Entrepreneurship in Wind Power and Solar Energy
4.3.5.3 Selected Quantitative Applications of the Bracket Model the Bracket Model
It is the mark of an educated person to look for precision only as far as the nature of the subject allows.
Aristotle (Stanford Encyclodia of Philosopy, Episteme and Techne) For a quantitative discussion of NTBFs’ growth we shall follow cybernetics (4.3.4) and use a heuristic approach of reinforcement for periods of growth of sub-states of a new firm. We use a phased process interrupted by brackets and associated transition states (Figure I.133, Figure I.134) and are guided by situations described by Equation I.16, Equation I.5 and Figure I.70. The heuristic will be verbally expressed by the well- known saying
Growth breeds growth.
In the same category of algorithm, but including growth or decline (Figure I.114 and Equation I.16) one can cite the “Matthew Effect.” The Matthew Effect [Merton 1968]
specifically refers to a statement in the Christian Bible (XXV:29):
“For unto every one that hath shall be given, and he shall have abundance;
but from him that hath not shall be taken away even that which he hath.”
In this context of “growth breeds growth” and emphasizing VC-based NTBFs Gompers et al. [2008] present evidence of “performance persistence” (definition of Gompers et al. [2008])) in entrepreneurship and that entrepreneurs with a track record of success are much more likely to succeed (“success breeds success”) than first-time entrepre- neurs and those who have previously failed.
Similarly, related to innovation persistence, Flaig and Stadler [1994] revealed a posi- tive impact of previous innovative success to further innovations in the following years.
They studied product and process innovations of private German firms from the manu- facturing sector (data between 1979 and 1986). And there is a combined internal and an external effect paralleling growth by innovation and investment persistence: As entrepreneurs build a track record, uncertainties about their ability and business pro- positions diminish; the entrepreneur gets more self-confidence, confidence in the firm, experience of raising money from external financial backers as well as getting suppli- ers and customers.
We focus on dynamic stability during firm development which will be expressed by
“equations of state” for a given “dynamically stable interval” of time <t0, tn> to describe time-dependent relations under a given set of conditions in terms of appropriate in- dicators, such as revenues. Concerning firm growth we look at its financial sub-states (Equation I.15) and regard these is indicative for the firm’s growth (ch. 4.3.5.1).
Let R(t) be the value (of revenue) at time t for the financial sub-state taken also as a growth indicator for the whole system and R(t+1) the value of one unit of time later.
Following conceptually and structurally “growth breeds grows,” the innovation and
investment persistence growth cycles (Figure I.127) and Equation I.13 for an equation of state for dynamically stable states of NTBFs, intensity ⊗ capacity, makes R(t+1) proportional to R(t): R(t+1) → nR⋅⋅⋅⋅R(t).
The factor nR is an intensity covering interacting firm-internal and external effects. In particular, it may include responsiveness of the startup toward (changes of) the mar- ket environment. To account for this effect and self-reinforcement of the capacity fac- tor we complement R(t) by an additional factor [1 + ( R(t+1) / R(t) )] to finally suggest Equation I.17 as a fundamental systemic reflection of “growth breeds growth” for dy- namically stable states of NTBFs. In particular, it shall cover dynamically stable states with a steady state condition as defined above (Figure I.132).
We interpret the capacity term R(t)⋅⋅⋅⋅[ 1 + ( R(t+1) / R(t) )] as being related to capability in the sense of RBV (ch. 4.3.3). The factor nR is assumed to be a constant for a spe- cific bracket interval and treated as an empirically determined factor.
Equation I.17:
R(t+1) = n
R⋅⋅⋅⋅R(t)
⋅⋅⋅⋅[ 1 + ( R(t+1) / R(t) )]
t = 0, 1, 2, …n; nR a state and environment characterizing (empirical) factor;
R(t+1) ≥ R(t) for all t; asymptotic behavior for nR = ½ gives R(t+1) = R(t)
The equation of state describes the state over a period of time beginning after a front bracket at the time when the firm has adapted a new “dynamically stable” state (t = 0, R(0) until the occurrence of a new bracket ends the development of this dynamic state (cf. right part of Figure I.133)). This new bracket may be identifiable.
On the other hand, based on an equation of state like Equation I.17 we regard a strong deviation of calculated values of an otherwise good fit between calculated and observed values as evidence for the impact of a bracket that cannot be directly ob- served on the basis of indicator data.
Equation I.17 is defined “recursively” – resembling to a certain degree the Fibonacci numbers [WolframMathWorld].
For Ri (i ≥ 2) we proceed taking R(0) and R(1) as the basis and then start with the mappings, e.g. R(1) → R1 and (R(1)/R(0) → R2/R1.
Then the factors nR and (R2/R1) are fitted to acceptably describe the whole se- quence of data on the interval <t0, tn>:
R2 = nR·R1⋅(1 + R(1)/R(0)) → R2 = nR⋅⋅⋅⋅R1⋅(1 + R2/R1) → Numerical fit for all data
We are primarily interested in an equation of state for monotonically increasing reve- nue growth R on a given interval <t0, tn> The numerical values of the state- characterizing factor nR and the starting quotient R(t+1)/R(t) of Equation I.17 will shape the curve, usually after a bracket.
Curves as determined by different values of the factor nR and fixed values for an initial growth rate (R(0) and R(1) may exhibit various appearances as are given in the below Figure I.155. Shapes range from exponential (hyperbolic) to asymptotic growth and non-growth shape (Figure I.107).
Equation I.17 reflects also a mechanism when early growth of revenues, due to an un- favorable situation and insufficient magnitude of the intensity factor nR, can turn into decreasing revenues, as is illustrated in Figure I.114, and ultimately disappearance of a firm:
Failure paths all begin with one or more fundamental problems;
They all lead to a situation where the symptoms of the worsening situation be- come visible in the financial situation.
Furthermore, Equation I.17 can simulate the case that the “thrust capacity” for a startup’s development does not suffice for a successful “lift-off” for (revenue) growth.
R(t+1) = n⋅⋅⋅⋅R(t)⋅⋅⋅⋅[ 1 + ( R(t+1) / R(t) )] (Equation I.17);
Start: R(0) = 1.5, R(1) = 2.5 units;
initial growth ratio: R(1)/R(0) = 1.67
(simulating very strong growth of the particular state)
An expansion of the curves for n = 0.50 and n = 0.52 is given on the next page.
Figure I.155: Prototypical development curves as shaped by relevant parameters.
Equation I.17 becomes extremely simple if the quotient R(t+1)/R(t) is very small com- pared to 1 and, hence, can be neglected or can be viewed as a non-negligible con- stant g’. Then either we have
R(t+1) = nR⋅⋅⋅⋅R(t) or R(t+1) = nR⋅⋅⋅⋅ (1 + g’)⋅⋅⋅⋅R(t)
Therefore, at least for a particular interval <t0,tn>, we propose a growth period of an NTBF to be described by the very simple formula Equation I.18 (nR⋅⋅⋅⋅(1 + g’) → (1 +g)).
Though we shall denote g as a “growth factor” it should be noted that its numerical closeness to a common growth rate, such as CAGR (Equation I.10), must be viewed as accidental.
Through its relation to interwoven company-internal and external effects the growth factor g is related to nR. If we interpret growth of a new firm to be determined es- sentially by growth of the market and firm-internal response (capacity) toward the market opportunity we can differentiate the development categories for new firms as:
Grow less than the market Grow with the market and Grow more than the market.
Equation I.18:
R(t+1) = (1 + g)
⋅⋅⋅⋅R(t) for <t
0,t
n>
The following examples calculating revenue values for dynamically stable financial sub-states do not intend to provide numerically optimized solutions, but solutions with
“simple” numerical relations that will suffice to illustrate the essentials of the bracket model.
The simplest case is provided by WITec’s closely linear growth (Figure I.123) or the linear growth periods of Nano-X (Figure I.137).
The example of WITec (founded 1997) can be used to illustrate the essentials of the bracket approach. It is not about (statistical) curve fitting for an interval <1999,2009>
or <2002,2009>. The bracket theory interconnects a time interval for development of states started by a front bracket as displayed in Figure I.123.
The first bracket is firm’s foundation (ch. 4.3.5.1) with the startup thrust phase ex- tending over the next three to four years being associated with rather unstable firm states and usually decreasing year on year growth rates (Table I.71).
In Figure I.156 Equation I.18 could be used as a numerically sufficient approximation for the interval <2002,2009>. But this interval is “perturbed” by the Dot-Com Reces- sion. In Figure I.156 it is argued why <2004,2009> would be most appropriate interval characterizing a dynamically stable growth state of WITec. The interval would proba- bly start at 2003, but there are no revenue data available for 2003.
Similar to WITec almost linear, parallel developments of revenues and number of em- ployees for Nano-X GmbH (Figure I.137) express almost constant productivities for the intervals <2000,2003) (productivity ca. €63,000 per employee) and <2005,2008>
(productivity ca. €118,000 per employee) and indicate these periods to be associated with dynamically stable organizational states.
Corresponding effects are observed for SAP productivity (Figure I.147) which show little changing for the interval <1976,1979>.
On the other hand, the Microsoft’s revenues (Figure I.144) between 1977 and 1982 as well as 1980 and 1989 with starting growth proportions of ca. 3.6 or 2.1, respectively, exhibit exponential growth. The recession of 1981/1982 separates both phases.
Moreover, the period <1981,1986> is associated with many positive brackets and the IPO which together lead (probably) to accelerated growth expressed by a larger nR- factor. The interval <1977,1982> of Microsoft’s early growth is organizationally rather unstable (Figure I.146).
Equation I.17 is sufficient for the calculations (Figure I.157) of two dynamically stable financial sub-states of Microsoft separated by a recession bracket which, however, is not observable and obviously more than balanced by several positive brackets.
Formula: R(t+1) = (1 + g)⋅⋅⋅⋅R(t) The tempting approach:
Growth factor g = 0.13; R(0) = €3.29 mio. at 2002; CAGR (2002,2009) = 14.5%
Comments: Actually, a very good fit is seen for the interval <2004,2009> (dashed line, g = 0.09) which is the most appropriate characterization of a dynamically stable financial state as the observed data for 2002 may still be affected by the Dot- Com Recession, data for 2009 by the Great Recession.
Furthermore, WITec exhibits almost constant productivity (Figure I.123). This means, for the related interval it has a financially dynamically stable state – and also a dynamically stable organizational sub-state showing a steady state condition with almost constant productivities (Figure I.123).
An approach with
growth factor g = 0.17; R(0) = €1.55 mio. at 1999; CAGR(1999,2009) = 18.6%
provides a numerically acceptable approximation for the 1999-2009 period. How- ever, it covers also the Dot-Com Recession. Such pure numerics are not in line with the bracket theory and, hence, are lacking explanatory power.
Figure I.156: Calculated and observed revenues of WITec GmbH for a “perceived”
dynamic stability interval <2002,2009> and contrasted with the theory-related
<2004,2009> interval (dashed line).
Microsoft Corp., founded 1975, IPO 1986; Equation I.17 used:
Above figure: R(0) = $0.38 mio. (1977), R(1) = $1.36 mio. (1978); nR = 0.585;
R(1)/R(0) = 3.57
After 1882 there are large deviations between calculated and observed revenues (calculated values much larger than observed ones) indicating a transition state.
Below figure: R(0) = $7.52 mio. (1980), R(1) = $16.06 mio. (1981); nR = 0.605;
R(1)/R(0) = 2.13
Figure I.157: Calculated and observed revenue growth of Microsoft.
When looking at the examples of Cisco and First Solar (Figure I.158.) Equation I.17 turned out to be inadequate and required modification. Cisco’s starting growth propor- tion was 3.63 and that of First Solar is 4.21.
For both cases it has turned out that Equation I.19 different from Equation I.17 by the reinforcement factor is more appropriate for calculations. Its applicability is also shown for Google in Figure I.159.
“Similar” to Equation I.10 for CAGR growth relates to a fixed starting value R(0) in the series of R(t) across the dynamically stable state. Disregarding the construction of some sort of geometric mean through the t-eth root the emphasis on a fixed value of R(0) for the state seems to show that for the (financial) growth a particular initial constellation exerts a decisive influence.
For Cisco a good fit is observed and its revenues for the 1991 recession are obviously more than balanced by sales (and more employees). On the other hand, referring to productivity Figure I.145 shows that Cisco’s organizational state is by no means stable across the early 1987-1994 period.
Due to the small number of cases treated by Equation I.19 one can only speculate about what is behind R(0) – a large capital injection at the start of the dynamically stable period adding to the “base line” development of each of the R(t)’s which will drive the growth over the whole period, a jump in other resources like employee num- ber, or an extremely successful shift to another source of revenue.
Equation I.19:
R(t+1) = n
R⋅⋅⋅⋅R(t)
⋅⋅⋅⋅[ 1 + {R(t) / R(0) }
1/t]
t = 1, 2, …n; nR a state-characterizing (empirical) factor
Calculations using Equation I.19
Cisco Systems: R(0) = $1.50 mio. (1987), R(1) = $5.45 mio. (1988); nR = 0.585;
R(1)/R(0) = 3.63
First Solar: R(0) = $3.21 mio. (2003), R(1) = $13.52 mio. (2004) ; nR = 0.650;
R(1)/R(0) = 4.21
Figure I.158: Calculated (Equation I.19) and observed revenues of Cisco Systems and First Solar.
The large deviation between observed and calculated revenues for First Solar after 2008 due to the Great Recession and the change of the EEG in Germany shows that the financial sub-state to grow regularly only between 2004 and 2008. This is different
from Cisco whose regular growth was obviously not affected essentially by a reces- sion.
As a summary, the preceding discussions have presented growth categories (Figure I.155) for early development states of young firms (NTBFs) in terms of equations of state which are essentially recursive relations (Equation I.17 - Equation I.19) and re- flect self-reinforcement verbalized as “growth breeds growth.” This means, a series of interacting internal sub-states’ changes leads to increases in size (observed for reve- nues) accompanied by not necessarily synchronous changes of the characteristics of other sub-states of the growing entity over a certain period of time (Figure I.149).
Based on the calculational results in this chapter it appears that irregularities ending an otherwise good fit when comparing calculated and observed reve- nues may provide a means to detect brackets which may not show up in observed curves.
The development patterns represent systemic features of different types of growth paths of new firms for a given time period starting at a specified point in time that may initiate the search for explanations on the firm level. In the above text it has been pointed out that after a perturbation, a bracket, the firm will not return to the original state, but rather develops further into a new state.
This shall be illustrated by different theoretical descriptions in terms of different formu- las to be used before and after a bracket for the German NTBF Nano-X (Figure I.137;
Table I.77) and Google (Figure I.159).
For Nano-X the discontinuity, the bracket R(2004) → R(2005) of +2.33 units for the interval <2002,2008>, is surrounded by two states differing by the g-factor of Equation I.18. Table I.77 provides a “full” theoretical description of the (financial) states of Nano-X. The jump suffices qualitatively to identify the different states. Admittedly, the quantitative approach of the early state with g = 1.52 is not satisfying.
The significantly more pronounced growth rate for the 2000 to 2004 period cannot be explained straightforwardly. This period covers the Dot-Com Recession with no ob- servable negative impact for the revenues. However, Nano-X financed the first years essentially via R&D projects (“Verbundprojekte”; ch. 1.2.6) of the German federal/state governments, the EU and NGOs as well as cooperation with other SMEs. As project money is usually counted as revenue, it can be hypothesized that the significant reve- nue growth during the first years, from almost its start, are due to larger contributions from projects to revenues.
Therefore, the theoretical description of the pre-2005 dynamically stable state of Nano-X should be confined to the interval <2002,2004>, though it cannot be ruled out that it covers also the 2000 and 2001 range.
On the other hand, more successful applications of Equation I.18 are discussed later (Figure I.163, Figure I.164, Figure I.165, Figure I.166).
Table I.77: Theoretical descriptions (Equation I.18, <2002,2004>) of two developing financial states of the German Nano-X GmbH separated by a bracket with pronounced discontinuity, a “jump.”
Year
Revenues (€, mio.)
Calculated Revenues (€, mio.)
R(t+1) = (1 + g) * R(t) g = 0.52 g = 0.05
2000 0.50 0.50
2001 1.00 0.76
2002 1.4 *) 1.16
2003 N/A 1.76
2004 2.50 2.67
R(2005) = R(2004) + 2.33
2005 5.00 5.00
2006 5.20 5.25
2007 5.40 5.51
2008 6.00 5.79
Average Productivities (€ /employee):
*) Estimated from average productivity (70,000)
70,000 118,000
It is interesting to note that in its early years also SAP’s growth follows Equation I.18 (g = 1.70 for <1972,1974> and g = 1.63 for <1975,1978>; Figure I.147).
For Google (Box I.24) two growth periods according to different theoretical ap- proaches can be identified with a transition state (“bracket”) at 2002/2003 (Figure I.159) when revenues “jumped” from $86 million (2001) to $440 million (2002).
Starting with the period 2000 – 2003 it turns out that after 2003 the gap between calculated and observed revenues widens drastically. Hence, by trial, it was found with which formula to describe the 2003 – 2008 period.
Early Growth of Contextual Advertisement on the Web.
After 2002 ca. 97 percent of Google’s revenues were from advertisement! In 2001 ad revenue accounted for 77% and in 2002 it was already 92%. Advertising income is earned as Google operates its own Web sites, but it distributes ads also to part- ner Web sites. Its software AdWords (launched in 2000, major overhaul in 2002) is Google’s unique method for selling online advertising. AdWords analyzes every Google search to determine which advertisers get each of up to 11 “sponsored links” on every results page. It is one component for the link between searching and advertising. The second part is AdSense which relates to semantics [Sullivan 2004].
In 2003 Google launched its AdSense contextual ad program and then greatly ex- panded AdSense, meaning ad serving application. AdSense placements are almost certainly the reason why Google has seen network-derived ad revenue rise so sharply.
There is another factor whose effects, however, cannot be assessed. Google bought Applied Semantics in 2003 and also three other companies [Sullivan 2004].
Google:
Founded 1998 by Larry Page and Sergey Brin primary focus:
a better search engine for the Web (Box I.24)
Calculated values according to
Equation I.18 (small chart), g = 3.75, R(0) = $19 mio.);
Equation I.19 (large chart), R(0) = $440 mio., R(1) = $1,467 mio., nR = 0.50
Figure I.159: Theoretical descriptions of two developing financial states of Google se- parated by a bracket with pronounced discontinuity, a “jump” (data from [Tech Crunchies 2010]).
So far, brackets were identified on the basis of revenues or employees/productivities.
In case of Google revenue data do not reveal a 2002/2003 bracket (cf. Figure I.136).
Fortunately, for Google the theoretical description is corroborated by other data: The pronounced “jump” characterizing the transition into the second state in clearly seen in the development of Google’s profit (Figure I.160).
Based on the limited number of cases, it has turned out that IPO brackets could not be detected consistently though they may represent drastic changes of a firm’s sub- states. Change will relate to ownership and potential change of control (change of leadership or management), respectively, to huge addition of capital (financial resources) which can be used, for instance, to pay back debts or increase of human resources in terms of R&D and marketing, sales and distribution (exploiting existing markets and enter new markets) or increase capacity of production.
IPO brackets are sometimes associated with increase of the number of employees which is observable by a decrease of productivity (Q-Cells, Figure I.153). But for Cisco (Figure I.145) in a steady state this does not show up.
Figure I.160: Timeline for development of Google’s profit (data from [Tech Crunchies 2010]).
Summarizing the presented aspects of the theoretical approaches to early growth of NTBFs one is led to suppose that for early unperturbed growth phases Equation I.17 and Equation I.18 are appropriate if the growth is essentially driven by cash flow from income (WITec, Nano-X, SAP, Microsoft and Cambride Nanotech and US LED given in ch. 4.3.6). This would be reflected by the cyclic process of innovation and invest- ment persistence in Figure I.127. On the other, if large capital comes from outside
sources, but not through an IPO (Cisco, First Solar, Google), Equation I.18 seems to be adequate. However, this is still a proposition.
The equations do not only provide numerically satisfying descriptions of dynamically developing growth states for NTBFs, but in certain cases various structures for dif- ferent intervals may differentiate states without prior identification of brackets. In so far, for the developmental processes of NTBFs’ patterns of growth are created that characterize growth states. However, a more detailed understanding of the observed effects must be bolstered by reference to the micro-level of the firm under considera- tion if common features for such states shall be revealed.
Specific initial startup configurations and random occurrences of largely conceivable bracket events and related proceedings concerning decision-making will lead to over- all differences in the growth of new firms. But there are periods, time intervals of growth, which are structurally comparable for the dynamics of developments of different firms.