Forecasting
- Delphi Method
- Analogy Method
- Extrapolation
- Applying a Statistical Association
- Applying a Causal Model
- Determining Limits
- Assessing and Describing Uncertainty
In the same way than when looking at present events and products, your view into the future can be either descriptive or normative. Their difference is that in the former case you accept the future as it comes and in the latter case you wish to change it.Views Into the Future
The descriptive vista to future usually aims at finding out the most probable future. This view is conventional when you cannot affect the future. You just want to know it so that you can prepare yourself to the inevitable, like tomorrows weather. This approach has traditionally been called forecasting, or predicting. It tries to answer the question: "What happens to object A in time B if the evolution continues without interventions?"
Another, less common variant of descriptive approach is the utopia, a detailed narrative of a possible or "hypothetical" future which need not be the most probable one. The writers of utopias, beginning from Plato and Thomas More, usually leave the question of probability to the discretion of their public. Utopias have also been written explicitly as warnings, as abhorrent examples to be avoided, in the style of the nightmare-like novel 1984 by George Orwell.
The novels of Jules Verne have shown that fiction can emit constructive ideas to the developers of new products. A few large companies have lately found out that they do not need to rely on sci-fi writers in the creation of utopias. They have themselves started concept design projects which do not aim at creating real products but just generate ideas for imaginable original products in the future, to be used in strategic planning, for internal education of personal and for publicity. These visions often include utopias of potential future ways of living, where completely novel products of the company may find a market.
A peculiar characteristic of descriptive forecasts is their tendency of self-fulfillment, i.e. a prediction that is well known by all people tends to come true. It seems that when people believe in a forecast, some of these people also want to adapt themselves to this seemingly unavoidable future, for example giving their vote to the predicted winner of an election. Ancient holy scriptures predicted that Jews would one day immigrate the promised land, and finally it indeed was done. A third example is the fashion predicted in fashion magazines - many people want to wear the newest fashion, they buy what is recommended by the magazines, and the predicted fashion thus becomes reality.
Normative approach to the future means that you think that you can affect the development. You perhaps know which kind of future you want, but you are not sure of the best method of obtaining it. Normative study of future tries to answer questions like:
- What follows if the object A is submitted to manipulation B?
- Which manipulations are necessary to obtain the state of things A in time B?
The normative view necessarily involves evaluations, and it becomes necessary to define the people whose point of view shall be used in evaluation, cf. Normative Study: Point of View. Usually only the final state of the predicted or planned development is evaluated, in other words the internal procedures of normative forecasting seldom include subjective evaluations. These phases of normative forecasting thus resemble the descriptive approach, and same methods can be used in both types of forecasting.
Quite a number of alternative methods are available for forecasting. The choice depends on the nature of the information that is available for underpinning the forecast.
Normally two types of information are combined when producing a forecast: recent observations from the phenomenon, and some knowledge about the normal pattern of long-term development of the phenomenon. The last-mentioned information can be presented in several alternative ways, typically as follows:
- A general theoretical model describing the normal development of the class of phenomena that shall be predicted. Especially useful are the models which define the dynamic invariance that usually is seen in the phenomenon.
- Data of the historical development of the particular phenomenon. The mode of presenting these depends on the type of data. For numerical data, time series is the normal method; for the developing shapes of products, a series of pictures (see example). Besides, written and verbal presentations are always possible.
In the case that both a model and a time series are available, you have the possibility of making two concurrent forecasts with different methods. Comparing them makes an excellent test of their reliability.
In any case, when using a theoretical general model as the basis of your prediction, you should keep in mind that the model has normally been produced by studying a certain population, which means that the model is really valid only in that context. You should not generalize too unscrupulously and believe that the model will unconditionally be valid in the future environment that you are forecasting. When generalizing, you should consider the following questions:
- Which aspects of the future environment are possibly going to differ from the context of the original object of study? See also Assessing the results of a study of literature.
- Which influence these differences will have for the forecast?
The logical structure of forecasting (the red lines in the diagram on the right) consists essentially of gathering, fitting together and manipulating available information about the phenomenon. As the diagram shows, this information has to be obtained from quite different sources: from empiria, from a conceptual level of case studies, and from the more sophisticated conceptual level of general models. The nature and presentation of information is quite different on each level, and it also largely prescribes the methods available for forecasting, as can be seen in the table below.| Information available for the basis of forecasting | Method of forecasting |
|---|---|
| Expertise knowledge (tacit knowledge may also be used) | Delphi Method |
A model based on another comparable system
| Analogy Method |
Trend, the recent development in the system to be predicted, as defined by a series of observations, e.g.:
| Extrapolation - from the last observations, or - from all the findings - perhaps within limits |
| A statistical association between the variables to be predicted | Applying a Statistical Association |
An explanation for the phenomenon to be predicted:
| Applying a Causal Model |
All the forecasting methods included in the table given above will be discussed in the following. In some cases it may be possible to combine some of the methods to improve the reliability of the prediction. Further, after the presentation of the methods there is a discussion on the available ways of assessing and describinguncertainty of forecasts.
Delphi Method
The most primitive method of forecasting is guessing. The result may be rated acceptable if the person making the guess is an expert in the matter. An important thing to note is that guessing is the only method where we can make use of tacit knowledge that the specialist has not been able to express as exact words or numbers. Generally the best method for eliciting such a forecast from the expert is an unstructured interview. The method of interviewing allows you to inquire into the reasons and explanations for the presented forecast which you might also choose to criticize and thus try to reach an improved forecast. When interviewing an expert you may also learn something that you can later use if you prefer to construct your own forecasts with other methods.Sometimes at least some of the experts live far away and they are thus difficult to interview. (Examples of potential sources of specialists can be found under the titlePopulations of Evaluators.) To consult such experts, you may resort to a questionnaire instead of an interview. If you wish to question several persons simultaneously, you may consider using the Delphi method.
In the Delphi method, the researcher directs identical questions to a group of experts, asking them to give their guesses on the future development of the specified topic. In the next step, the researcher makes a summary of all the replies he has received. He then sends the summary to the respondents and asks if any of them wants to revise his original response. If the respondents are amenable to the extra effort, they may be asked to justify their opinion, especially if it differs from that of the majority.
As it is difficult to make summaries of other than quantitative responses, the questions that are used in the Delphi process are usually quantitative, e.g. "What will the price of crude oil be in 20 years?" On the basis of this type of responses, the researcher will be able to calculate e.g. the means and the ranges. One advantage of the method is that you can readily use the range of the responses as a first measure of the reliability of the forecast.
Nothing prevents using qualitative or any other type of questioning if the nature of the object so requires. If you wish to do so, it is advisable to plan in advance the method of making the summary of responses, otherwise it will be difficult to arrange the second round of the questionnaire.
The Delphi procedure is normally repeated until the respondents are no longer willing to adjust their responses.
The Delphi method is not very reliable. Results of Delphi questionnaires are often later found to have predicted the real course of events remarkably badly. Wrong guesses are often made by renowned specialists and sometimes even by a majority of them, and the odd person who is later found to have predicted right would perhaps never have been elected to the Delphi group of experts. "If you had predicted the collapse of the Berlin wall one year before it happened, that would have proved that you are no expert in politics."
Analogy Method
Most forecasting methods make use of some kind of a model that is assumed to portray the relations between the various aspects and attributes of the "system" whose development shall be predicted. A very simple method of acquiring such a model is available when you can find an earlier "foreign" system that is similar to the now existing "home" system whose future you want to forecast. The foreign system must have already undergone the phase of development that you are trying to predict for the home system. In other words, it must have reached a more mature stage in development than the home system.Usually it will be impossible to find a foreign system that would be absolutely identical to your home system. At least the environments of the systems differ. The systems are then just "analogous". An obvious difference is that the foreign system has been observed in the past and the home system is to be continued into the future. This is a difference that you cannot help, but many other divergences like for example dissimilarities between the sizes of the systems can be eliminated by making suitable corrections.
The process of forecasting through analogy is:
- Explore the recent development of the home system.
- Find a foreign system that has undergone similar development. Get data from it.
- Find out the point of time t when the foreign system was in a similar state than the home system is now.
- Compensate for the differences between the home and foreign systems, eliminating e.g. the size difference by multiplying the data of the foreign system by the ratio of the sizes.
- The data of the foreign system, from the time t until present, can now be taken as a forecast for the home system, beginning from the present.
The analogy method is not restricted to quantitative data. In fact, it can handle any format of description of a temporal development. An example of qualitative forecasting can be found in Oswald Spengler's (1880 - 1936) book Untergang des Abendlandes (1918, 1922) which explains the typical development of the ancient cultures of China, Egypt, Rome and a few others which have flourished in their time and then withered away. Spengler found that cultures are processes that share a common model of development. He then made the prediction that the Western culture which is still under development will follow the same pattern. In this part of his treatise, Spengler thus created an analogy between objects of the same category (i.e. between cultures).
Moreover, Spengler (and likewise Arnold J. Toynbee in the book A Study of History, 1935-39) drew the analogy further and asserted that the pattern of cultural development is also analogous to the succession of seasons, i.e. spring, summer, fall and winter, and even to the lives of plants and animals which include the phases of birth, maturing, bloom, decline/decadence and death. In other words, Spengler extended the analogy from one species of systems (cultures) to another (to seasons, or to animals).
Another example of an analogy between objects of different category is Alvin Toffler's book The third wave (1980), where an analogy to waves is used to describe and predict the evolution from agricultural to industrial society, and later to the information society.
Indeed, bold analogies between objects of different types (see examples of these) can sometimes generate fertile hypotheses for discussion, but if you just wish to make a plausible forecast it is usually safer to restrict the analogy to a single class of objects only. Think if you tried to predict the development of cars by making an analogy to computers? You might conclude that cars should soon run at 10.000 mph, while their weight dwindled to a few grams?
Even those analogies that keep to a restricted class of objects often suffer from various irregular factors that affect the home system differently than the foreign one. You can try to diminish their influence by using more than one foreign system, if available. In other words, you make parallel forecasts and combine the results. Still better, if you can find the general pattern that all the systems follow; if it is possible you can shift to the more reliable forecasting methods of Applying a Statistical Model or Applying a Causal Model.
One more weakness of the analogy method is that it is difficult to assess the uncertainty of its results.
Extrapolation
Extrapolation is the most usual method of forecasting. It is based on the assumption that present development will continue in the same direction and with unvarying speed (or alternatively, with steadily growing or diminishing speed, i.e. a logarithmic extrapolation).The basis of an extrapolation will be knowledge on the recent development of the phenomenon. You will need at least two (although usually you have more) sequential observations made at known points of time. The principle is shown in the figure on the right.
- you start from two or more observations that were made in different points of time (on moments t= -1 and t= 0, in the diagram),
- you note the difference d between them. The difference can be quantitative or qualitative,
- you modify the latest observation by implementing (i.e. adding) once more the difference d, and get thus your forecast (the red line in the diagram).
If you have more than two observations, you have the option of choosing the number of observations that you will base the extrapolation on. If you feel that the very last observations have better predicting capacity than the earlier ones, you may prefer to disregard the earlier observations. An alternative is to give more weight to the later observations than to the earlier ones. If you decide to use a large number of observations (in other words, you are extrapolating the trend) you will probably wish to make the calculations with a regression analysis program if your data are quantitative.In the examples above the observations that will be extrapolated are recorded as quantitative variables. In other words, a time series is extrapolated. In addition, a numerical forecast is often explained in qualitative verbal terms as well, to make it easier to comprehend. An example is the book Megatrends by Naisbitt (1982).
Nevertheless, nothing prevents to extrapolate trends that are described partially or entirely in qualitative terms. Moreover, it is often practical to describe existing products with the help of pictures and other icon models, and this mode of presentation is serviceable even for extrapolations. In the book Industrial Design, Raymond Loewy (1979) thus combined two directions of view: historical and predictive. On page 74 of the book we find an "Evolution Chart of Design" which shows the development from 1900 to 1942. The last picture is Loewy's forecast which he created on the basis of the trend in the entire series, the main trend being here a gradual shift to more streamlined design.The innate weakness of all extrapolation is that it only can comply with such processes or forces which already are in operation. It always ignores those new impacts that begin to apply only in the present or in the future. Often there will be gradually more and more such new impacts in the future, and in such a case extrapolation can give useful results only for a relatively short period.
Another weakness is that it is almost impossible to assess the probable error of an extrapolation. A rough notion of it can be obtained by studying the consistency and homogeneity of the series of the original observations.
Applying a Statistical Association
| high pressure | fine weather |
| low pressure | foul weather |
The descriptive models that are used in forecasting are often quantitative, but qualitative ones are used as well. Indeed, any of several model languages can be used.
For example, verbal proverbs were used as the basis of weather forecasting already long before barometers: "Red sky at night, sailor's delight; red sky in morning, sailors take warning." And likewise: "Ring around the moon, brings a storm soon." In the development of products there are many aspects that can best be expressed graphically, for example when forecasting trends of fashion.
Quantitative descriptive models consist of variables and an expression which defines their relation to each other. This relation is sometimes called statistical association in order to emphasize that it originates from statistics i.e. from a series of observations. For quantitative models this association can be expressed as a curve or as an equation, e.g. of the type y = ax + b. In the section on extrapolation we already discussed two types of associations: the linear and the logarithmic trends.
The method of predicting on the basis of a descriptive model is simple, as can be seen in the diagram on the right. If one of the variables in the model is time, it is only necessary to insert any chosen future time point in its place and then read the "value" of the desired variable in the model. ("Value" is in quotes here because in qualitative models its contents are not numerical.)There is also another way of using a model: to study the direction of change of the variables or factors like in the table on the right.
| rising pressure | improving weather |
| sinking pressure | worsening weather |
In this method we need a recent observation on the direction of change of the factor that is not being predicted (here: air pressure). From it we can deduce the anticipated change of the factor to be predicted (the weather).
Forecasting on the basis of statistical models is feasible even when you do not know the reason or explanation of the mathematical association you have found in the historical data.
For example, the life span of an animal follows usually the same pattern that is typical for the species. If you know this pattern, it may even be possible to forecast without having any explicit theoretical formulation of the pattern: you simply regard an earlier observed specimen of the pertinent class as a model of development. Already in Hippocrates' time physicians knew the typical process of many illnesses, and when observing the initial symptoms of such a process in a patient the physician could predict the progress of the disease.
The method of statistical model might give a right prediction even in such a case when your assumed explanation of the existing statistical association is quite erroneous! Famous historical examples of successful predicting on the basis of statistical models only were astronomical calculations in ancient Mesopotamia, and those of Ptolemy the Greek. Most, perhaps all, of these early scientists believed that the Earth was the centre of the universe, and the sun, the moon and the planets were just moving around it. Nevertheless, the mathematical models of these apparent movements were accurate and yielded correct predictions of the rises and sets of the sun and the moon, and even of their eclipses.
Many companies habitually present a new model of their principal product in the beginning of each year or season. Often the evolution of sales of each model more or less follow the same pattern which can be presented as an "average sales curve" as in the figure on the right. Without knowing the reasons why the sales quantity follows this pattern, it nevertheless can be used for predicting.If you additionally have statistics about the previous models' sales, say, a fortnight after the beginning of the publicity campaign, you have the possibility of calibrating your forecast. For this, you have to note the factor by which present sales deviate from the earlier average at this moment, and then you simply multiply the rest of the curve by this same factor.
Sometimes the time series that you wish to extrapolate includes simultaneously several types of variation. Beside the trend, which was discussed above, there are often one or more kinds of seasonal variation. If that is the case, the normal method is first to analyse the time series, dividing it into its discernible components. After this, you continue by making separate forecasts for all the components (the trend and the different seasonal variations if applicable) and only in the last phase do you recombine the components.
If you, for example, wish to forecast the heating energy consumption of an industrial building, your analysis of past data will probably reveal that the variation of the consumption has been following several patterns simultaneously. Some patterns are due to the working rhythm of the business, which normally varies in three frequencies: along yearly, weekly and daily rhythms, and perhaps with respect to business conjunctures. Moreover, there may be linear trends, caused perhaps by a gradual shift to larger machines, or alternatively to more energy conserving methods and machines. -- The forecast is now made by estimating all the different cyclic variations, one at a time, then by calculating their continuations, and finally by combining all of these together with an extrapolation of the trend.
There are great risks in forecasting on the basis of a statistical association, without knowing the reasons for it. For example, scientific forecasts of national economies are notorious for their low reliability, which is due to a missing insight in the factual relationships of the variables of economy. Besides, this method invariably fails to detect the weak signals that exist in data but remain unnoticed until somebody afterwards understands that it was these factors that finally determined the direction of the evolution.
Generally, you should always try to find out the rational explanation behind the statistical association that you are using as the basis of your forecast. It is always safer to forecast on the basis of a causal model (described below), than to forecast on the basis of statistical associations only.
Applying a Causal Model
A reliable method of prediction becomes possible if you have, through research, obtained a model which not only describes (as in the previous section) the development of the phenomenon to be predicted but also explains it, in other words enumerates the reasons why it happens. In the best case the reasons and their outcomes are assembled as a model defining the dynamicinvariance of change in the process to be predicted.The weather, for example, need today no more be predicted on the basis of a statistical association of air pressure and weather only. The science of meteorology has lately advanced so much that we now know and can make use of the invariable structure of moving cyclones (Fig. on the right) which explains the changes both in air pressure and in weather. Even the proverb aboutred skies has now been given an explanation:
"Because the weather patterns in North America generally move from west to east, when clouds arrive overhead at sunrise the sky will appear red, signalling a storm "moving in". When the storm eventually passes, the sky will clear in the western sky. If sunset occurs simultaneously, the light will cast a red glow on the clouds above, now moving towards the east." (Cited from Gene Rempel and Mike Hanson.)The most elementary method of forecasting on the basis of a causal model is to use the model just like a statistical association, explained earlier. This is particularly easy when one of the variables in the model is time: then you just insert the right year into the model, and it immediately becomes the desired forecast.
If time is not included in the causal model, the model may still be helpful, because you can often predict the development of the independent variable easier than the future of the dependent variable or of the entire system - not least because of the fact that a reason normally precedes its result and it is thus not so distant in the future as the outcome will be.
When you know the causal relationships between the variables you will be able to use much more advanced methods of forecasting than mere statistical models would permit. These include:
- You can better assess whether the model remains valid also in the future.
- You can assess with sensitivity analysis the probable error of the forecast.
- You will also be able to modify the model, with a high degree of reliability, according to the requirements of the situation.
- If you want not only to forecast but also to change the future, you can pinpoint those changes in the independent variables that are needed to cause the desired change in the dependent variables.
The causal model is often so complicated that it is best managed by using a computer. Even then, you will usually need an illustrative presentation of your model to clarify your thinking and finally to be presented in the report. In such an illustration you will need a notation system to describe the various logical relations between the variables. The computer program will often be able to print out the model, using its in-built notations. If you can find no suitable ready made notation systems, you can devise one.A legendary example of a large causal model was fabricated by the so-called Club of Rome in 1972. This model, published in the book The Limits to Growth, consists of dozens of variables, including the world population, birth rate, industrial and agricultural production, the non renewable resources, and pollution. In the model, the levels, or physical quantities which can be measured directly, were indicated with rectangles
, rates that influence those levels with valves
, and auxiliary variables that influence the rate equations with circles
. Time delays were indicated by sections within rectangles
. Real flows of people, goods, money, etc. were shown by solid arrows
and causal relationships with broken arrows
. Clouds
represent sources or "sinks" (exits of material) that are not important to the model behaviour.The Club of Rome started building their "World Model" by first constructing five sub-models. These concentrated on the five "basic quantities": population, capital, food, non-renewable resources remaining (measured as now remaining fractions of the 1900 reserves), and pollution. One of the sub-systems included the causal relations and feedback loops between population, capital, agriculture, and pollution (fig. on the right). Finally the researchers combined all the five sub-models and thus created the final World Model, part of which is illustrated below.
Examples of the forecasts produced by the Club of Rome can be seen later on.
The examples above were from quantitative models; however the same principle can be applied when predicting on the basis of qualitative models which have explanatory power. Examples of these are found in Explaining a Development.
Note that although the causal type of explanation was discussed above and it is most often used for prediction, the function or motive types of explanation could also be used as a basis of a forecast.
Weak signals. The models that are available for forecasting are usually simplified so that they contain only the most important factors that affect the phenomenon to be predicted. Beside these, most empirical phenomena are influenced by a great number of minor factors, but their influence is so small that it disappears among errors or measurement and in the random fluctuation of the cardinal factors. Researchers often call these minor factors "noise" and simply disregard them.
Nevertheless, you cannot always count on that the relationships expressed in your model remain constant during the whole time span of your forecast. Sometimes it happens that a factor that until now has played only a marginal role will suddenly gain importance and will finally change the direction of the development. Such factors which initially seem unimportant but finally become crucial are sometimes called weak signals. In order to identify such factors you can try various approaches, such as:
- Try to contemplate the context of the phenomenon in a wider perspective.
- Consult other experts (perhaps with the Delphi method) and ask their opinion about the model that you are using. If your original model was made in cooperation with experts that have a long experience of the phenomenon to be predicted, it is possible that they have been accustomed to overlook a factor that nevertheless can be important quite soon. Other people, less experienced with the phenomenon, may sometimes find new and surprising vantage points to look at the phenomenon.
- If you can find another system that has already undergone the development that you are trying to predict, you can use the analogy method and inspect whether the factors that have caused this development are present in the system to be predicted, albeit perhaps in an embryonic degree.
- If your model is quantitative, you can try to find out which independent variables have had an gradually increasing influence on the phenomenon to be predicted. This can be done, for example, by calculating twice the correlation between such a factor and the phenomenon: one correlation with data measured at an earlier time, and another with latest measured data.
Determining Limits
It is often advantageous to use one method for the short time part of the forecast and another one for the long-term period. For the near future linear extrapolation is often useful, while it often happens that common sense, research, or other source of general knowledge tells you that the evolution that you are forecasting is subject to pre-set limits or laws which dictate not the nearest events but rather a more distant future. You may, for example, be studying the growth of a plant knowing that the steady growth will eventually reach an end.If that is the case, you may combine two forecasting methods: you extrapolate just the nearest values, while basing the forecast of the later values on a general law. Typical examples of such long term developments are:
The s-curve is usual if the growth has natural limits, as is the case with plants and with the natural resources of the earth,- the "catastrophe curve" describes the end of a development which has had a smooth, gradual start but is nevertheless eventually expected to reach a complete extinction, perhaps an abrupt one.
Assessing and Describing Uncertainty
Predicting the reliability of your prediction will be necessary, if you intend to describe not only a few possible futures but also the most probable of them. For this task there are not too many methods. In the following are enumerated some of the most usual ones.One of the best is triangulation: making parallel forecasts with different methods if it is possible. If different methods lead to dissimilar forecasts, it gives an idea of the range of the uncertainty.
Sensitivity analysis is another method which, however, works only with numerical models. Most forecasting methods allow you to calculate what the result will be if one of your starting assumptions or one variable in the input data is varied. This is a useful method of measuring whether the weight of a so called weak signal is growing.
Moreover, if you believe that you know the probable error of one of your assumptions, you may use this knowledge to calculate the probable error of the resulting forecast.
Once the researcher has developed for himself an approximation of the likelihood of the forecast, the next task is to disclose this likelihood to his public as well. Many usual methods of presenting the forecast (like diagrams) are very exact, indeed their very exactness often badly corresponds to the uncertainty of the forecast. Instead, the researcher should select such a presentation of the forecast which gives the right impression of the degree of uncertainty. There are, indeed, a number of methods which can be used to describe the probable error or likelihood of a forecast:
- The likely variation. This is indeed the normal fashion when using the Delphi method which nearly always produces a large set of different prophecies made with various methods. If the questions in the Delphi questionnaire are quantitative, you will also be able to calculate an expression for the dispersion of the responses, e.g. their range. Sometimes, it will also be possible to calculate the probability that the factual outcome will not exceed the range.
- A fuzzy curve. This can be accomplished by drawing the curve as a thick line or by free hand.
A fuzzy scale. For example, the Club of Rome deliberately chose to omit the vertical scales of the variables and also made the horizontal scale somewhat vague (consisting of just the values 1900 and 2100). This was because they wanted to indicate that the numerical values were approximate (ibid., p. 123).- Verbal explanation, even when the forecast has been quantitative, can be used to describe the likelihood of the forecast. The drawback is that people have quite differing notions of what is meant by "probable" or "likely", for example.
- Parallel scenarios.
Their "standard" scenario, shown above, assumes that all the variables follow their historical values from 1900 to 1970. Food, industrial output, and population grow exponentially until the rapidly diminishing resource base forces a slowdown in industrial growth. Population growth is finally halted by a rise in the death rate due to decreased food and medical services.
There is another scenario from the same book (fig. 36) on the left. Here the assumed resource reserves were doubled, while all the other assumptions were kept identical to the "standard" scenario. Industrialization can now reach a higher level. The larger industrial plant releases pollution at such a rate, however, that the absorption mechanisms of the environment become saturated. Then pollution causes an immediate increase in the death rate and a decline in food production.
A third scenario from The Limits to Growth (fig. 44, on the right) is identical to the "standard" one, except that the population is assumed to stay constant after 1975. The industrial output continues to grow exponentially until the depletion of non renewable resources brings a sudden collapse of the industrial system.WWW sites on forecasting:
On the questions of reporting, a separate chapter is included.
