revenues growth is required to turn a profit for the

year ending 2012 or 2013, see also Figure 9.

Alternatively, since the rate of conversion of revenues (in excess of the breakeven

or cut-off value x0) into profits, is given by the slope h of the profits-revenues

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

0 12 24 36 48 60

Time, t [months since Dec 2008]

An

nu

alized

Re

ven

ues,

x [

$,

milli

on

s]

∆x = 1,297.5

∆t = 1 year

Page 25 of 33

graph, the higher the slope h, the higher will be the profits y. Hence, it is important

to understand the factors that control the two parameters b and p that enter into the

equation for the slope h = 1 – (b/p), the unit variable cost b and the unit price p.

Additionally, reducing the cut-off revenue x0, i.e., the fixed cost “a” would also

enhance the profits for a fixed value of h.

Page 26 of 33

Table 1 (Revised): Profits and Revenues Data for Groupon

Quarter

ending

Revenues,

x

$, millions

Profits, y

$,

millions

Profit

margin,

y/x %

Change,

∆x

Change,

∆y

Slope, h

= ∆y/∆x

31-Mar-09 0.252 -0.309

30-Jun-09 3.301 0.021 0.6

30-Sep-09 9.998 0.850 8.5

31-Dec-09 16.920 -1.903

31-Mar-10 44.24 8.028 39.6 Higher revenues in S-1 filings

restated in the subsequent 10-K 31-Mar-10 20.27 8.551 42.2

30-Jun-10 87.3 -35.929 S-1 filing was for the IPO

30-Jun-10 38.67 -35.929 launch in November 2011

30-Sep-10 185.23 -49.032

30-Sep-10 81.78 -49.032

31-Dec-10 396.60 -313.23

31-Dec-10 172.22 -313.23

31-Mar-11 644.73 -146.48 As given in the S-1 filing June 2, 2011

31-Mar-11 292.52 -146.48 Revenues were restated in 10-Q filings

30-Jun-11 392.58 -101.24

30-Sep-11 430.16 -10.57

31-Dec-11 492.16 -64.95

31-Mar-12 559.28 -11.695 264 135 0.511

30-Jun-12 568.34 28.39 4.99

Notice the slight discrepancy in the March 2011 and 2012 revenues and profits, which were used

to calculate the Type I equation in Figures 2 and 3. The change in revenues ∆x and profits ∆y

between March 2011 and 2012, yield the slope h = ∆y/∆x = 135/264 = 0.511. The discrepancy is

due to the restated values in 10-K (post IPO)and the figures in the S-1 filing in June 2011 (prior

to IPO) and also the three line items called net loss, net loss attributed to common shareholders

and net loss to Groupon. In some cases only net loss to Groupon figures are available.

Data sources:

http://www.sec.gov/Archives/edgar/data/1490281/000104746911005613/a2203913zs

-1.htm#do79801_management_s_discussion_and_an__man03466 SEC Registration

statement S-1 filed on June 2, 2011. (The March 2011 and March 2010 values are different.)

http://files.shareholder.com/downloads/AMDA-E2NTR/2021598481x0xS1445305-12-

1731/1490281/filing.pdf and http://files.shareholder.com/downloads/AMDA-

E2NTR/2021598481x0xS1490281-12-15/1490281/filing.pdf and

http://files.shareholder.com/downloads/AMDA-E2NTR/2021598481x0xS1445305-12-

922/1490281/filing.pdf

Page 27 of 33

Table 2 (Revised): Annual Profits and Revenues Data for

Groupon in the S-1 Filing of June 2, 2011

Year

ending

Revenues, x

$, millions

Profits, y

$, millions

Change,

∆x

Change,

∆y

Slope, h =

∆y/∆x

31-Dec-10 713.365 -456.32 713.271 -454.162 -0.637

31-Dec-09 30.471 -6.916

31-Dec-08 0.094 -2.158

Prior to the IPO, the registration statement with the United States Securities and

Exchange Commission (SEC) contained above information. Figures were restated

later in the 10-K filing for 2011. Although revenues increased between 2008

(partial year of operation) and 2010, losses also increased, yielding a negative

slope h = -0.637. Since Costs = Revenues – Profits, the negative profits means that

costs were increasing as revenues increased. The situation reversed itself and a

positive slope was established between 2010 and 2011, see Table 2.

Page 28 of 33

Appendix II: Bibliography

Related Internet articles posted at this website

Since the Facebook IPO on May 18, 2012

The first article listed below discusses a little known mathematical property of a

straight line. Figures 1 to 3 in this article provide the philosophical basis for

considering the significance of the significance of a nonzero intercept c as it

applies to many problems in the real world. We make observations (x and y values

of interest to us) to deduce y/x, usually called “rates”, “ratios”, or percentages.

1. http://www.scribd.com/doc/102000311/A-Little-Known-Mathematical-

Property-of-a-Straight-Line-Strange-but-true-there-is-one Published August 4,

2012.

Financial data (Profits-Revenues) analysis and Generalization of Planck’s law

beyond physics.

2. http://www.scribd.com/doc/95906902/Simple-Mathematical-Laws-Govern-

Corporate-Financial-Behavior-A-Brief-Compilation-of-Profits-Revenues-

Data Current article with all others above cited for completeness, Published

June 4, 2012 with several revisions incorporating more examples.

3. http://www.scribd.com/doc/94647467/Three-Types-of-Companies-From-

Quantum-Physics-to-Economics Basic discussion of three types of

companies, Published May 24, 2012. Examples of Google, Facebook,

ExxonMobil, Best Buy, Ford, Universal Insurance Holdings

4. http://www.scribd.com/doc/96228131/The-Perfect-Apple-How-it-can-be-

destroyed Detailed discussion of Apple Inc. data. Published June 7, 2012.

5. http://www.scribd.com/doc/95140101/Ford-Motor-Company-Data-Reveals-

Mount-Profit Ford Motor Company graph illustrating pronounced maximum

point, Published May 29, 2012.

Page 29 of 33

6. http://www.scribd.com/doc/95329905/Planck-s-Blackbody-Radiation-Law-

Rederived-for-more-General-Case Generalization of Planck’s law,

Published May 30, 2012.

7. http://www.scribd.com/doc/94325593/The-Future-of-Facebook-I Facebook

and Google data are compared here. Published May 21, 2012.

8. http://www.scribd.com/doc/94103265/The-FaceBook-Future Published May

19, 2012 (the day after IPO launch on Friday May 18, 2012).

9. http://www.scribd.com/doc/95728457/What-is-Entropy Discussion of the

meaning of entropy (using example given by Boltzmann in 1877, later also

used by Planck to develop quantum physics in 1900). The example here shows

the concepts of entropy S and energy U (and the derivative T = dU/dS) can be

extended beyond physics with energy = money, or any property of interest.

Published June 3, 2012.

10. The Future of Southwest Airlines, Completed June 14, 2012 (to be

published). http://www.scribd.com/doc/102835946/The-Future-for-Southwest-

Airlines-The-Unknown-Story-of-Rising-Costs-and-the-Maximum-Point-on-

Profits-Revenues-Curve Published August 14, 2012.

11. The Air Tran Story: An Important Link to the Future of Southwest Airlines,

Completed June 27, 2012 (to be published).

http://www.scribd.com/doc/102832984/The-Air-Tran-Story-The-Merger-and-

Maximum-Point-on-Profits-Revenues-Graph Published August 14, 2012.

12. Annie’s Inc. A Single-Product Company Analyzed using a New

Methodology, http://www.scribd.com/doc/98652561/Annie-s-Inc-A-Single-

Product-Company-Analyzed-Using-a-New-Methodology Published June 29,

2012

13. Google Inc. A Lovable One-Trick Pony Another Single-product Company

Analyzed using the New Methodology.

http://www.scribd.com/doc/98825141/Google-A-Lovable-One-Trick-Pony-

Another-Single-Product-Company-Analyzed-Using-the-New-Methodology,

Published July 1, 2012.

14. GT Advanced Technologies, Inc. Analysis of Recent Financial Data,

Completed on July 4, 2012. (To be published).

Page 30 of 33

15. Disappearing Brands: Research in Motion Limited. An Interesting type of

Maximum Point on the Profits-Revenues Graph

http://www.scribd.com/doc/99181402/Research-in-Motion-RIM-Limited-Will-

Disappear-in-2013 Published July 5, 2012.

16. Kia Motor Company: A Disappearing Brand

http://www.scribd.com/doc/99333764/Kia-Motor-Company-A-Disppearing-

Brand, Published July 6, 2012.

17. The Perfect Apple-II: Taking A Second Bite: A Simple Methodology for

Revenues Predictions (Completed July 8, 2012, To be Published)

http://www.scribd.com/doc/101503988/The-Perfect-Apple-II, Published

July 30, 2012.

18. http://www.scribd.com/doc/101062823/A-Fresh-Look-at-Microsoft-After-its-

Historic-Quarterly-Loss Microsoft after the quarterly loss, Published July 25,

2012.

19. http://www.scribd.com/doc/101518117/A-Second-Look-at-Microsoft-After-the-

Historic-Quarterly-Loss, Published July 30, 2012.

20. http://www.scribd.com/doc/103027366/Groupon-Analysis-of-Profits-Revenues-

Data-and-its-Business-Model, Published August 16, 2012. The current article is

the latest revision to clarify the concept of a work function.

21. http://www.scribd.com/doc/103265909/A-Brief-Analysis-of-Groupon-s-Profits-

Revenues-Data Published August 19, 2012.

******************************************************************

The Unemployment Problem: Evidence for a Universal value of h in the

unemployment law.

22. http://www.scribd.com/doc/100984613/Further-Empirical-Evidence-for-the-

Universal-Constant-h-and-the-Economic-Work-Function-Analysis-of-

Historical-Unemployment-data-for-Japan-1953-2011 Single universal value of

h for US, Canada and Japan in the unemployment law y = hx + c, Published

July 24, 2012.

Page 31 of 33

23. http://www.scribd.com/doc/100939758/An-Economy-Under-Stress-

Preliminary-Analysis-of-Historical-Unemployment-Data-for-Japan, Published

July 24, 2012.

24. http://www.scribd.com/doc/100910302/Further-Evidence-for-a-Universal-

Constant-h-and-the-Economic-Work-Function-Analysis-of-US-1941-2011-and-

Canadian-1976-2011-Unemployment-Data Published July 24, 2012.

25. http://www.scribd.com/doc/100720086/A-Second-Look-at-Australian-2012-

Unemployment-Data, Published July 22, 2012.

26. http://www.scribd.com/doc/100500017/A-First-Look-at-Australian-

Unemployment-Statistics-A-New-Methodology-for-Analyzing-Unemployment-

Data , Published July 19, 2012.

27. http://www.scribd.com/doc/99857981/The-Highest-US-Unemployment-Rates-

Obama-years-compared-with-historic-highs-in-Unemployment-levels ,

Published July 12, 2012.

28. http://www.scribd.com/doc/99647215/The-US-Unemployment-Rate-What-

happened-in-the-Obama-years , Published July 10, 2012.

****************************************************************

Traffic-fatality and Teen pregnancy problem

29. http://www.scribd.com/doc/101982715/Does-Speed-Kill-Forgotten-US-

Highway-Deaths-in-1950s-and-1960s Published August 4, 2012.

30. http://www.scribd.com/doc/101983375/Effect-of-Speed-Limits-on-Fatalities-

Texas-Proofing-of-Vehciles Published August 4, 2012.

31. http://www.scribd.com/doc/101828233/The-US-Teenage-Pregnancy-Rates-1

Published August 2, 2012.

32. http://www.scribd.com/doc/102384514/A-Second-Look-at-the-US-Teenage-

Pregnancy-Rates-Evidence-for-a-Predominant-Natural-Law Published August

8, 2012.

******************************************************************

Page 32 of 33

About the author

V. Laxmanan, Sc. D.

Email: vlaxmanan@hotmail.com

The author obtained his Bachelor’s degree (B. E.) in Mechanical Engineering from

the University of Poona and his Master’s degree (M. E.), also in Mechanical

Engineering, from the Indian Institute of Science, Bangalore, followed by a

Master’s (S. M.) and Doctoral (Sc. D.) degrees in Materials Engineering from the

Massachusetts Institute of Technology, Cambridge, MA, USA. He then spent his

entire professional career at leading US research institutions (MIT, Allied

Chemical Corporate R & D, now part of Honeywell, NASA, Case Western Reserve

University (CWRU), and General Motors Research and Development Center in

Warren, MI). He holds four patents in materials processing, has co-authored two

books and published several scientific papers in leading peer-reviewed

international journals. His expertise includes developing simple mathematical

models to explain the behavior of complex systems.

While at NASA and CWRU, he was responsible for developing material processing

experiments to be performed aboard the space shuttle and developed a simple

mathematical model to explain the growth Christmas-tree, or snowflake, like

structures (called dendrites) widely observed in many types of liquid-to-solid phase

transformations (e.g., freezing of all commercial metals and alloys, freezing of

water, and, yes, production of snowflakes!). This led to a simple model to explain

the growth of dendritic structures in both the ground-based experiments and in the

space shuttle experiments.

More recently, he has been interested in the analysis of the large volumes of data

from financial and economic systems and has developed what may be called the

Quantum Business Model (QBM). This extends (to financial and economic

systems) the mathematical arguments used by Max Planck to develop quantum

physics using the analogy Energy = Money, i.e., energy in physics is like money in

economics. Einstein applied Planck’s ideas to describe the photoelectric effect (by

treating light as being composed of particles called photons, each with the fixed

Page 33 of 33

quantum of energy conceived by Planck). The mathematical law deduced by

Planck, referred to here as the generalized power-exponential law, might actually

have many applications far beyond blackbody radiation studies where it was first

conceived.

Einstein’s photoelectric law is a simple linear law, as we see here, and was

deduced from Planck’s non-linear law for describing blackbody radiation. It

appears that financial and economic systems can be modeled using a similar

approach. Finance, business, economics and management sciences now essentially

seem to operate like astronomy and physics before the advent of Kepler and

Newton.

Cover page of AirTran 2000 Annual Report