S Shaped growth followed by decline

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"John Gunkler"
Member
Posts: 31
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by "John Gunkler" »

I work with many businesses that experience a kind of S-shaped growth
followed by exponential decline. I know some simple dynamic structures that
can generate the initial S-shaped growth but I have been unable (so far) to
create a relatively simple structure that captures the entire time course of
behavior.

A little more detail:

1. The (simplified) "best" case scenario: The system experiences
apparently slow growth initially, then apparently accelerating growth, then
decelerating growth approaching a "carrying capacity" kind of limit. So
far, so good (this is the S-shaped growth phase.) But then, when the limit
is neared, the behavior changes into something that looks like exponential
decay (or maybe S-shaped decline to zero.)

Some of the most obvious apparent causes of the decline include: entry of
more (and more successful) competitors (resulting in low levels of customer
loyalty); the resulting price competition and quality competition (having to
provide the customer more and more for less and less); the difficulty of
innovation (the market is mature and has relatively fixed expectations; the
"easy-to-find" innovations have already been found); the small amount of
growth that is possible as you approach the limit (and the fact that many
more companies are competing for it.)

2. The (simplified) "typical" case scenario: The system experiences
S-shaped growth for a time but stops far short of the limit, going into
exponential decline earlier than in the "best" case scenario. This
transition to decline can occur almost anywhere on the S-shaped growth
curve.

Some of the most obvious apparent causes of this include: displeasing,
disappointing, or irritating the market (poor service; failure to innovate;
product failures; PR disasters); entry of a competitor providing perceived
market value significantly better than yours; or failure to meet any of
several other typical challenges of growing a firm (solving the distribution
efficiency problem; kicking the entrepreneurial founder upstairs and
installing operational managers; etc.)

My first challenge to you: Is there a relatively simple dynamic structure
that can throw S-shaped growth into exponential decay (or S-shaped decline)
when it nears the upper asymptotic limit?

My second challenge: Could this structure also throw the behavior into
decay at some other "trigger" point (or simply at an earlier time) based on
what is going on endogenously?

My third challenge: I dont know if a relatively simple model can be built
that would incorporate both scenarios within the same dynamic structure. If
we can meet the second challenge would we have it? What do you think?

I would appreciate any speculations -- you neednt feel you have to build a
model before replying. Thanks for any ideas.

From: "John Gunkler" <
jgunkler@sprintmail.com>
LNUSTC1.LZTP67@gmeds.com
Newbie
Posts: 1
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by LNUSTC1.LZTP67@gmeds.com »

The simplest model I can think of now consists of one stock and inflowoutflow
rates. The inflow rate represents the s-shaped growth phase of a business and
the outflow rate represents the s-shaped decline phase with some time delay.
The time delay can be thought as late entry on a market by competitors.

The model I developed is enclosed: The easy way to model the s-shaped growth
phase is to take a first order derivative of s-shaped curve. Then you have a
bell-shaped curve, which may represent annual sales. The s-type decline can be
modeled the same way with market entry time delay. The basic model shows the
s-shaped growth and decline phases. And I guess you have to incorporate
interactions with competitors for other causes.

Best wishes.

Sangdon Lee
GM Tech Center
Sangdonlee@yahoo.com


The iThink model:

sales(t) = sales(t - dt) + (sales_increase - sales_decrease) * dt
INIT sales = 0
sales_increase = derivn(s_shape_growth,1)
sales_decrease = competitor_sale_increase

s_shape_growth = GRAPH(time )
(0.00, 0.00), (2.00, 1.50), (4.00, 4.50), (6.00, 10.5), (8.00, 17.5), (10.0,
30.0), (12.0, 50.5), (14.0, 75.0), (16.0, 91.0), (18.0, 97.5), (20.0, 100)

competitors_sales(t) = competitors_sales(t - dt) + (competitor_sale_increase)
* dt
INIT competitors_sales = 0

competitor_sale_increase =
delay(derivn(s_shape_growth,1), market_entry_delay)
market_entry_delay = 20 (time units)
"Phil Odence"
Junior Member
Posts: 15
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by "Phil Odence" »

John,

I believe there are some pretty simple explanations. Ill try to articulate
them below. If it would help, I could throw together a simple ithink model
which would do a better job than the words.

Phil Odence
HPS

A typical s-shaped growth model is structured as a customer stock built by a
gaining customers flow that is drawing down a potential customers stock.
Customers bring in new customers through word of mouth, thus exponential
growth. However, their productivity in bringing in their friends declines as
the potential stock declines. The behavior of the customer stock is an
s-shape.

However, when we talk about growth of a business, we are usually taking
about revenues, not customer base. Many businesses (product businesses for
example) gain revenue on the inflow (i.e. purchasing). If you generate
revenue in the model based on the inflow of new customers (times a price),
the revenue will appear to grow exponentially, level out and then decline to
an asymptote of 0. I think this is one of the behavior modes you were
looking for.

Many businesses also collect money from customers in their customer base
(software maint fees, add-on product sales, service, etc.) so you can add
another revenue component based on the stock (times an annual rev per
customer), you will see that for a small value of annual rev per customer
the revenue will decline but not to zero. For larger values, youll see more
of an s-shape. The value crudely represents how good the business is at
selling new stuff to their existing customers. If you saturate the market
and arent very good at selling to your existing customers, your business
declines.

This is all pretty simplistic. The next enhancement would probably be to
look at the loss of customers. Issues like service levels due to
undercapacity, etc. could drive that, all of which could contribute to a
decline in business. Certainly a hot competitor sniping your customers could
do it as well.

I hope this is helpful.
From: "Phil Odence" <
podence@hps-inc.com>
"j-d"
Junior Member
Posts: 5
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by "j-d" »

In reply to Johns thought-provoking questions on S-shaped growth followed
by a decline:

(First, for a realistic example of how this may happen, please check out the
story Jesse Liberty provides in his book "Clouds to Code - A Case Study in
Application Development with UML, Design Patterns, and C++". It is a
fascinating story of how Ziff Davis failed to capitalize on the same ideas
that made Mosaic/Netscape later a phenomenal success. A classic case of
overshoot and collapse - though John here is referring more to S-shaped
growth and then collapse).

John, could you perhaps provide more real-world examples from your own
experience.

To the 3 specific questions on how a SINGLE, SIMPLE structure may model the
S-shaped growth followed by an exponential decline, I have the following
proposal. It seems to work.

The typical sigmoidal S-shaped growth may be represented by the following
equation.

dN/dt = -r(1-N/K)N

where
N = population (or sales or revenues or spread of infectious cases etc.) [N0
= initial value]
r = intrinsic growth rate (ie, rate in the absence of environmental limits)
K = environmental carrying capacity, which N asymptotically approaches for
all N0 (>< K)
t = time

The above may be extended to

dN/dt = -r(1-N/K)(1-N/T0)N

where T0 represents a threshold level below which growth does not occur (all
N approach ZERO as t --> infinity when N0 > T0)

However, above T0, the usual sigmoid growth occurs with the limiting
environmental constraint set at K.

(Note: The above is a traditionally accepted logistic growth model with a
threshold - for application, see Oliver Austin, Birds of the World, 1983 - a
story of extinct passenger pigeons in America, where N0 was brought below T0
due to excessive hunting in the 1880s).

The above model captures the sigmoidal growth and exponential decay
SEPARATELY for various combinations of T0 and N0. To satisfy Johns
reference mode where BOTH must occur together, we will start with T0 < N0 <
K, and K sufficiently high (so that we dont reach environmental limit
before seeing anything interesting). This gives us a nice sigmoidal growth
approaching K.

NOW THE TRICK THAT SATISFIES THE EXP DECAY REQUIREMENT OF JOHNS REFERENCE
MODE - I assume that at a later time t* > t0, the threshold T changes from
T0 to T* where T* > "current value" of N. This is a feedback causal
connection existing at time t* > t0.

NOW, because the threshold T has changed from T0 < N0 to T* > (current) N >
N0, we have a situation of exponential decay right AFTER this sigmoidal
growth in ONE SIMPLE STRUCTURE/MODEL. In Powersim, SAMPLEIF function may be
used to effect this time-based change. The toy model that I made for trying
this shows exactly what John wanted - a sigmoid growth followed by exp
decay in a simple, single structure. Please check my URL below if you want
to download the model - it is really a toy model made only for
proof-of-concept.

The question as to where exactly the exp decay will start in this structure
(near K or much earlier than K, the concerns in questions 1 and 2 in Johns
email) can be easily handled by a suitable choice of parameters and initial
values.

As to what a change of T means in "real-world" terms: actually Johns email
itself provides an answer to this one - "Some of the most obvious apparent
causes of this include: displeasing,
disappointing, or irritating the market (poor service; failure to innovate;
product failures; PR disasters); entry of a competitor providing perceived
market value significantly better than yours; or failure to meet any of
several other typical challenges of growing a firm ETC ETC" The way I have
used and changed T reflects how entrepreneurical start-up companies find low
barriers to entry in the current US market (venture capital available,
frontier mentality, creative minds etc), BUT find big obstacles later -
please refer to Scott Adams Dilbert comics for examples here (I am
serious;-)

The above threshold change has to be empirically determined for particular
markets - what firms and industries should focus on is that WHATEVER THEY
DO, THEY MUST NOT GO BELOW THE "EVER-CHANGING" THRESHOLD T VALUE, which
leads to a downward death spiral (-ve feedback loop).

Ziff-Davis did in 1992-1993, Netscape did not - and the rest is history.

Regards,

Jaideep Mukherjee, Ph. D.
From: "j-d" <j-d@technologist.com>
713 523 2713
Virtual Office http://www.netopia.geocities.com/shunya/
"Jim Hines"
Senior Member
Posts: 88
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by "Jim Hines" »

John,

I think that Jay Forresters Market Growth as Influenced by Capital
Investment can do what you request and more.

Regards,
Jim Hines
From: "Jim Hines" <jhines@MIT.EDU>
Bill Braun
Senior Member
Posts: 73
Joined: Fri Mar 29, 2002 3:39 am

S Shaped growth followed by decline

Post by Bill Braun »

Hi John,

Im not sure this fits the bill but it does produce the curve you refer to.
Simple model to show the effect of marketing promotion. Run the model with
promotion taking place in the first 180 days, then stop all promotion for
the remaining 180 days. The "Promotional Effectiveness" variable produces
the curve you ask about. If you put PromoAwareness and PromoEffect in the
same graph you can see their interaction, the impact of delays and their
respective curves.

init TotalPromo = 0
flow TotalPromo = +dt*EffectiveDailyPromo
doc TotalPromo = Stock to capture total expenditures
unit TotalPromo = Dollars
aux EffectiveDailyPromo = MonthlyPromo/30
doc EffectiveDailyPromo = Feedback
unit EffectiveDailyPromo = Ratio
aux Budget = InputFrequencyPerMonth*PromoExpenditurePerTimePeriod
doc Budget = Total expenditures committed
unit Budget = Dollars
aux CumulativeAwareness = OrgMeanAwareness+PromoAwareness
doc CumulativeAwareness = Combines general org awareness with product or
service promotional awareness
unit CumulativeAwareness = Ratio
aux CumulativeEffect = PromoEffect+OrgMeanAwareness
doc CumulativeEffect = Combines general org awareness with product or
service buy consideration
unit CumulativeEffect = Ratio
aux ExpectedPromoImpact = Budget/(RequiredToMoveByOnePercent*100)
doc ExpectedPromoImpact = Expected impact of promotional activity
unit ExpectedPromoImpact = Ratio
aux MonthlyPromo = (PromoExpenditurePerTimePeriod*InputFrequencyPerMonth)
doc MonthlyPromo = Feedback; calculates total monthly expenditures
unit MonthlyPromo = Dollars
aux PromoActivity =
PULSE(ExpectedPromoImpact,1,(30/InputFrequencyPerMonth))*PromoThis_Period
doc PromoActivity = Actual promotional activity expenditures
unit PromoActivity = Dollars
aux PromoAwareness = MIN(1,DELAYINF(PromoActivity,AwarenessDelay,2,0))
doc PromoAwareness = Target audience awareness of promotional activity
unit PromoAwareness = Ratio
aux PromoEffect = DELAYINF(PromoAwareness,EffectDelay,2,0)*PurchaseRate
doc PromoEffect = Target audience buy decision consideration
unit PromoEffect = Ratio
const AwarenessDelay = 30
doc AwarenessDelay = Delay before target audience becomes aware of product
or service
unit AwarenessDelay = Days
const EffectDelay = 60
doc EffectDelay = Delay before target audience considers a buy decision
unit EffectDelay = Days
const InputFrequencyPerMonth = 30
doc InputFrequencyPerMonth = The number of promotional activities per month
(one equals once per month, two equals every two weeks, four equals every
week, etc.)
unit InputFrequencyPerMonth = Occurences
const OrgMeanAwareness = .5
doc OrgMeanAwareness = The mean awareness the target audience has of the
organization exclusive of the promotion for a particular good or service
unit OrgMeanAwareness = Ratio
const PromoExpenditurePerTimePeriod = 10000
doc PromoExpenditurePerTimePeriod = The promotional budget for the time period
unit PromoExpenditurePerTimePeriod = Dollars
const PromoThis_Period = 1
doc PromoThis_Period = Boolean to trigger promotional activity on or off
unit PromoThis_Period = Boolean
const PurchaseRate = .5
doc PurchaseRate = Rate at which target audience makes purchase decision at
point of purchase
unit PurchaseRate = Ratio
const RequiredToMoveByOnePercent = 5000
doc RequiredToMoveByOnePercent = Promotional expenditure required to move
purchase decisions by one percent
unit RequiredToMoveByOnePercent = Dollars
spec start = 0.00000
spec stop = 360.00000
spec dt = 1.00000
spec method = Euler (fixed step)

P.S. In PowerSim, DELAYINF(<input>,<delay>,<order>,<init>)
The Health Systems Group
- Physician Leadership Training
- Simulation Modeling for Healthcare
http://www.hlthsys.com
From: Bill Braun <medprac@HLTHSYS.COM>
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