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The Greiner Curve

Understanding and overcoming the crises of growth.

By the Mind Tools Content Team

Fast-growing companies can often be chaotic places to work.

As workloads increase exponentially, approaches which have worked well in the past start failing. Teams and people get overwhelmed with work. Previously-effective managers start making mistakes as their span of control expands. And systems start to buckle under increased load.

While growth is fun when things are going well, when things go wrong, this chaos can be intensely stressful. More than this, these problems can be damaging (or even fatal) to the organization.

Growth can be painful but you can make it easier by preparing. Learn how with our video and transcript .

The "Greiner Curve" is a useful way of thinking about the crises that organizations experience as they grow.

In this article, we'll learn more about the Greiner Curve, and how you can use it to understand the root cause of the problems you're likely to experience in a fast-growing business, as well as how to prevent them.

What Is the Greiner Curve?

The Greiner Curve (shown in figure 1, below) describes the different phases that organizations go through as they grow. All kinds of organizations – from design shops to manufacturers, construction companies to professional service firms – experience these.

Each growth phase is made up of a period of relatively stable growth, followed by a "crisis" when major organizational change is needed if the company is to carry on growing.

Figure 1. The Greiner Growth Model

business growth curve model

Reprinted by permission of Harvard Business Review . From " Evolution and Revolution as Organizations Grow " by Larry E. Greiner, May 1998. Copyright © 1998 by the Harvard Business School Publishing Corporation; all rights reserved.

Although the word "crisis" is often linked to a state of panic, it can also mean "turning point." While companies certainly have to change at each of these points, if they properly plan ahead, there is no need for panic, and so we will call them "transitions."

The Six Phases of Growth

Larry E. Greiner originally proposed the Greiner Curve (also known as the Greiner Growth Model) in 1972 with five phases of growth. In 1998, he added a sixth phase in an updated version of his original article. [1]

The six growth phases are described below:

Phase 1: Growth Through Creativity

Here, the entrepreneurs who founded the firm are busy creating products and opening up markets. There aren't many staff, so informal communication works fine, and rewards for long hours are probably through profit share or stock options.

However, as more staff join, production expands and capital is injected, there's a need for more formal communication .

This phase ends with a Leadership Crisis , where professional management is needed. The founders may change their style and take on this role, but often someone new will be brought in.

Phase 2: Growth Through Direction

Growth continues in an environment of more formal communications, budgets and focus on separate activities like marketing and production. Incentive schemes replace stock as a financial reward.

However, there comes a point when the products and processes become so numerous that there are not enough hours in the day for one person to manage them all, and they can't possibly know as much about all these products or services as those lower down the hierarchy.

This phase ends with an Autonomy Crisis in which new structures based on delegation are needed.

Phase 3: Growth Through Delegation

With mid-level managers freed up to react faster to opportunities for new products or new markets, the organization continues to grow. Meanwhile, top management focuses its efforts on monitoring and dealing with the big issues (perhaps starting to look at merger or acquisition opportunities).

Many businesses flounder at this stage because the manager whose directive approach solved the problems at the end of Phase 1 finds it hard to let go of the control they've assumed. This can mean that the mid-level managers begin to struggle with their roles.

This phase ends with a Control Crisis . A much more sophisticated organizational design is required, so the separate parts of the business can work together more effectively.

Phase 4: Growth Through Coordination and Monitoring

Growth continues with the previously isolated business units re-organized into product groups or service practices. Investment finance is allocated centrally and managed according to Return on Investment (ROI) and not just profits. Incentives are shared through company-wide profit share schemes aligned to corporate goals.

Eventually, though, work becomes submerged under increasing amounts of bureaucracy, and growth is stifled as a result.

This phase ends on a Red-Tape Crisis: a new culture and structure must be introduced.

Phase 5: Growth Through Collaboration

The formal controls of Phases 2-4 are replaced by professional good sense, as staff group and re-group flexibly in teams to deliver projects in a matrix structure which is supported by sophisticated information systems and team-based financial rewards.

This phase ends with a crisis of Internal Growth: further growth can only come by developing partnerships with external, complementary organizations.

Phase 6: Growth Through Extra-Organizational Solutions

Greiner's recently added sixth phase suggests that growth may continue through mergers, outsourcing, networks, and other solutions involving external companies.

Growth rates will vary between and even within phases. The duration of each phase depends almost totally on the rate of growth of the market in which the organization operates. The longer a phase lasts, though, the harder it will be to transition to the next phase of growth.

This is a useful model, however not all businesses will go through these crises in this order. Use this as a starting point for thinking about business growth, and adapt it to your circumstances.

Applying the Greiner Curve

The Greiner Growth Model helps you to think about your own organization's growth trajectory, and plan ahead so you can overcome each growth crises that affects it.

To apply this model, use the following five steps:

By thinking this through, you can start to plan and prepare yourself for the inevitable changes, and perhaps help others to do the same.

The Greiner Curve (also known as Greiner's Growth Model) was first developed by Larry E. Greiner. It illustrates six key phases of growth that organizations typically go through – from start-up phase to multinational corporation.

After each stage of growth, organizations tend to hit a crisis, which they must adapt and overcome to in order to continue to grow.

The six phases of growth are:

The Greiner Curve can help organizations to understand their own personal trajectory of growth, and to plan ahead more effectively, so that when they reach a growth crisis, they are in a better position to overcome it and continue to grow.


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What Is a Growth Curve?

Understanding a growth curve.

Growth Curve: Definition, How They're Used, and Example

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

business growth curve model

A growth curve is a graphical representation that shows the course of a phenomenon over time. An example of a growth curve might be a chart showing a country's population increase over time.

Growth curves are widely used in statistics to determine patterns of growth over time of a quantity—be it linear, exponential, or cubic. Businesses use growth curves to track or predict many factors, including future sales .

Key Takeaways

The shape of a growth curve can make a big difference when a business determines whether to launch a new product or enter a new market . Slow growth markets are less likely to be appealing because there is less room for profit. Exponential growth is generally positive but it also could mean that the market could see a lot of competitors.

Growth curves were initially used in the physical sciences such as biology. Today, they're a common component of social sciences as well.

Digital Enhancements

Advancements in digital technologies and business models now require analysts to account for growth patterns unique to the modern economy. For example, the winner-take-all phenomenon is a fairly recent development brought on by companies such as Amazon, Google, and Apple . Researchers are scrambling to make sense of growth curves that are unique to new business models and platforms.

Growth curves are often associated with biology, allowing biologists to study organisms and how these organisms behave in a specific environment and the changes to that environment in a controlled setting. This is used to help with medical treatments.

Shifts in demographics, the nature of work, and artificial intelligence will further strain conventional ways of analyzing growth curves or trends.

Analysis of growth curves plays an essential role in determining the future success of products, markets, and societies, both at the micro and macro levels.

Example of a Growth Curve

In the image below, the growth curve displayed represents the growth of a population in millions over a span of decades. The shape of this growth curve indicates exponential growth. That is, the growth curve starts slowly, remains nearly flat for some time, and then curves sharply upwards, appearing almost vertical.

This curve follows the general formula: V = S * (1 + R) t

The current value, V, of an initial starting point subject to exponential growth, can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of t, or the number of periods that have elapsed.

In finance, exponential growth appears most commonly in the context of compound interest.

The power of compounding is one of the most powerful forces in finance. This concept allows investors to create large sums with little initial capital. Savings accounts that carry a compounding interest rate are common examples.

What Are the 2 Types of Growth Curves?

The two types of growth curves are exponential growth curves and logarithmic growth curves. In an exponential growth curve, the slope grows greater and greater as time moves along. In a logarithmic growth curve, the slope grows sharply, and then over time the slope declines until it becomes flat.

Why Use a Growth Curve?

Growth curves are a helpful visual representation of change over time. Growth curves can be used to understand a variety of changes over time, such as developmental and economic. They allow for the understanding of the effect of policies or treatments.

What Is a Business Growth Model?

A business growth model provides a visual representation for businesses to track various metrics and key drivers, allowing businesses to map out growth and adjust the businesses accordingly to foster these metrics.

Curran, Patrick J., Obeidat, Khawla, and Losardo, Diane. " Twelve Frequently Asked Questions About Growth Curve Modeling: Abstract ." Journal of Cognition and Development , vol. 11, no. 2, 2010.

Sigirli, Deniz and Ercan, Ilker. " Examining Growth with Statistical Shape Analysis and Comparison of Growth Models ." Journal of Modern Applied Statistical Methods, vol. 11, no. 2, November 2012, pp. 1.

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Guide to Greiner’s Growth Model

Where do you sit on the growth curve and what risks do you face…🌱

5 flowerpots with different stages of growth to represent the phases in Greiner's Growth Model

Table of Contents

Growing your company is exciting, stressful, at times tiring, but always fun! During the growth you’ll encounter numerous crises that will jeopardise the success. As with most of life’s business problems, there’s a framework to explain or help map out this scenario…

Greiner's Growth Model

So, let’s take a look at Greiner's Growth Model.

What is Greiner’s Growth Model?

Greiner's Growth Model is a framework that shows the different phases a company goes through to achieve growth and the different types of crisis that may occur during those milestones.

Greiner's Growth Model

The graph shows time on the X axis and size of the business on the Y axis, with both increasing as the company goes through the different phases.

The model is helpful in showing companies the different approaches to growth, as well as highlighting the different challenges. It is commonly used by businesses to self-identify obstacles they are facing that will hamper their efforts to achieve their full potential.

What are the phases of growth in the Greiner’s Growth Model?

Let’s firstly look at the different phases a company goes through based on this model.

Growth Through Creativity

All businesses start from a spark of an idea, one that is fostered and developed over time. It’s a truly creative stage of a company's life as they attempt to develop a new product or service, pull together a team, a route to customers, and get that sometimes elusive 'product-market fit'.

There are some common traits of companies at this stage:

Growth Through Direction

At this point in a company's life the owner/founders begin to hire managers, releasing some of the control of the resources and direction of the business. This is normally a 'growing up' period of a company's life, when processes become slightly more formal, departments may be developed, and a culture is set within the business. That’s not to say the founders/owners aren’t still actively involved, they are indeed ultimately running the company, but it’s a collaboration of managers that drive the direction.

A company can be considered in the Growth Through Direction phase if:

Growth Through Delegation

The Delegation phase of growth occurs when key staff members are given accountability and responsibility to deliver in areas where they are better equipped to than the manager. At this point in a company life there will be specialist employees, focused on specific roles.

Delegating jobs to more specialist, skilled employees means you’ll get a better result, with the added benefit that the executive team have time to focus on the market data, their strategic decisions, and business planning.

A company may be in this phase if:

Growth Through Coordination

This is now a mature stage of growth, one that focuses on the company core competencies and all departments working in line with each other to output a product or service. Growth comes from the whole business being greater than the sum of its parts.

Growth Through Collaboration

The final stage of growth in this model is deemed to be Collaboration. This is an evolution of Coordination, one where all parts of the company work together in a trusted, effective manner. Systems are simplified for efficiency, learning and development is prominent, and all aspects of the business contribute towards ways to continue success.

Growth Through Alliances

The final stage of growth is a new one introduced more recently to the curve, and it focuses on strategic alliances. The idea being that companies may merge, acquire, partner or work with other companies in order to grow themselves.

Are the phases of growth linear in the Greiner’s Growth Model?

The model suggests that is the case, but in real life it is not necessarily always linear. For example, a start-up company focused on Direction may also embark on Strategic Alliances. It’s important to note the real importance and value of this model lies in the crises that may impact a company at the different stages… so let’s take a look at those.

What are the crises in the Greiner’s Growth Model?

Each phase in the Growth Model has an associated potential crisis that may disrupt the trajectory of growth.

Crisis of Leadership occurring during Creativity

This is a common issue for start-ups and young companies that find themselves growing via creativity and innovation. Initially with a small and informal team it’s possible for founders to manage the business in a relaxed manner, but over time this becomes a challenge.

Growth will lead to increasing difficultly around coordinating processes, communicating and motivating the team, or driving the company forward. This can be fatal for a business as it can result in key people departing (remember, people leave managers as much as they leave jobs) and founders becoming increasingly frustrated.

At this point a more defined management style is required in the company to take it to the next level.

Crisis of Autonomy occurring during Direction

This is a really interesting crisis. As a company develops in direction then managers may become more interested in their own unit than the business as a whole. This can result in conflict between management where a decision may be good for one department or area but bad for another.

The balance to strike is giving managers and employees autonomy but ensuring everyone is on the same page around decision that are best for the business as a whole. Ensuring everyone is on the same page around their strategy is key in that balance.

Crisis of Control occurring during Delegation

The crisis around the Delegation phase can be summed up with two factors:

The latter can sometimes be a reason to reinforce the behaviour of the former, with founders citing concerns that if they do not do something it won’t be done well. It ultimately will result in a sub-par outcome though, with founders struggling to ‘do everything’ and teams feeling unmotivated.

Crisis of Red Tape occurring during Coordination

Another very relatable crisis within the life of a company is that of 'Red Tape' or bureaucracy. The addition of extra reports, processes, functions, all of which contribute to additional work for employees and can risk the wider culture of the business.

This can slow down decision making, resulting in a less agile company that cannot respond to market changes while also suffering a wider loss of efficiency/reduced margins.

Of course, this is a risk at all points in a company life, but it has more chance of arising when coordination is required and thus processes are needed within a company.

Crisis of Growth occurring during Collaboration or Alliances

The final crisis is one of how to grow. In the framework we’ve moved through each phase, so the company is now successful and mature. The question becomes how does it continue to grow, given the success?

If you’re in Collaboration, then perhaps Alliances are your way forward. If you are already developing partnerships then perhaps diversification is the route to growth? There are lots of potential options here, it’s a good point to evaluate your industry and develop a new strategy.

What are the advantages of Greiner’s Growth Model?

There are lots of advantages to this model including:

What are the disadvantages of Greiner’s Growth Model?

The few limitations of this model include:

What frameworks go well with Greiner’s Growth Model?

As a company using Greiner’s Growth Model you may also want to use a SWOT Analysis, which should include strengths & weaknesses from this model.

Who invented Greiner’s Growth Model?

The Greiner's Growth Model was invented by Larry E. Greiner in 1972 with the five phases of growth. In 1998 he updated the model to add the sixth phase around Alliances.

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Greiner's Growth Model

Last updated 25 Mar 2021

Greiner's Growth Model attempts to predict the six phases and five crises that businesses may experience as they grow.

The phases of the Greiner Growth Model are illustrated below:

business growth curve model

The five predicted crises of growth according to the model are:

Growth Phase: Direction - Crisis of Leadership

Growth Phase: Delegation - Crisis of Autonomy

Growth Phase: Coordination - Crisis of Control

Growth Phase: Collaboration - Crisis of Red Tape

Growth Phase: Alliances - Crisis of Growth

Key Messages from Greiner's Growth Model

What can we learn about the challenges of growing a business if, for a moment, we assume that Greiner's Growth Model is valid?

Criticisms of Greiner's Growth Model

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Twelve Frequently Asked Questions About Growth Curve Modeling

Longitudinal data analysis has long played a significant role in empirical research within the developmental sciences. The past decade has given rise to a host of new and exciting analytic methods for studying between-person differences in within-person change. These methods are broadly organized under the term growth curve models . The historical lines of development leading to current growth models span multiple disciplines within both the social and statistical sciences, and this in turn makes it challenging for developmental researchers to gain a broader understanding of the current state of this literature. To help address this challenge, the authors pose 12 questions that frequently arise in growth curve modeling, particularly in applications within developmental psychology. They provide concise and nontechnical responses to each question and make specific recommendations for further readings.

A foundational goal underlying the developmental sciences is the systematic construction of a reliable and valid understanding of the course, causes, and consequences of human behavior. Consistent with this goal, longitudinal studies have long played a critically important role in developmental psychology, and these designs are becoming increasingly common in contemporary research practices. However, consistent with the old adage be careful for what you ask — you might just get it , once longitudinal data are obtained, they must then be thoughtfully and rigorously analyzed. And as any developmental researcher can attest, statistical models for longitudinal data can become exceedingly complex exceedingly quickly, both in terms of fitting models to data and properly interpreting results with respect to theory (e.g., Curran & Willoughby, 2003 ; Nesselroade, 1991 ; Wohlwill, 1991 ). Further, during the past decade, a host of powerful analytic methods have been developed that allow for the empirical evaluation of theoretically derived research hypotheses in ways not previously possible. Given the rapid onslaught of new methods, it can often be a significant challenge for researchers to stay abreast of ongoing developments and to incorporate these new techniques into their own programs of research. As quantitative psychologists who conduct substantive programs of research, we feel these very same pressures ourselves.

In an attempt to help organize the constantly shifting sands of new information, we have posed 12 specific questions that frequently arise with respect to growth curve modeling. We are under tight space constraints, so our rather modest intent is to provide brief and nontechnical responses to these questions and to recommend specific resources for further reading. The questions we pose are by no means exhaustive nor are our associated responses. Importantly, given our quest for brevity, we offer only a subset of available citations; the inclusion of one citation at the expense of another should be taken to mean nothing more than that we ran out of space. We hope that our brief foray through the intriguing yet sometimes bewildering topic of growth modeling might entice readers to consider ways in which these approaches might be incorporated into your own program of research. So let’s give it a go.


Growth curve modeling is a broad term that has been used in different contexts during the past century to refer to a wide array of statistical models for repeated measures data (see Bollen, 2007 , and Bollen & Curran, 2006 , pp. 9–14, for historical reviews). However, within the past decade or so, this term has primarily come to define a discrete set of analytical approaches, particularly as applied within the social sciences. More specifically, the contemporary use of the term growth curve model typically refers to statistical methods that allow for the estimation of inter-individual variability in intra-individual patterns of change over time (e.g., Bollen & Curran, 2006 ; Browne & du Toit, 1991 ; McArdle, 2009 ; Preacher, Wichman, MacCallum & Briggs, 2008 ; Raudenbush & Bryk, 2002 , pp. 160–204; Singer & Willett, 2003 ). In other words, growth models attempt to estimate between-person differences in within-person change. Often these within-person patterns of change are referred to as time trends, time paths, growth curves , or latent trajectories . These trajectories might take on a variety of different characteristics that vary from person to person: They might be flat (i.e., showing no change over time), they might be systematically increasing or decreasing over time, and they might be linear or curvilinear in form. In many applications, the trajectories are the primary focus of analysis, whereas in others, they may represent just one part of a much broader longitudinal model.

The most basic growth model is composed of the fixed and random effects that best capture the collection of individual trajectories over time. Loosely speaking, a fixed effect represents a single value that exists in the population (e.g., the population mean height for men), and a random effect represents the random probability distribution around that fixed effect (e.g., the population variance in height for men). Consistent with these definitions, in the growth model, the fixed effects represent the mean of the trajectory pooling of all the individuals within the sample, and the random effects represent the variance of the individual trajectories around these group means. For example, for a linear trajectory, the fixed effects are estimates of the mean intercept (i.e., starting point) and mean slope (i.e., rate of change) that jointly define the underlying trajectory pooling of the entire sample; in contrast, the random effects are estimates of the between-person variability in the individual intercepts and slopes. Smaller random effects (i.e., smaller variances of intercepts and slopes) imply that the parameters that define the trajectory are more similar across the sample of individuals; at the extreme situation where the random effects equal 0, all individuals are governed by precisely the same trajectory parameters (i.e., there is a single trajectory shared by all individuals). In contrast, larger random effects (i.e., larger variances of intercepts and slopes) imply that there are greater individual differences in the magnitude of the trajectory parameters around the mean values; that is, some individuals are reporting higher or lower intercepts, or steeper or less-steep slopes relative to others. Taken together, the fixed and random effects capture the general characteristics of growth for both the group as a whole and for the individuals within the group.


There is a long and rich history in the analysis of repeated measures data, and many methods have been proposed for use within the social sciences. Key traditional approaches include repeated measures analysis of variance and multivariate analysis of variance, as well as various methods for analyzing raw and residualized change scores (see Hedeker & Gibbons, 2006 , chaps. 2 and 3, for a review). The history of these methods has at times been quite contentious with strongly worded recommendations supporting or refuting particular approaches (e.g., Cronbach & Furby, 1970 ; Rogosa, 1980 ; Rogosa & Willett, 1985 ). Despite the disagreements over the use of one approach over another, growth models differ from traditional methods in several key respects. Most importantly, current approaches to growth modeling are highly flexible in terms of the inclusion of a variety of complexities including partially missing data, unequally spaced time points, non-normally distributed or discretely scaled repeated measures, complex nonlinear or compound-shaped trajectories, time-varying covariates (TVCs), and multivariate growth processes. All of these issues routinely arise in developmental research, yet all present significant challenges within traditional analytic approaches. Further, both analytical and simulation results show that growth models are typically characterized by much higher levels of statistical power than comparable traditional methods applied to the same data (e.g., B. O. Muthén & Curran, 1997 ). To stress, traditional methods for analyzing repeated measures data remain a powerful tool in many research applications when the underlying assumptions are met. However, these methods become increasingly limited under conditions commonly encountered in social science research, whereas growth models typically are not.


There are two general approaches used to fit growth models to observed data that share certain similarities but are also characterized by certain distinct differences (e.g., Bauer, 2003 ; Curran, 2003 ; Raudenbush, 2001 ; Willett & Sayer, 1994 ). The first approach is to fit the growth model within the multilevel modeling framework ( Bryk & Raudenbush, 1987 ; Raudenbush & Bryk, 2002 ; Singer & Willett, 2003 ). The multilevel model was originally developed to allow for the nesting of multiple individuals within a group, such as children nested within classroom or siblings nested within family. However, the model can equivalently be applied to multiple repeated measures nested within each individual that allows for the direct estimation of a variety of powerful and flexible growth models. The second approach is to fit the growth model within the structural equation modeling (SEM) framework (e.g., Bollen & Curran, 2006 ; Duncan, Duncan, & Strycker, 2006 ; McArdle, 1988 ; McArdle & Epstein, 1987 ; Meredith & Tisak, 1990 ). The SEM incorporates the observed repeated measures as multiple indicators on one or more latent factors to characterize the unobserved growth trajectories. In many situations, the multilevel and SEM approaches to growth modeling are numerically identical, yet in others, there are important differences. For example, the multilevel model naturally expands to estimate higher levels of nesting (e.g., repeated measures nested within child, and child nested within classroom); the SEM approach is currently more limited in these situations. In contrast, the SEM is well suited to the estimation of latent variables that estimate and remove the effects of measurement error that might exist in the predictors or the outcomes; the multilevel model is currently more limited with respect to the estimation of comprehensive measurement models. However, the similarities between the multilevel and SEM approaches often outweigh the differences, and the optimal approach should be selected as a function of the particular research application at hand ( Raudenbush, 2001 ).


Although there are few strict requirements for the types of data that might be analyzed using growth models, there are a number of general data characteristics that are particularly amenable to these methods. First, an adequate sample size is needed to reliably estimate growth models. However, what constitutes “adequate” cannot be unambiguously stated, because this depends in part on other characteristics of the research design (e.g., complexity of the growth model, amount of variance explained by the model). For example, growth models have successfully been fitted to samples as small as n = 22 ( Huttenlocher, Haight, Bryk, Seltzer, & Lyons, 1991 ), although sample sizes approaching at least 100 are often preferred. Further, there is a close relation between the number of individuals and the number of repeated observations per individual (e.g., B. O. Muthén & Curran, 1997 ); as such, the total number of person-by-time observations plays an important role in model estimation and statistical power as well. Second, growth models typically require at least three repeated measures per individual, although this requirement is also rather vague. For example, in the presence of partially missing data, some individuals might have just one or two observations, whereas others have three or more. However, three repeated measures over-identifies a linear trajectory (that is, there is more observed information than estimated information) and is thus preferred for at least a sizeable portion of the cases. Third, for the typical method of estimation called maximum likelihood (ML), it is assumed that the repeated measures are continuous and normally distributed. However, alternative methods of estimation allow for measures that are continuous and non-normally distributed ( Satorra, 1990 ) or even discretely or ordinally scaled (e.g., Mehta, Neale, & Flay, 2004 ). In sum, growth models may be fitted to many types of sample data structures, although care must be taken in the selection of proper models and methods of estimation that maximally correspond to the characteristics of the given data set.


Growth models can be estimated in the presence of partially missing data, although certain assumptions regarding the mechanism of “missingness” must be invoked for valid results. There are two general approaches to estimating models with partially missing data ( Allison, 2001 ; Schafer, 1997 ; Schafer & Graham, 2002 ). The first is direct ML ( Arbuckle, 1996 ; Little & Rubin, 1987 ). Under direct ML, the growth model is estimated by summing the individual contributions of each case such that observations with a larger number of data points are weighted more heavily than observations with a smaller number of data points. The second approach is called multiple imputation, and the growth model is estimated in a two-stage sequence ( Rubin, 1987 ; Schafer, 1999 ). In the first stage, the missing data points are imputed based upon the characteristics of the non-missing data points, and this is done multiple times (typically 5 to 10 times). In the second stage, the growth model is fitted separately to each of the imputed data sets, and the results are pooled into a final set of estimates. Although extremely flexible, both approaches invoke explicit assumptions about the nature of missing data. Specifically, the missing data must be characterized as missing completely at random (e.g., cases are truly missing at random) or missing at random (e.g., cases are missing as a function of measured characteristics such as gender or ethnicity). Importantly, data that are missing not at random (e.g., cases are missing as a direct function of unmeasured characteristics such as the very value that is missing) cannot be included in standard growth modeling applications, and much more complex procedures are required (e.g., Heckman, 1976 ; Rubin, 1988).


A critically important first step in any growth model is the identification of the optimal functional form of the trajectory over time; that is, it must be established exactly how the repeated measures change as a function of time. If the incorrect functional form is used as the basis for the initial growth model, then expanding this model to include complexities such as predictors of growth or multiple group analysis will likely lead to biased results. The most basic form of growth is a random intercept-only model; this implies that there is a stable overall level of the repeatedly measured construct over time and individuals vary randomly around this overall level at any given time point. It may seem an oxymoron to call an intercept-only model “growth,” but this is consistent with the notion of a trajectory that is simply flat with respect to time. This intercept-only model can then be expanded in a variety of directions. The most straightforward method is to consider the family of polynomial functions; examples include a straight line, a quadratic curve, and a cubic curve. Polynomials are widely used given that these can be easily estimated within either the SEM or multilevel frameworks. Other more complex functional forms are possible including entire families of interesting exponential trajectories (e.g., monomolecular, logistic; Cudeck & Harring, 2007 ). However, a variety of complications arise when incorporating these types of trajectories, because the parameters enter the model nonlinearly, making model estimation substantially more difficult, if not at times impossible. A flexible alternative is to use piecewise linear modeling to approximate complex nonlinear functions in which two or more linear trajectories are joined together to correspond to a potentially intractable nonlinear function ( Bollen & Curran, 2006 , pp. 103–106; Raudenbush & Bryk, 2002 , pp. 178–179; Singer & Willett, 2003 , pp. 207–208). A final option is a fully latent curve model available within the SEM framework in which some or all of the loadings on the slope factor are freely estimated so that change optimally corresponds to the unique characteristics of the data under study ( McArdle, 1988 ; Meredith & Tisak, 1990 ).


It is as essential to establish the adequate fit of the hypothesized model within the growth modeling framework as it is in any other statistical model (but see Coffman & Millsap, 2006 , for an alternative perspective). How this is best done directly depends upon the specific analytic strategy used to estimate the growth models. Within the SEM, it is possible to judge the fit of a hypothesized model relative to a saturated baseline model allowing for the estimation of standalone indices of overall fit for a given model. Examples include the model chi-square test statistic and fit indices such as the RMSEA (root mean squared error of approximation), CFI (comparative fit index), and TLI (Tucker-Lewis index), among many others. Within the multilevel framework, it is not possible to estimate a saturated baseline model to which to compare the hypothesized model. As such, there are no standalone measures of overall fit for a hypothesized model (although other indices of appropriate fit can be used such as residuals and Wald tests). Instead, comparisons of competing alternative models are required (which we believe is a strategy that could be used to a much greater extent within the SEM framework). If two comparison models are nested (i.e., if the parameters of one model are a direct subset of the parameters of the second model), then formal likelihood ratio tests can be calculated based on the differences between model deviance (see, e.g., Raudenbush & Bryk, 2002 , pp. 283–284). For models that are not nested, informal comparisons can be made using indices such as the Bayesian Information Criterion or the Akaike Information Criterion to rank order models (e.g., Bollen & Long, 1993 ). Regardless of approach, it is extremely important that clear evidence be presented that supports the adequacy of fit of the hypothesized model to the observed data prior to drawing theoretical inferences from the results.


Once the optimal baseline growth model has been established, this can then be expanded to include one or more predictors of growth. The inclusion of predictors in the model results in what is often called a conditional growth model because the fixed and random effects are now “conditioned on” the predictors. There are generally two types of predictors to consider: time-invariant covariates (TICs) that do not change in value as a function of time and TVCs that at least in principle can change as a function of time. TICs typically predict the random components of growth directly with the goal of determining what variables are associated with individuals who report higher versus lower intercepts or steeper versus flatter slopes. For example, say that a linear trajectory is deemed to be the optimal functional form over time, and there is evidence of significant random effects in both the intercept and slope components of the trajectory. TICs can then be incorporated to predict this random variability in starting point and rate of change. This would directly evaluate hypotheses about whether characteristics of the individual (e.g., gender, treatment condition) are predictive of higher or lower starting points or steeper or less steep rates of change over time (e.g., Curran, Bauer & Willoughby, 2004 ).

Importantly, TICs are assumed to be independent of the passage of time. In other words, the given value of the TIC could in principle be assessed at any time point as this is constant over time. This assumption is sometimes strictly true (e.g., biological sex, country of origin), and at other times, the construct might in principle vary with time but is only assessed at a single time period (e.g., baseline anxiety or initial reaction time). However, growth models can easily be expanded to include the effects of covariates that do vary as a function of time; these are TVC models ( Bollen & Curran, 2006 , pp. 192–198; Raudenbush & Bryk, 2002 , pp. 179–186; Singer & Willett, 2003 , pp. 159–188). Whereas TICs directly predict the growth factors (e.g., Bollen & Curran, 2006 , Figure 5.1), TVCs directly predict the repeated measures while controlling for the influence of the growth factors (e.g., Bollen & Curran, 2006 , Figure 7.1). Thus, any given repeated measure is jointly determined by the underlying growth factors and the impact of the TVC at that time period. The TVC model can then be expanded to include interactions between the TVCs and time (to assess differences in the magnitude of the TVC effect as a function of time) and interactions between the TVCs and the TICs (to assess differences in the magnitude of the TVC effect as a function of between-person characteristics such as gender or ethnicity). Taken together, models can be constructed that simultaneously evaluate within-person influences (via TVCs) and between-person influences (via TICs) on stability and change of the outcome over time.


Although the TVC model allows for covariates to change in value over time, it is assumed that the covariates themselves are not characterized by a systematic growth process. For example, say that the repeated outcome was reading ability, and the TVC was number of days of instruction that were missed in a given academic year. It would be reasonable to assume that days of instruction may influence reading ability at a given time point but that there is not a systematic growth process underlying days of instruction missed (that is, children would not be expected to show consistent developmental trends in days missed). However, say instead that the outcome was again reading ability, but the TVC is substance use; in this case, developmental theory would predict that the onset and escalation of substance use itself is characterized by some type of systematic growth function. If so, then the TVC model may be mis-specified and result in biased effects.

Both the multilevel and SEM growth frameworks can be expanded to allow for the simultaneous growth of two constructs over time, and this is commonly called a multivariate growth model ( Bollen & Curran, 2006 , chap. 7; MacCallum, Kim, Malarkey, & Kiecolt-Glaser, 1997 ; McArdle, 1988 ). Each construct can be characterized by a unique functional form (e.g., one may be linear, the other quadratic), and their relation is examined at the level of the growth factors (e.g., direct estimates of the relation between the intercepts and slopes within and across construct). Finally, these multivariate models can themselves be extended to include one or more TICs to predict the set of growth factors. There are several variations of the multivariate model that attempt to simultaneously examine bidirectional effects between two constructs both at the level of the growth trajectories and at the level of repeated measures. Two examples include the latent difference score model ( McArdle, Ferrer-Caja, Hamagami, & Woodcock, 2002 ) and the autoregressive latent trajectory model ( Bollen & Curran, 2004 ; Curran & Bollen, 2001 ), although several other approaches exist as well. The systematic study of the bidirectional relation between two or more constructs is a topic of much ongoing research, so we can expect additional multivariate methods to become available soon.


It is important to realize that when estimating the growth models described thus far, strong assumptions are made about the equivalence of the model parameters across all individuals within the sample (e.g., Bollen & Curran, 2006 , chap. 6). As a simple example, consider fitting a model to data that consist of responses from males and females. If an unconditional growth model is fitted to the pooled sample (i.e., the usual single-group analysis), it is explicitly assumed that all of the parameters that define the growth model are precisely equal for both gender groups. If gender differences were hypothesized, the growth model can easily be expanded to include gender as a time-invariant predictor; however, this only introduces differences in the conditional means of the growth factors (e.g., on average, males may start higher or lower compared with females and increase more or less steeply). Gender thus serves to shift the conditional means of the intercept and slope to higher or lower values, yet all other parameters that govern the model are assumed to be equal between the two groups.

Whereas in many situations these assumptions are perfectly reasonable, in others, they may be distinctly questionable. For example, a potential outcome of a treatment intervention is to decrease variability in the expression of certain behaviors within the treatment group but not the control group over time (e.g., an intervention designed to decrease antisocial behavior in preschool children will also likely decrease the variability of types of disruptive behavior in the children exposed to the treatment). If these estimates of variability are markedly different across groups, yet a model is fitted that assumes these to be the same, then biased parameter estimates are expected. Both the SEM and multilevel approaches address this issue through the simultaneous estimation of growth models across two or more groups in what are called multiple-groups growth models. If all model parameters are set equal across all groups, this is equivalent to estimating a single-group growth model. Alternatively, if all parameters are allowed to freely vary across all groups, this is equivalent to estimating a growth model within each group separately. The typical application will fall somewhere between these two extremes in which some parameters are equated and others are not. This framework provides yet another option for maximally understanding growth processes both within and across groups.


In the multiple-groups growth model described above, any grouping variable of interest must be directly observed within the data. That is, groups based upon biological sex, treatment condition, or ethnic heritage must be unambiguously identified for each observation in the data set. This group identification measure is used to assign each case to its associated group, and the growth models are then simultaneously fit to the set of groups. However, there may be situations in which it is hypothesized that two or more groups exist in the sample, yet the grouping variable was not directly observed. For example, when studying lifetime trajectories of delinquent behavior, developmental theory may dictate that specific subgroups exist that are indirectly defined by the pattern of behavior over development, and thus, group membership is not an observed variable in the data set (e.g., Moffitt, 1993 ). That is, there is some latent group that was not directly observed yet whose existence must be estimated from the characteristics of the data. There has been a flurry of recent developments in the estimation of models such as these, and a number of terms are used to describe these types of models. Examples include growth mixture models, latent class growth models, and semi-parametric groups-based trajectory models, among others (e.g., B. O. Muthén, 2004 ; B. O. Muthén & Shedden, 1999 ; Nagin, 2005 ). These techniques are being applied with increasing frequency in many areas of developmental research including the study of criminology, alcohol use, parenting, and reading difficulties (e.g., Boscardin, B. Muthén, Francis, & Baker, 2008 ; B. O. Muthén & L. K. Muthén, 2000 ; Nagin & Land, 1993 ; Stoolmiller, 2001 ). Importantly, a number of nontrivial differences exist across these various approaches, and care must be taken in selecting the optimal strategy for a given research application. Further, although growth mixture models are both intriguing from a theoretical perspective and powerful from an analytical one, a number of concerns have been identified about the use of these techniques in practice (e.g., Bauer, 2007 ; Bauer & Curran, 2003 , 2004 ). As with any statistical procedure, it must be clearly established that the growth mixture model is the most appropriate analytical approach available for testing the specific research hypotheses at hand.


We hope that we have been able to help guide you through at least an initial foray into the exciting collection of growth curve models that can be used with great effectiveness in many areas of developmental research. A logical final question is: Where does one go from here? An initial step is to turn to existing written work in this area. First, there are a number of more pedagogically oriented papers that walk the reader through different aspects of the application and interpretation of growth models; examples include Curran (2000) , Curran and Hussong (2002 , 2003) , Duncan and Duncan (2004) , Preacher et al. (2008) , Singer (1998) , and Willett, Singer, and Martin (1998) . Second, there are several recently published textbooks that cover more comprehensive aspects of these techniques; examples include Bollen and Curran (2006) , Duncan et al. (2006) , Hedeker & Gibbons (2006) , Raudenbush & Bryk (2002) , and Singer and Willett (2003) . Finally, there are a growing number of quality applications of various types of growth models within the developmental sciences; several recent examples include Brown, Meadows, and Elder (2007) , McCoach, O’Connell, Reis, and Levitt (2006) , Owens and Shaw (2003) , and Williams, Conger, and Blozis (2007) . Next, there are many well-developed online resources available that provide fully worked examples with empirical data and associated computer code; specific Web site addresses come and go, so the best strategy is to enter relevant terms in any major search engine and proceed from there. Finally, there are an increasing number of workshops available around the country that are focused on the theory and application of growth modeling within the social sciences; again, specifics change with time, but a bit of careful online searching will provide a current summary of available workshops. And if all else fails, send one of us an e-mail and we’ll try to point you in the right direction.

We have only briefly touched on just a few of the many interesting topics associated with the potential for growth models to help us gain a better understanding of individual differences in developmental change. Important remaining issues include growth models with binary or discrete outcomes (e.g., Mehta et al., 2004 ), incorporating alternative metrics of time (e.g., Mehta & West, 2000 ), using growth trajectories as predictors (e.g., B. O. Muthén & Curran, 1997 ; Seltzer, Choi, & Thum, 2003 ), estimating statistical power for growth models (e.g., B. O. Muthén & Curran; L. K. Muthén & B. O. Muthén, 2002 ), and the estimation of hybrid autoregressive and change score models ( Bollen & Curran, 2004 ; McArdle, 2001 ). Growth models offer a plethora of exciting opportunities for testing theoretically derived hypotheses in ways not previously possible. Despite the strength and flexibility of these methods, even greater care must be taken to ensure that the estimated growth model maximally corresponds to the underlying developmental theory (e.g., Curran & Willoughby, 2003 ). Any disjoint that exists between the theoretical model and the statistical model only serves to undermine our ability to draw empirically informed conclusions about our theory under study. Despite this caveat, growth models have a tremendous amount to offer to a broad array of developmental research endeavors and represent a powerful set of tools to help us continue to propel forward as a science.


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S-Curve In Business And Why It Matters

The S-Curve of Business illustrates how old ways of doing business mature and then become superseded by newer ways. The S-Curve itself is based on a mathematical concept called the Sigmoidal curve. In the context of business , the curve graphically depicts how an organization grows over a typical life cycle.

Understanding the S-Curve of Business

A key argument of the curve is that sooner or later, most businesses will reach a period of stagnation – no matter how successful they were in the past.

At the point of stagnation, the business reaches an inflection point.

At this point, it will be forced to innovate to grow and remain competitive.

For executives, understanding where their business lies along the S-Curve is crucial.

If the business has already reached an inflection point – also referred to as a “stall point” – it has less than a 10% chance of fully recovering.

In the next section, we’ll discuss these terms at various points of the life cycle in more detail.

The stages of the S-Curve life cycle

Initially, start-up companies begin at the bottom of the curve with a product or service they are taking to market. 

If they are lucky, their offering gains traction – albeit very slowly at first and then gradually quickening as more consumers become aware.

This is the first inflection point, where sales and revenue increase rapidly after an initial period of stagnation or low growth .

While growth will continue for some time, a host of internal or external factors will eventually cause growth to decrease and then taper off.

These factors include:

Here, the business encounters the second inflection point. At this point, a critical decision must be made.

For the growth curve to start anew and begin trending upward, the business must innovate and ride the wave of technological advancement.

Ultimately, a business at the second inflection point that then tries to innovate is already too late.

Inflection points must be identified before they occur so that businesses have adequate time to develop new products that have a high chance for success.

How do businesses commonly reach stall points?

External factors

Most businesses will find it hard to maintain growth during recessions since consumers are spending less.

When state or federal laws are enacted to regulate or ban certain products or services, businesses must have the ability to pivot quickly.

This is particularly prevalent in technology where trends shift quickly.

Examples of companies unknowingly reaching inflection points because of technology include Nokia, Blackberry, Xerox, and Kodak.

Internal factors

Dilution of focus.

Many start-ups have visionary leaders whose sole intent is to serve their customers well.

But when companies become larger, focus and effort can become diluted – particularly as management becomes more convoluted.

Talent shortage

For whatever reason, some companies are hindered in their growth because they cannot source the required talent to make it happen.

Examples of S-Curve

Population growth of a country.

As a country’s population grows, the growth rate typically builds momentum slowly.

Yet it accelerates during the middle of the S-curve while leveling off as the population reaches its maximum capacity.

The adoption of a new technology

When a new technology is introduced, it might take time before this technology becomes adopted by the masses.

In the initial stage of the technology adoption curve , its path it’s very steep. Yet when it does take off, it does that very quickly.


Thus, here the slowly then suddenly saying works exceptionally well.

business growth curve model

As the adoption rate increases rapidly, thus enabling technology to reach the masses, it eventually reaches a plateau as the technology won’t have any more market penetration.

An example is how smartphones took off and how today, they have become a saturated market, as there are billions of smartphones across the world.

The evolution of a market

Take the example of the iPhone; when it was launched, it didn’t pick up right on.

Indeed, Apple first launched the iPhone in 2007, and only when by 2008, when Apple launched the App Store in combination with the iPhone, the store worked as a jet engine for the iPhone to take off very quickly.


Yet, Apple’s iPhone success was built on the premise that the smartphone market had already been developed by other players like BlackBerry.

Thus, Apple wasn’t a first mover, but when it did enter the market, it took off very quickly.

Key takeaways

Read Next: Business Model Innovation , Business Models .

Related Innovation Frameworks

Business Engineering


Business Model Innovation


Innovation Theory


Types of Innovation


Continuous Innovation


Disruptive Innovation


Business Competition


Technological Modeling


Diffusion of Innovation


Frugal Innovation


Constructive Disruption


Growth Matrix


Innovation Funnel


Idea Generation


Design Thinking


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What Is The Experience Curve And Why It Matters In Business

About The Author

business growth curve model

Gennaro Cuofano


An Entrepreneur’s Guide to Surviving the “Death Valley Curve”

business growth curve model

Two key questions: Do you have the right business model, and do you have growth ambitions?

The so-called “death valley curve” represents a crucial early phase of new ventures, when substantial work on a new enterprise has begun but no sufficient revenue has been generated. During this period, companies deplete their initial capital in their quest to establish the business. To help navigate this tricky time, the authors have created a matrix with four phases of new entrepreneurial ventures and the strategic challenges in each phase.

According to recent estimates, around 90% of start-ups fail . With the global start-up economy valued at $3 trillion , much is at stake.

Our research has focused on a crucial initial phase of new ventures, known as “ the death valley curve ,” when substantial work on a new enterprise has begun but no sufficient revenue has been generated. During this period, companies deplete their initial capital in their quest to establish the business.

How do successful companies navigate this tricky period? The steps entrepreneurs should take depend on the strategic situation in which they find themselves. We have identified four phases of the death valley curve and created a matrix on which entrepreneurs can place their business to identify the key challenges going forward.

Our matrix is based on two key challenges that all new ventures face: 1) Do they have the right business model? and 2) Do they have growth ambitions?

To determine whether they have the right business model, entrepreneurs should use the two business-model tests suggested by Joan Magretta: the narrative test and the numbers test. An enterprise passes the narrative test when there is logic and alignment in the business model — in other words, when the story of the business model makes sense. The numerical test focuses on the financial performance of the business model and whether that business model can produce a profit. When turnover exceeds costs, the numerical test is passed.

So-called “growth ambitions” describe a new enterprise’s projected growth targets in terms of customers and financial performance. It is often these growth ambitions that attract investors to fund the costs in the beginning of the journey. Hence, they comprise an important dimension in the decision-making of new enterprises.

The Four Phases of New Enterprises

When we plot business model success and growth ambitions on a matrix, we can identify four phases of new enterprises: shape-ups, stand-ups, start-ups, and scale-ups. Each comes with strategic challenges.

business growth curve model

These new enterprises have already reached their growth objectives but have failed to maintain a well-functioning business model. The reasons might include a logic that doesn’t make sense any longer as the market has changed (e.g., Tamagotchi ), outdated technology (e.g., investing in personal digital assistants before smartphones emerged), value propositions that are challenged by competitors (e.g., Uber challenging the taxi industry ), or significant changes in customers’ demands (e.g., trends toward non-smoking, veganism, or do-it-yourself). In the latter case, the problem is not that a competitor offers something better, but that customers are disappearing from the existing market altogether.

All these situations have one thing in common: The business model has become irrelevant after significant growth, and the business is now in a declining market. Therefore, these new enterprises need to shape up to survive. Thus, shape-ups face the significant challenge of (re)inventing their business models, whether through innovation, business development, strategic re-positioning, or divestment. At the same time, these companies must restore investor trust as they are managing through a disappointment. Simply put, these enterprises need to reinvent their business models and themselves as entrepreneurs.

After firms have reached their envisioned size, entrepreneurs’ attention should shift toward stabilizing the business model and securing returns on investment. Stand-ups have momentarily left the valley of death, but that doesn’t mean that their troubles are over. They must do everything they can to remain relevant among consumers, outperform competitors, and fight any complacency that might creep in. Put differently, all their effort must be applied thoughtfully to continue to stand up.

The challenges in this phase are to protect the business model and safeguard related investments. These aims can be achieved by forcing competitors out of the market, optimizing processes and earnings, or gradually developing the business model. Simply put, these enterprises need to protect their business models — both today and in the future.

These new ventures have an ambitious growth target, but have yet to find a well-functioning business model. Their defining elements are their search for a business model and their constant experimentation, often in the form of trial and error.

Start-ups may, for instance, shift focus from one customer segment to another, develop new products and services, or change their payment options from fixed to subscription to on-demand, and back again. They often also try different means of sales and marketing to find customers. Moreover, they develop new capabilities to support all of these mentioned changes.

In short, in a start-up, nothing is fixed and everything is in flux in the quest to find a profitable — and sustainable — business model.

Of course, the search for an excellent business model is not free of charge. However, as everything is small in scale, total investments are typically low. The strategies applied typically include “fail-fast,” “trial-and-error,” “co-creation,” and “crowd-funding” — some of the most popular start-up principles. Simply put, the strategic challenge for start-ups is to find the right business model.

After a start-up has created a suitable business model, it may choose to scale up in volume, usually following one of two paths. First, scale can come from onboarding an increasing number of customers. In this case, the business model already encompasses the necessary capabilities and value propositions — the focus is on obtaining as many customers as quickly as possible. This is typical for digital, platform-based business models. Second, scale can come from replication of the original business model, as seen in franchise systems. Think of a restaurant chain: Apart from back-office functions (such as supply chain, human resources, and IT), identical copies of the business model are established. Scale in customers therefore necessitates scale in resources and capabilities.

For scale-ups, the challenges entail quickly onboarding customers and finding the resources needed to enlarge the business model’s volume so that capabilities grow in line with the number of customers. Simply put, scale-ups need to fund expansion and limit innovation in their quest to live up to the projected growth expectations.

Companies can fall in each of the four phases, but do not have to go through all phases. Consider Amazon, which went rather abruptly from start-up to scale-up . Jeff Bezos found a business model adequate for the advent of the internet, founded a company with a vision of becoming “the earth’s biggest bookstore” from the beginning, and focused relentlessly on long-term growth at the expense of short-term profits.

Yet, sometimes companies do actually go through all phases at different points of time. Consider the trajectory of Facebook. Initially, they were a start-up that had to find a business model. Then Facebook evolved into a scale-up, seeking to obtain growth by scaling their model. When they went public , they essentially turned into a stand-up trying to secure their model. But with the longstanding criticism of their business model and data usage, they may now have fall into the shape-up phase, where they need to reinvent their existing business model and essentially be entrepreneurial again.

Lessons for Entrepreneurs

Our work suggests that there are three key lessons for entrepreneurs:

The bottom line is that it’s important to honestly assess the organization’s situation and to craft a corresponding strategy. A failure to understand the situation may result in a significant loss of investor trust and investments. The road out of the valley of death is paved with situational awareness and transparent communication — one phase at a time.

business growth curve model

Partner Center

Alex Birkett

Growth models.

How do you model and predict growth (and growth opportunities)?

There’s all this talk about “growth models” and “growth modeling” but not much talk about how to build them and get value from them.

As with many things, it’s easy to see why they’re important, but hard to put them into action.

This post covers how I think about growth modeling, how several impressive companies do growth models, and how you can build your own (even if you’re just starting out).

What’s a Growth Model? An Introduction

Growth models are a representation of the underlying mechanisms, levers, and reasons for your company’s growth.

They seem to have become popular recently, notably with the rising trend of ‘growth hacking,’ ‘growth marketing,’ ‘growth,’ or whatever we’re currently calling data-driven startup marketing ( “a marketer by any other name…” ).

Here are a few definitions of growth models from various sources…

“Growth models…are feedback loops that project how one cohort of users leads to the acquisition of the next cohort of users. Viewing growth with this reinforcing model simplifies a complex system with tons of moving pieces to a set of functions and assumptions.”

HackerNoon :

“Every startup needs a framework/model for growth; a focused approach for scaling its organisation and user base.”

GrowthHackers :

“The concept of a growth model is both an old and a new one. It has a lot of similarities and connections to what’s traditionally called a “business model”, but companies and teams now focus much more specifically on growth and take a much more data-driven and experimental approach. At its core, a growth model boils down to a way to conceptualize and summarize your business in a simple equation, which allows you to think about growth in a holistic and structured way.”

Translation: it’s a new concept for an old idea about creating a simplified high level model in order to make better business decisions. You’re trying to explain “how does this company grow?” What levers exist as inputs that contribute to “growth” as an output (however you define growth)?

In growth, there tends to be two different types of models:

Qualitative models are going to be more descriptive in nature. They’ll be high level descriptions of how your business is growing and plans to grow. You can draw pictures of these:

business growth curve model

Image Source

For example, it’s easy enough to analyze the growth of an ecommerce business on a qualitative level: look at your Google Analytics source/medium report. How are you currently acquiring visitors and customers?

business growth curve model

You can say, at a high level, that you’re acquiring users from some main traffic sources, you’re converting them to subscribers, users, or customers at some rate, and some amount go on to repeat purchase (or tell their friends). You can get a pretty darn good idea of how your business is growing just from these numbers.

Even without considering attribution or looking at data, you probably have a good finger tip feel for how you’re getting customers. Is it virality? Word of mouth? Content marketing? PPC? Write all this down, and you’ve got a good idea of your primary growth levers.

Eventually, when you learn more about how you’re growing, particularly when you justify the use and implementation of a good analytics setup and can get granular data, then you can build out a quantitative model from this qualitative one.

Things can get pretty nitty gritty here, and every model is different (it really depends on your business concept). Here’s an example from Sidekick, a team that existed within HubSpot a few years ago that made features like Email Tracking and Documents that now exist within the HubSpot Sales suite:

business growth curve model

As a hypothetical example, let’s say your ecommerce business is growing primarily from content marketing. In this case, you’d itemize the different steps of that customer journey and fill in corresponding metrics.

Some important variables in the context of content marketing might be sessions to the blog (overall top of funnel traffic), how many people click over to the store, how many people add something to their cart, how many people start checkout, how many people purchase, what the average purchase size is, how often people make return purchases on average, and possibly email subscriptions as well (which you could also build a micro-model out of).

You should make your spreadsheet prettier than mine, but here’s an example of modeling out that one growth channel over a few months.

business growth curve model

Semi-related side note: I covered, in depth, how to model content growth and analyze results in this post on content marketing analytics as well as in my course on content marketing strategy . 

business growth curve model

The important part with a growth model is that you project out possible results to the future using your current trends. That way, you can tweak different variables to see what the biggest impact levers could be.

You can predict, based on current trends, where you’ll be in 3, 6, and 12 months, and if you’re okay with that projection. If you’re not okay with the projection, you can run sensitivity analysis to see which levers may be the most effective places to put your focus.

Is the add to cart to checkout page step low? Conversion optimization can help solve that. Do you simply need to lift your organic sessions because they’re stagnating? It’s easy enough to see that in the model.

Your model is only worth building if you’re going to use it to help you make decisions.

Growth Models Examples: 5 Ways to Model Growth

While most growth models are spoken of in general terms, they usually have similar ingredients. What acquisition channels are bringing in customers? How much is each customer worth? How long does each customer last? How many friends do they invite to your product or service?

In addition, they usually boil down to a few variables, which most of the time are further broken down into sub-variables. The main things are usually:

Acquisition Channels * Value of a User/Customer * Retention

One of the most lucid growth models I’ve seem comes from Drew Sanocki , who explains ecommerce growth in terms of three levers.

1. Drew Sanocki’s Three Levers of Ecommerce Growth

Drew Sanocki , ecommerce growth legend, teaches a concept in CXL Institute’s Ecommerce Growth Masterclass that I really like.

He explains that there are really only three growth levers we can pull to improve ecommerce growth:

Here’s his full quote from the course (also, take the course):

“We get caught in tactical maneuver hell, where we look at all these tactical opportunities and get stressed out about optimizing this entire thing when it really only boils down to these three multipliers. And the power of these three is that improving any one of them is good, but if you can improve all three, the results multiply. For example, in a year, do you think you can increase your retention by 30%? Can you increase your average order size by 30%? Can you increase your total number of customers by 30%? Any one of these in isolation, I think, is really doable. The trouble people get in is when they try to find the silver bullet that will double your total number of customers in a year. It’s really hard. But if you look at only moving each of these only 30%, you’re going to more than double the business.”

Now, you can further break down each of these categories, right? Customer acquisition can be broken down into several smaller factors actually:

Each of those is sort of a mini-lever that you can pull within the number of customers category. That’s the magic of building out a quantitative model, too. Once you see that, while you’re bringing tons of customers to your site with SEO, but none of them are converting, you can begin to work on conversion rate optimization. It’s a great way to prioritize high impact growth opportunities.

Similarly, you can break down average order value into different factors:

This, I suppose, could be also put under the category of conversion optimization. But in reality, with this lever you’re optimizing for increased order size, instead of optimizing for increased purchase percentage.

Finally, the last category is around retention. How long do customers stay around (and in ecommerce terms, how frequently do they purchase from you?). To this end, you can break that down into channels like:

If you’re a subscription commerce company, this step is even more apparent and even more important.

Note: you can use models like this for things outside of ecommerce as well. For example, Shanelle Mullin used this concept to create a model for content marketing growth :

business growth curve model

As Drew mentioned in the quote above, if you can increase one of these levers by a few percentage points, that’s great. But if you can increase every one of these levers by 10%, that’s compound value. That’s where growth happens.

This is a good macro-model for ecommerce. Let’s look at a similar model for SaaS (traditionally B2B, but works for B2C as well).

2. SaaS Growth Modeling

Similar to ecommerce, you’ve really got a few growth levers for SaaS:

The only real difference here is how you define customer retention, and the steps that it takes to become a customer.

Often, in B2B, you’re going to break down your “number of customers” lever into distinct pieces:

In this model, someone may find you via a search query (“customer feedback software”), and perhaps they land on a blog post. They find that you offer an ebook that explains how to accurately measure customer satisfaction, so they download that and they become a Marketing Qualified Lead.

Next time they visit, they sign up for a webinar on customer success, and they give you their company info and phone number. Now we can refer to them as a Sales Qualified Lead.

Because your model will be higher touch, most customers will require a sales touch, so you separate your stages into two distinct lead classes to reflect that.

If your B2B model is a lower touch model, like Dropbox , or if you run a B2C application, your model may look like this:

In this scenario, a customer may hear about HeadSpace on podcast advertisement, check out the website, and return later to give it a try. They sign up for the free app, use it for a couple days, then never returned. They dropped off before upgrading to becoming a paid customer.

For all intents and purposes, these B2B growth models have pretty much the same levers as ecommerce models, you just define stages differently and break steps down into micro-models differently (though how you break these down also depends on your acquisition strategy).

business growth curve model

Image source

One of the better worksheets I’ve seen was built out by Candace Ohm , data scientist and improv comedy legend. She offers an Excel spreadsheet ( here ) where you can  learn how funnel metrics, customer churn, user demand, and virality effect your growth curve. It’s worth playing around with:

business growth curve model

We’ve got the high level models down now, so let’s dive into a few micro models that we can use to improve given channels or parts of the funnel, like word of mouth/referral and conversion optimization.

3. Optimizing Referral Marketing: A Model

Referral is usually a growth lever, regardless of business type or size. People telling other people about your business is one of the best ways to grow. Most smart businesses try to incentivize this in some way (in addition to building something worth talking about).

Though a lot of word of mouth is frankly un-trackable (if I tell a friend how awesome MeUndies is, they won’t know how to attribute that), you can track and optimize a good portion of referral traffic, especially if it’s incentivized (i.e. you give a discount or tracking code).

Like any other model, you’ll want to break it out step-by-step:

Then it loops back, because you can offer that person a referral code as well. This is essentially known as a viral loop, and you measure its effectiveness using a ‘viral coefficient.”

We can also vastly simplify this model, like the following graphic shows:

business growth curve model

Depending on your technology stack, you may have to pull this data and build the model by yourself. But you might also just be able to get the reporting from your tool, especially if you use something like Referralcandy .

4. Conversion Optimization Modeling

Conversion optimization should be an inevitable part of your model, because no matter what business you’re in, you’re going to bring a some amount of visitors to the website, and a certain percentage of them are not going to buy or become users.

If you can increase the percentage of visitors that convert, you get a compound effect over time (and increasing conversion rate increases the effectiveness of your other channels, which lets you bid more on ads, spend more on content, etc.).

business growth curve model

Now again, if you’re in ecommerce, a lot of this is simplified due to simple prototypicality. In other words, almost all ecommerce sites follow a very similar pathway to conversion. Everyone has a cart, a checkout flow, a thank page after conversion, etc.

The simplest growth model you can construct for ecommerce CRO, then, is a sort of funnel. You can start at the broadest level and go all the way to the home run: Homepage -> product page -> add to cart -> checkout -> purchase conversion.

Every time you can increase a step of this is a step in the right direction, though the end conversion is of course the most important (as well as the order size). But if you can systematically improve your funnel, all other marketing efforts will be improved by extension.

Everyone has their own approach to auditing and modeling conversion opportunities, as well.

I reached out to Luiz Centenaro , Optimization Manager at Optimizely & an eCommerce consultant , to see how he approaches CRO growth models, and he explains that he sets up a baseline with A/A testing and Google Analytics analysis:

“I typically approach an eCommerce site by running an A/A test on every touchpoint. A/A test the Homepage, Category Pages, Cart Pages, Product Pages and Checkout and track clicks on everything. If you have a good marketing team this can all be accomplished within Google Analytics or Google Tag Manager but you can also do this with your A/B testing platform such as a Optimizely . After you run your A/A test you’ll have a baseline with revenue per visitor and conversion rate for every page and you can segment to see the difference between mobile and desktop too. Simultaneously while the A/A test is running you can research the demographics of the visitors using Google Analytics. One of my favorite reports is demographics by age. Age and demographics should be taken into account when crafting hypothesis to A/B test. You won’t market to a 65 year old the same way you market to an 18 year old and they show significantly different user behaviors.”

It’s not much more difficult in other types of websites either, so long as you can define the discrete stages of your customer journey. With a B2B SaaS company, that may look something like: Homepage -> Pricing Page -> Get Started Page -> Signup Flow (several steps?) -> New User Created -> Activation Event -> Upgrade to Paid

Conversion optimization can help optimize the steps on a site that lead to a visitor becoming a user, and possibly even a user becoming an activated or engaged user.

5. Virality: A Micro-Model

Viral growth is one of the better developed channels for building out models. While some viral mechanisms may look different (Apple iPods, Hotmail, Dropbox, and Bird scooters are all examples of virality), they do include similar variables that allow us to model the system:

The Viral Coefficient is just a name for the number of new users a current user brings in through virality. The formula is stupid simple: K = i * conv%, where i is the number of invites sent and conv% is the conversion rate of those invites.

That’s really all you need to model out the first part. ForEntrepreneurs even offers a free spreadsheet to help you build that out ( here ):

business growth curve model

The other part of the equation is how fast you can acquire new users through viral loops. Clearly, the faster you can go through the viral loop, the better.

As with the Viral Coefficient, your Cycle Time includes several sub-variables as well. David Skok draws out an example here:

business growth curve model

To the extent you can shorten the time length between any of those steps, you can increase your growth rate.

It’s important to model the time factors as well (which is included in the ForEntrepreneurs spreadsheet . Really, if you’re interested in viral growth, I’d just read that post, as this section is clearly just a summarized version of it):

business growth curve model

With this, and every other model, the powerful part is that you can tweak different variables to see what happens to the output. Increase or decrease conversion rate of invites by 5%. What happens? Increase the number of invites sent. Does that move the needle?

This can help you make important decisions on what drivers to focus on when optimizing viral loops.

Everyone wants virality.

As David Skok put it , the perfect business model is “Viral customer acquisition with good monetization. However viral growth turns out to be an elusive goal, and only a very small number of companies actually achieve true viral growth.”

In my experience, this area has been the most over-exposed to bad content publishing and bad thought leadership. Most of the core of viral growth is a noteworthy product. It’s hard to make a bad or a commodity product viral (though not impossible, just probably not worthwhile).

Limitation With Models and What to Expect

As the statistician George E.P. Box famously said, “All models are wrong; some models are useful.”

No matter what model you use to represent and predict growth, it won’t be completely accurate once the rubber meets the road, once your plan meets the messiness of reality. As the great philosopher Mike Tyson once said , “Everyone has a plan until they get punched in the mouth.”

This echoes the common wisdom, probably first said by Helmuth von Moltke the Elder : “No plan survives contact with the enemy.”

That’s all to say: be fluid, and update your model as you get new information and insight.

When you look at, say, an inbound marketing funnel, you don’t expect it to exactly and linearly reflect reality, do you? Funnels are growth models; they’re ways to simplify the concept of how you’re growing, provide directions for measurement, and allude to opportunities for effort and optimization.

business growth curve model

These types of things are most useful in planning; the execution of those plans is still to be determined. I’ve built out complex models only to find once I hit the ground that I didn’t actually have the resources to carry out some of my top prioritized plans. Whoops.

So, don’t expect these models to be perfect. Expect them to be useful and actionable.

Growth models give you an imperfect, yet helpful model to show you how you’re growing, what kind of growth numbers you can expect in the future, and some possible opportunities to impact that growth in a positive direction.

In most businesses, there are really three levers you can pull, at least at a high level:

To each of these levers, you can break them down into, really, an unlimited array of possible channels and tactics. That’s where things get complicated (and there’s always a tradeoff between cost and complexity of modeling). A good model is both useful and accurate, to some degree of each, but no model can be both perfectly accurate and comprehensible/useable.

Build out models to give you a better understanding of your growth, and also a better way to communicate that with others. Don’t expect a perfect vision of reality, but expect them to help you prioritize and find opportunities you otherwise may not have.

business growth curve model

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What Is the S Curve in Business?

The Disadvantages of Diversified Business

Growth stages of industry, what does reorganization mean in a corporation.

The very best businesses understand the concept of growth, and they apply ideas formed by this concept so that they can anticipate and counter growth challenges as they come up. In fact, the process of growth for business, any business, is so predictable that there is a special name for it: The S Curve of Business. Of course, we can find the S-curve in economics and lots of other fields as well. But what, exactly, is this curve and what does it mean for your business?

What Is the S Curve of Business?

The ‘S’ in the ‘S-Curve’ definition stands for ‘Sigmoidal’, which is a mathematical term related to the way the curve is derived. You can, however, think of it as an S-shaped curve that predicts how a business will grow over its life cycle. It’s pretty easy, in retrospect, to see how your business grew following the S-curve. However, it can be a bit more challenging to navigate this curve as you’re moving through it.

The most challenging points in the curve are the so-called inflection points, where your growth stagnates. In the moment, you might feel like your competitors are overtaking you and everything might look hopeless. You might want to make some appealing but bad decisions that would potentially take your company under.

With a good understanding of the S-curve and of the general process of growth, however, you will be able to navigate such times and you will push your business toward new heights and better times.

The Stages of the S-Curve

Every business starts the S-curve model at the bottom. The new business has a new product or service, and they try to sell it to the mass market. As their product gains traction in the market, the business begins to grow. At first, the growth is slow, and then it develops more rapidly, as consumers begin to warm up to the product. As the business expands, that growth continues. Eventually, a host of factors, both internal and external, cause the growth rate to decline and then gradually, they taper off. It could be anything, really; it could be competitors that have begun to adjust and target your customers; it could be that you have saturated the market and that there are no more markets to grow into with your product; it could be that your company has internal issues that affect its ability to move forward.

That tapering-off point is also a turning point. It leads to a slight decline in growth, with growth actually being negative for a while. This is a critical point for the business. If the business innovates afresh and finds a way to stay relevant, the growth curve turns back up and growth becomes positive again. It will be a true inflection point. If the company makes some bad decisions and does nothing to renew its relevance in the market, then the turning point will be a permanent one and the company will find itself plunging for the depths of the ocean.

The success of your business depends on your ability to recognize these inflection points and take the right steps to put your business back on track with the right kind and amount of momentum to fuel further growth.

If you don’t recognize when your business is at an inflection point, you could put it in jeopardy. Not only will your business be denied the opportunity to develop strategically, but you will also be unable to meet an important need. The best solution is to prepare for the inflection points before they happen and make sure you know how to recognize one when you see it.

One thing you should note is that an inflection point isn’t necessarily an indicator of crisis. What it simply means is that your business is faced with an important decision, and the kind of decision you make will determine what happens at that inflection point.

What Contributes to an Inflection Point?

There are countless factors that affect growth. However, in the initial stages of a company’s growth, some common factors can be found to almost always lead to the first inflection point. These can be subdivided into internal and external factors.

Internal Factors

A lack of sense of ownership among the founders: The founding members of a business, including the initial employees, usually have a very strong sense of belonging when it comes to working to meet the objectives of the company. However, as the organization grows and more levels of management are added, or when external investors come in with different mindsets, this sense of ownership is lost and the initial goals of the company can get blurry.

A shortage of talent: The companies that grow the fastest in the initial stages often face a problem with scaling their teams fast enough to keep up with the pace. The growth of your revenue can only really happen when you have enough skilled employees to support the growth of your company.

A ceiling for founders: The zeal and passion of a founder can be enough to get a new company to grow exponentially. However, if the founder does not adapt his or her leadership style to scale with the company, and tries to get involved in every single company decision, it can cause bottlenecks to the growth of the company.

The customer becomes silenced: Small organizations tend to be effective in responding quickly to the needs of their customers. This is usually what makes them grow so fast in the early stages. However, as they grow, a greater gap appears between the executives at the top and the employees that deal directly with the customer. Innovations related to the customer, therefore, may not get communicated fast enough to the top executives to be implemented in time. In fact, they may not be implemented at all.

Issues with innovation: In order to grow, you must focus your strategy around the customer and innovate effective and efficient solutions to their problems. This is what gets most businesses growing in the early stages. However, when an organization grows too large, it has a tendency to get caught up in the details, such as perfecting the processes that occur in production lines and departments, rather than driving innovation by using data obtained from the consumers.

External Factors

The economy: Whether locally, regionally, nationally, or even globally, the economy affects all businesses to different extents and a recession can make a dent in the growth curve of the most promising company.

Financial issues: Every company needs some financial institution or other to succeed. These institutions are involved in transactions, control interest rates, credit, and even the loans the consumers can access. Their solvency and stability of the finance sector will affect all businesses.

Infrastructure: Businesses that rely on physical locations to attract talent or new customers or to improve their operations will be affected by zoning laws, housing developments, and construction activity.

The political climate: Any change in laws and regulations at whatever level can have an effect on a business when some service or product of theirs is affected or even becomes illegal.

Public trends: While it would be unfortunate, it is possible for a business to spend vast amounts of time and money, just so they can position themselves, but then they discover that they are on the wrong side of public trends.

How to Survive an Inflection Point

The short answer is simple: Innovate and optimize.

When you reach a saturation point, you need to make your processes more efficient so that you can deliver more efficiently to your customers, or you need to innovate by expanding your product and service offering, so that you can give your organization further room for growth.

You can also look for a solution for some of the internal issues that your company might be facing or that you can find ways to turn external factors into opportunities for your company's growth. It is only when you adapt quickly to changing circumstances that you can maintain the momentum of your company, so that it will continue to grow long term.

Nicky is a business writer with nearly two decades of hands-on and publishing experience. She's been published in several business publications, including The Employment Times, Web Hosting Sun and WOW! Women on Writing. She also studied business in college.

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Growth Curve - Explained

What is the Growth Curve?

business growth curve model

Written by Jason Gordon

Updated at March 10th, 2022

Table of Contents

What is a growth curve.

A growth curve is a graphical representation of the increase in a particular quantity over time. 

Growth curves can be typically classified into two types -

A growth curve has different applications in different fields of study. Growth curves are extensively used in finance, especially by businesses, in order to create a mathematical model to analyze the growth in sales or profits, and also to predict future sales.


What is an Exponential growth curve ?

Exponential growth, also referred to as unrestricted growth, usually occurs in an ideal environment with unlimited resources. The growth is slow in the beginning but increases rapidly with the passage of time. 

Exponential growth curves are most commonly used to denote population growth, growth of wealth and investments, business growth, and growth in website traffic as well as followers on social media. 

A great example to illustrate exponential growth would be that of living bacteria in a petri dish in a laboratory under ideal conditions, the bacteria will reproduce by binary fission, i.e by splitting in half, roughly once every hour. Now assuming that the petri dish originally contained 1 million bacteria cells, it would end up with 2 million cells after an hour. After two hours, there would be 4 million bacteria cells. The number of bacteria cells in the petri dish would increase to 8 million cells after the passage of three hours, and so on. However, it should be borne in mind that it is usually not possible to sustain exponential growth over long periods of time since there is a limit to the availability of resources in the real world. 

What is a Logistic growth curve ? 

Logistic growth, or restricted growth, occurs when the numbers begin to approach a finite carrying capacity. The growth is typically fast in the initial stage but drastically slows down with the passage of time. 

Logistic growth patterns are most prominent in graphical representations of increase in literary skills or language proficiency, weight loss regimes and musical skills. Logistic growth also occurs in populations that begin to experience environmental resistance while approaching the carrying capacity. 

A good example of logistic growth would be that of a person partaking in mass building or strength training at the gym the initial muscle or strength gains will be quick and fairly noticeable. However, once the individual attains a certain degree of fitness, the gain will typically slow down and become much less noticeable as time passes. In business, the shape of the growth curve essentially determines the direction that the company will be required to take in the market. For many businesses, logistic growth markets are not the most desirable places for product launches because such saturated markets do not leave much room for profits. On the other hand, exponential growth markets do provide opportunities for fast growth and handsome profits, but they also typically lure in a lot of competitors.

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    The S-Curve of Business allows a company to determine where it is on a typical growth life cycle, and adjust its strategies accordingly. The S-Curve of Business life cycle consists of two inflection points. The second is the most critical, as it signifies that a business has reached a growth ceiling. Inflection points are caused by a variety of ...

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  15. Growth Curve

    Growth Curve is the graphical representation of a process or phenomenon changes. The curve reflects outcomes such as exponential growth or maximization of growth over time. The x-axis represents the time, long-term or short-term period. In contrast, the y-axis represents changes in quantity or other growth variables such as revenue, sales ...

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    Every business, whether it's big or small, goes through the 4 stages of business growth: Startup. Growth. Maturity. Renewal or decline. Each of the stages of the business life cycle, also known as maturity phases, growth phases or growth stages, have unique challenges and your business will need to find creative approaches to overcome them.

  17. The Sigmoid Curve: A Model for Constant Business Growth and ...

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    The S curve, a mathematical model also known as the logistic curve, describes the growth of one variable in terms of another variable over time. S curves are found in fields from biology and ...

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