Abstract
The title of this entry requires some explanation. I use the term ‘stochastic models’ to distinguish those theoretical models which include one or more stochastic variables from ‘determinist models’ which do not. I shall confine attention to some stochastic models which are obtained by introducing into a determinist model a single stochastic variable (which can be multivariate, but will in illustrative examples be univariate). I shall use the term ‘generating system’ to mean a determinist model in which from an initial state of the system an unending sequence of successive states of the system can be exactly predicted by means of a set of rules such as lagged equations. It is convenient to distinguish generating systems from stochastic models rather than extend the former class to include some or all of the latter. The important feature of stochastic models is that they can make allowance for wide margins of uncertainty and ignorance.
Access provided by CONRICYT-eBooks. Download reference work entry PDF
Similar content being viewed by others
Introduction
The title of this entry requires some explanation. I use the term ‘stochastic models’ to distinguish those theoretical models which include one or more stochastic variables from ‘determinist models’ which do not. I shall confine attention to some stochastic models which are obtained by introducing into a determinist model a single stochastic variable (which can be multivariate, but will in illustrative examples be univariate). I shall use the term ‘generating system’ to mean a determinist model in which from an initial state of the system an unending sequence of successive states of the system can be exactly predicted by means of a set of rules such as lagged equations. It is convenient to distinguish generating systems from stochastic models rather than extend the former class to include some or all of the latter. The important feature of stochastic models is that they can make allowance for wide margins of uncertainty and ignorance.
By a ‘divided population’ I shall generally mean a frequency distribution most of which is closely clustered around two or possibly more peaks, but fairly empty elsewhere: an extreme case would be that where the peaks were completely separated by an unoccupied stretch. However, the term ‘divided population’ can occasionally be extended to refer to a society which is divided into groups with contrasted living conditions, prospects and aims.
The term ‘crisis’ refers to an unstable situation where a small disturbance could tip the scales between the prospects of two widely different eventual outcomes. In a determinist model the representation of such a crisis would be a point of unstable equilibrium, and in a stochastic model based on that determinist model, one would still regard the point of instability as indicating crises facing that part of the population found close to it, by ‘crises’ meaning here the crises that chance might play a predominant part in determining their future prospects.
As a preliminary to the main discussion, it will be helpful to consider some standard tools for use with determinist models involving divided populations and crises.
Some Standard Methods for the Study of Unstable Situations
A standard method of constructing a model of the response of an economic system to the passage of time, or to possible changes of policy or of outside influences, is to set up a generating system giving a set of initial conditions containing the present and recent values of a set of economic variables and policy parameters, together with a set of rules for calculating the set of the same variables one time-unit later and repeating this operation successively for any required number of time-units. Such rules would normally take the form of a number of equations giving the values of each variable as functions of the values of other variables, mainly at earlier dates, taking account of the present values assumed for any policy parameters. We may confine attention to very simple examples of such models.
It is quite usual to find in simple models that given the initial information, the application of the system of rules with fixed policy parameters will generate a sequence of sets of values of the variables which tend to a long-run equilibrium set, apart possibly from one or more constant common growth-rates. But it is also possible to frame fairly simple rules which lead to oscillations which persist at a constant amplitude. Often these will be smooth and sinusoidal, but there is another possibility which is the one relevant to crises, where there are periodic jumps from one smooth steady path (which we might call boom) to another smooth steady path (which we might call slump) alternately to and fro indefinitely.
A convenient tool for the representation of such systems when there are sufficiently few equations involved is the phase diagram. If we are dealing with difference equations of the kind just described, the axes of the diagram could measure the values of one important variable along the horizontal axis as independent variable and the change of that same variable over the next time-unit along the vertical axis as dependent variable. In such diagrams the curve relating the change of the variable as a function of the value itself will reveal points of equilibrium by its intersections with the horizontal axis: however, where the curve cuts the axis from below on the left, the equilibrium will evidently be unstable, and we shall call such equilibrium points crisis points. Figure 1 is a phase diagram applicable to the determinist model described below in section “Rules for a Model Generating Lines of Bequests”.
Phase diagram with crisis point K
In Fig. 1 the horizontal axis is cut by the graph ITJKLSB in three points J, K and L denoting equilibrium levels of the index of prosperity, but the point K is a ‘crisis point’ indicating an unstable equilibrium value. The arrows following the paths starting from A and from near K illustrate how, given the level of the index in an initial period, the chart may be used to predict its values in later periods assuming the rules of the model to be obeyed. For example, to follow the changes from the initial value of 1000 at A on the horizontal axis, measure horizontally the same (negative) distance AA1 as the vertical distance of the graph from A. Having marked A1, for the value after one unit of time, repeat the operation from A1, to mark in A2 and so continue as illustrated in Fig. 1. It is apparent that the series of such values must converge to the value marked by S, and similarly that starting from C, near K on the right, when the distances concerned will now be positive, (to the right or upwards), it should again be clear that the series obtained must converge to the same stable equilibrium value S. Finally, starting from any point to the left of K, the usual procedure must result in a series arriving at the other stable equilibrium point T: (this is so because the gradient of the line ITJ is −1 in this example, which entails that as soon as the path hits ITJ it leads directly to T). This illustrates why the equilibrium at points, such as K, where the axis is cut by the curve from below on the left are unstable, while the points such as T and S mark stable equilibrium values.
Fig. 1 will be used again in section “Rules for a Model Generating Lines of Bequests” to illustrate the numerical example there, which involves equations (3) to (6): Table 1 in that section provides the first few values of the sequences that would be obtained by applying the rules, starting from values 1000, 100 and 70 respectively.
An early example of a determinist economic model involving oscillation between two points of stable equilibrium across a gap containing a crisis point of unstable equilibrium, due to the interventions of a disturbing force moving the phase-curve, was the model of the trade cycle published by Kaldor in the March 1940 issue of the Economic Journal. This contained a diagram closely related to a phase diagram of the elementary type shown here in Fig. 1, and which relied on the property that the curve itself moved upwards or downwards, depending on whether the currently relevant point representing equilibrium was on the right or left of the diagram.
Figure 2 is a transposition of Kaldor's diagram into a phase diagram of the type outlined above. Three positions of the curve are shown marked 0, + and *. Initially the relevant point of intersection is B a stable prosperous stable equilibrium point: K and S mark the currently irrelevant crisis and slump equilibrium points on this curve. During the boom the curve moves down to the position + + + at which B and K meet at the point K+ of tangency and the curve loses contact with the horizontal equilibrium axis so that the relevant equlibrium shifts rapidly to D*, the slump stable equilibrium, and now the curve moves upwards past position 0 to position +, at which S and K coalesce at the new point K* of tangency, and again the curve loses contact with the equilibrium line, so that now the relevant stable equilibrium moves rapidly to the right to B*, the boom stable equilibrium. Then the curve moves downwards again through the position 000 and the story is repeated again and again with alternative stable equilibria B and S in boom and in slump.
Phase diagram: three curve positions
Since World War II a number of models have been based on such non-linear differential or difference equations to produce fairly regular switching between temporarily stable situations of slump and boom. An early and particularly neat example was provided by R. M. Goodwin in a paper delivered in June 1955 to a meeting of the International Economic Society in Oxford. An account of that and other such early models will be found in chapter 8 of Mathematical Economics by R. G. D. Allen (London: Macmillan; New York: St Martin's Press, 1956).
The model in section “Rules for a Model Generating Lines of Bequests” with its crisis point has much in common with those of Kaldor and Goodwin and later writers, but in section “Easy Rules for a Stochastic Model of a Divided Population” we shall develop it in a different direction by introducing stochastic disturbances so that it may be used for modelling the development of bimodal distributions. The point which that model is intended to illustrate is that a few simple ingredients which may underlie a number of complex situations in which bimodal frequency distributions may alone be sufficient to produce bimodality, without any of the many further influences which may also be possible explanations of it. It is quite plain that such a model is not in fact a complete explanation, but it may be helpful as illustrating a method of taking a first step in a variety of investigations of situations where divided populations are observed.
The simple ingredients alluded to above are as follows:
- (1)
A set of largely unidentifiable and unexpected disturbances to each member value of the population whose distribution is being generated. This may well increase the dispersion.
- (2)
A set of influences encouraging the growth of large member values and the declines of small member values.
- (3)
Opposing these influences, 1 and 2: specific measures taken to discourage further growth of very large member values and reverse the fall of very small ones.
These three ingredients will often be sufficient to produce a bimodal distribution. We have not included in (3) the many influences that there may be operating to diminish or reverse the effects of ingredient (2) at intermediate levels: where this omitted set of influences is strong, a unimodal distribution is likely to be found.
Rules for a Model Generating Lines of Bequests
In this section we consider a population consisting of family lines within which bequests are passed down generation after generation according to a mechanical system of rules governing inheritance, earnings, consumption, taxation, subsidies and dividends, and which lead all family lines eventually to ruin or to considerable wealth.
The same rules apply to each and every line of bequests, which differ only in the level of the initial bequest. We may denote the level of the initial bequest in a representative line as B0 and the level of the bequest in that line t generations later as Bt. We shall set out in the next paragraph a set of rules which entail that the following bequest, Bt+1 in the line is always obtainable from one or two linear equations from the current bequest Bt. For reference these equations (1), (2), are set out below and followed by an explanation of the notation and by a description of the rules governing the accumulation of wealth for bequests and implying these equations.
When Bt < WEX,
if Bt exceeds P but otherwise
or if this is <0, Bt+1 = 0.
When Bt> WEX,
Both (1) and (2) operate if Bt = WEX.
The meanings of the symbols T, E, C, R, TAX and WEX are as follows: T = length of generation in years: we take T = 25 for examples, E = level of earnings per annum: we take E = 10 for examples, C = consumption expenditure per annum: C = 12 for examples, R = interest rate for dividends per annum: R = 2.5% for examples, P = level up to which bequests less than it are subsidized, P = 50 for examples, TAX = rate of tax of bequests starting at exemption level WEX for tax on bequests: TAX = 2/3 or 3/5; WEX = 250 or 400, in examples.
The four rules which lead to the equations (1) and (2) are:
Rule 1. So long as any of a bequest remains it constitutes a fund attracting interest at the rate R per annum and provides a source from which the excess expenditure (C − E) can be maintained.
Rule 2. If the whole of a bequest gets used up before the end of a generation, consumption is cut from C to E, (debt is ruled out) and in this case the bequest (before subsidy) must be zero.
Rule 3. Every bequest consists of the accumulated fund at retirement (before tax or subsidy), which fund may be zero.
Rule 4. The tax or subsidy on the bequest Bt is applied at the moment of payment to the heir, so that the heir receives Wt out of Bt< P where Wt = Bt − TAX. (Bt− WEX) if Bt > WEX, Wt= P if Bt< P and Wt= Bt otherwise.
If we denote the value of the fund after u years by F(u), the derivation of equations (1) and (2) follows directly from the rules by solving the differential equation dF/Fu = R ⋅ F(u) − C + E by standard methods to obtain F(T) given F(0) = Wt which may be found by Rule 4.
The equations (1), (2) and a knowledge of the values of the parameters T, E, C, R etc. and of the initial bequest B0 of any line now enable us to derive the whole line of bequests B0, B1, B2, … as far as we wish and to find the limiting value in long-run equilibrium, by repeated application of the relevant equations.
The operation of the model can be illustrated by a phase diagram if we select values for the parameters.
Putting
we obtain, when Bt < 250,
but if Bt < 40;
But when Bt > 250,
and can derive the phase diagram, Fig. 1 in which the curve cuts the horizontal axis in the three equilibrium points at T, K and S at which Bt = 23.95, 80 and 641.26. The values of the turning points J and L are those of P and WEX, 50 and 250 (see section “Some Standard Methods for the Study of Unstable Situations”).
Table 1 covers ten generations and gives the values of the bequests in three lines starting at 70, 100 and 1000.
Statistical Methods for Studying Distributions with Many Peaks
Twin-peaked distributions often arise in situations where there are three equilibrium points of which two are stable, but the third is unstable, and lies between them. In such unstable situations the fact that an initial distribution contains individuals on both sides of the unstable crisis point will ensure that the population will eventually be divided into two groups at or close to the two stable equilibrium points.
The same mathematical device that underlies the crisis models generating alternative progressions to the two stable equilibrium points, or in some models regular switching from one to the other across an unstable one, may be adapted to represent situations which produce a frequency distribution consisting of two peaked distributions each centred on stable equilibrium points on either side of an unstable one. The adaptation may consist of the introduction of rules for moving the curve that indicates the equilibrium points, as in Kaldor's models, or by the introduction of rules disturbing the point indicating the current state of affairs off that curve: that is the line we shall investigate.
The whole frequency distribution may either continue strictly positive across the neighbourhood of the unstable point between the two peaks, or be split into two entirely separate distributions, with the unstable point left in the gap between. In the class of models which will be discussed in section “Easy Rules for a Stochastic Model of a Divided Population” of the entry, the split version can only emerge if the rules governing the stochastic disturbances to the movements of the points (representing the individual values whose frequency distribution is being generated) do not enable individuals to arrive at or cross the unstable equilibrium point. This is a very stringent condition, but it represents an intermediate case between the stochastic model generating the unbroken two-peaked equilibrium distribution and the cruder determinist models generating a long-term equilibrium with all individual points concentrated at the two stable equilibrium points. More elaborate determinist models with lagged variables may lead to undamped regular oscillations about a single equilibrium point, but these will not be further discussed in this entry.
In the past, economists were largely concerned with the study of equilibrium positions towards which market competition and other social and economic forces would drive the economic individuals and conglomerations involved. The particular concerns of the statisticians and econometricians were more often with the movements of those equilibrium points and with the dispersion of the individuals or groups around these points. In the simple cases where there was just a single equilibrium they might for example study the shapes of the frequency distributions of the one or more coordinates of the point, and suggest and test various theories to explain how such shapes could arise, as well as what caused the movement of the equilibrium point itself. Thus there have been theories to account for and predict the age-distributions of the populations of various territories, the size-distributions of their cities and the distribution of the shares of votes cast for a particular party in the various constituencies, and again the distribution of income, wealth and other measures of prosperity, both between individuals and between various groups of persons.
However, if stochastic disturbances can interfere with the equilibrating forces or even shift the three equilibrium points, there may be preserved a considerable spread of distribution around each stable equilibrium point, and if the stochastic disturbances are strong, there may still be movement between the two groups across the unstable equilibrium point. In the former case we should expect two separated equilibrium distributions, whose relative sizes would depend on the nature of the initial distribution, but in the latter case a single continuous but probably bimodal equilibrium distribution whose shape could well be independent of the initial distribution.
In those cases where a considerable valley between the two peaks of the long-term equilibrium distribution is preserved, the stochastic mechanism is quite different from the simple determinist explanation for such a bimodal equilibrium distribution: this determinist explanation is simply that two quite distinct populations have been juxtaposed and counted together as one population. For example, if a wealthy island were to annexe an impoverished island with roughly the same population and then compiled wealth-or income-distribution figures for the two combined, one might expect a fairly stable bimodal wealth-or income-distribution. However, with good transport between the two islands one might expect eventually that the later generations sprung from the impoverished island would acquire gradually some of the cultural and other advantages of the descendants of the wealthy islanders and, vice versa, some of the descendants of the wealthy islanders and, would be impoverished as a result of the competition of the more gifted immigrants from the other island. Thus in the long run the stochastic intermingling of the two races might make the stochastic model more relevant than the determinist analysis of the equilibrium distribution to be expected.
In section “Easy Rules for a Stochastic Model of a Divided Population” we shall explain how to introduce a stochastic variable into our determinist model of lines of bequests so as to change it into a stochastic model of the distribution of bequests in successive generations and in the following section will provide some numerical examples to suggest some questions which such models might be helpful in answering if they were suitably elaborated. These questions will be related to situations featuring an apparent contrast between two overlapping groups, ‘poor’ and ‘rich’, where the ‘persons’ to whom the distributions refer may be individuals or households or localities or larger groups such as whole economies.
Easy Rules for a Stochastic Model of a Divided Population
In our stochastic model we shall consider the distribution of bequests in each generation over a series of value-ranges of equal proportionate extent g. We shall suppose the top of range 0 to be P, and we shall number the ranges so that for each integer i, positive or negative, the top of range i is Pgi. We assume that initially all bequests are at the centres of the ranges, and we impose rules to ensure that the same is true in every ensuing generation. This simplification makes rather narrow ranges desirable so as to avoid introducing considerable inaccuracy, but in illustrative examples we shall have to use wide intervals with g = 10.2 = 1.585 or 10.1 = 1.26 so as to be able to set out the results in the space available.
The stochastic model differs from the deterministic one of section “Rules for a Model Generating Lines of Bequests”, in modifying the value of Bt+1, there calculated from Bt. Denote that value by Bt+1(i) when Bt is at the centre of range i, then in the stochastic model we multiply it by a stochastic multiplier mi, which when N = 2 scatters the bequests of Bt+1(i) to the centres of 4 (or more generally, of N + 2) consecutive ranges, in proportions that leave their arithmetic mean equal to Bt+1(i).
In section “Rules Setting the Probabilities of the N + 2 Values of mi” the rules for choosing the ranges and the proportions of bequests moved to each of them will be set out but nonspecialists may prefer to skip that section and be content with Figs. 3 and 4 below, which illustrate typical effects of the multiplier with (i) g = 1.585, N = 2 and (ii) g = 1.26, N = 7.
Stochastic disturbance about 400
Disturbance about 400 (N = 7; g = 1.26)
We may then work out for each range i from 0 upwards, the following bequest Bt+1(i) by the formulae (1) and (2) of the determinist model and apply the stochastic multiplier to the bequests in lines from each range i, so as to split them into sets going, when N = 2 to the 4 (or more generally, N + 2) appropriate consecutive ranges. By using the information for every range containing at least one bequest where we round off to the nearest integer, assuming a total number of one million bequests in each generation, it is then a matter of arithmetic, to find the size-distribution of bequests in generation t + 1 from that of bequests in ranges with non-negative i in generation t.
We still have to explain how to handle bequests in the ranges with i < 1, namely the ranges below the level P. It is again assumed that all bequests in such ranges are subsidized up to the level P. Indeed, some such egalitarian measure as this is needed if we are to avoid all bequest lines eventually becoming permanently zero or in a range well below P, except possibly for a wealthy set all considerably above the crisis, level (C − E)/R. So in the calculations we merely have to lump all the bequests in ranges with i less than 1 into range 0. This need not prevent there being bequests before subsidy in each generation in other ranges below P, and it is the distribution of ranges before subsidy that we shall calculate in examples and which are relevant to the distributions of wealth and dividends which are all available towards the retiring age.
We shall give a very few numerical examples of such distributions in section “Model Generating 2-Peaked-Distribution: Illustrative Cases” to which those uninterested in the details of the rules for the stochastic multiplier are advised to skip. Those rules will now be outlined in section “Rules Setting the Probabilities of the N + 2 Values of mi”.
Rules Setting the Probabilities of the N + 2 Values of mi
These rules will be illustrated by the case N = 2; where normally mi may take 4 values gj, gj+1, gj+2, gj+3, where j is an integer. There are, however, two simple special cases where only three consecutive integers are taken, the fourth value having no bequests dispersed to it. The rules ensure that these two cases give probabilities 0.25, 0.5, 0.25, 0 and 0, 0.25, 0.5, 0.25; the three non-zero terms are those of the binomial expansion (0.5 + 0.5)2. Similarly, where N is any positive integer there are two special cases where N + 1 ranges only may be occupied with probabilities given by the N + 1 terms of the binomial expansion (0.5 + 0.5)N. Returning to the special case N = 2, the rules further provide that the value of j and the proportions of the bequests should be so chosen that the four proportions are a weighted average with weights 1 − p and p (0 < p < 1) of the two special cases with three terms each, and that the arithmetic mean of the bequests should equal the value Bt+1(i) obtained in the determinist model. This entails that the four proportions should be the following: (1 − p)/4, (2 − p)/4, (1 + p)/4 and p/4. The arithmetic mean of the bequests must then be (1 + pg). Pgj (1 + g)2/4 so that our rules require
where the right-hand side is known. This uniquely determines j and p and they may easily be derived.
In the general case where N is any positive integer the main modifications are that the binomial expansions in the special cases are now (0.5 + 0.5)N and that in equation (7) and the preceding line, 4 must be replaced by 2N.
Model Generating 2-Peaked-Distribution: Illustrative Cases
In this section we shall illustrate the kinds of two-peaked distributions that are generated by such simplified stochastic models and the ways one might use them, by a few numerical exercises involving an imaginary set of a million lines of bequests. We shall mainly use arithmetic and diagrams for the exposition.
Let us start with the standard values for the parameters given in section “Rules for a Model Generating Lines of Bequests” equation (3) as
and in section “Easy Rules for a Stochastic Model of a Divided Population” as
The long-run equilibrium distribution obtained with this set-up will depend on how widely the stochastic multiplier disperses the bequests from each range in a single generation, and this is set by the parameter N. Figures 5, 6 and 7 show that widening the dispersion in this example through the first four values, N = 0, 1, 2 and 3 already exhibits a wide variety of types of solution. Case N = 0. Two separated distributions: in ranges −2 and −1; and in ranges 10 and 11. The proportion of bequests in the two distributions will equal the initial proportions that were in ranges up to and including 1 and in ranges 2 and above. Cases N = 2 and N = 3. All bequests are in ranges immediately below P; 4 of them when N = 2 and 5 when N = 3. Cases N = 4 and over. All bequests are in a single distribution extending over many ranges from well below P up to well above WEX the tax exemption limit. These are the most interesting cases. Some have a pair of peaks separated by a valley, but those with N taking higher values have only a single peak.
Equilibrium bequest–distributions
Equilibrium bequest distributions
Equilibrium Pareto Curve (non-cumulative)
Figure 5 illustrates the case N = 0 where we have assumed that half the bequests in the initial generation were in ranges 3, 4, 5 etc. … and half in the ranges 0, −1, −2 etc. The result, due to the minimum value of N, is a very divided distribution little different from the complete division that would be found in the determinist model: the case with N = 1, not shown, is less stark, allowing a spread over five ranges in the upper peak and over three in the lower peak. Figure 6 shows the unusual cases where in the long term there are no bequests in any range above P. With the particular values we took for the other parameters this unusual feature only occurs when N = 2 and N = 3. Figure 7 compares the typical bimodal form when N = 4 with the typical unimodal form when N = 18. The logarithmic scale used may given the deceptive impression that when N = 4 the valley between the peaks is not deep and therefore easily crossed, but a more careful inspection will reveal that it is very deep, since the valley floor indicates a range with roughly 10,000 bequests, whereas even the lower peak indicates one containing roughly 100,000 bequests.
Figures 8 and 9 are mainly concerned with a potentially instructive use of stochastic models for investigating the effects of altering one or more of the policy parameters. They do this for two examples of reflationary fiscal policies: raising WEX the tax-emption level from 250 to 400 (Fig. 8) and altering TAX from 3/5 and 2/3 to 50 per cent (Fig. 9). The effects of the higher exemption level are to deepen and widen the valley and shift the tail to the right along with the level of the exemption limit, without affecting its Pareto slope. The effect of the tax
Effect of higher tax-exempt limit
Effect of lower tax on bequests
reduction is mainly to lessen the Pareto slope of the tail. Each measure greatly increases the total number of bequests above 5000; the tax reduction by four or fivefold and the higher exemption level by more than tenfold. This is perhaps the right moment to reiterate the warning that such examples are not meant to be more than indications of elementary methods for testing hunches of what are the probable logical effects of such changes given any set of artificial rules being mechanically obeyed.
All our discussion has been concerned with equilibrium distributions. However, as in so many branches of economic theory, knowledge of the eventual equilibrium corresponding to the present state of the economy and current policy decisions is virtually useless unless one knows how rapidly that equilibrium will be approached and particularly what will happen during the reasonably near future.
This also can be illustrated with our elementary example. It is to be expected that with low values of N, approach to equilibrium may be very slow and that is only too well confirmed by a wide variety of examples not further reported here. It is more interesting to give the stochastic disturbances considerable scattering influence by choosing quite a high value for N and then following the pace of approach towards equilibrium from an initial distribution of bequests chosen so as to differ considerably from that equilibrium distribution.
But in Fig. 10 we have taken our standard example, with N = 7, and shown for each range the difference between the equilibrium number of ranges that would result from altering TAX from 3/5 to 2/3: we also show by the two intermediate curves how much of the approach to the new equilibrium would in each range be achieved after five and after ten generations of 25 years each. It will be seen how far from completed the transition is even after the 250 years.
Short-term moves to equilibrium
Space forbids showing further examples of the short-run effect of altering policy or other parameters in such models, by methods similar to those used in studying long-run effects: naturally such short-term enquiry can be more important for obtaining conclusions remotely relevant to the real world, yet the theoretical approach to such investigations need differ little from that to the long-term ones discussed above.
Concluding Observations
In the later sections of this entry we have provided an illustration of how, without introducing any of the more detailed causes of divided, i.e. of bimodal, frequency distributions, one can obtain a skeletal model of the development of such distributions by merely including the dispersive elements: (a) an ability and willingness on the part of the richer, to save a higher proportion of their income than on the part of the poorer, and (b) stochastic disturbances; both (a) and (b) tending to increase inequality; and the egalitarian elements: (c) to curtail (a) and (b) on the part of the very rich; and (d) subsidies to set a limit on the poverty of the very poor. We have hinted how models including at least these four basic elements, and thereby generating biomodal distributions, can provide some insight into possible long-run and short-run effects of altering policy parameters: in particular we have argued that considerations of long-run equilibrium can be very poor guides to short-run effects.
The next step is, naturally, to introduce into the model the more obvious and significant other causes acting to modify distributions of wealth, income, health, nourishment and other measures of well-being. That step is far too long for inclusion in an entry of this nature. Moreover, since although situations of the critical and divisive kinds, on which such research aims to throw some light become more frequent every year, statistics of these phenomena are patchy and unreliable. The development of stochastic methods to make allowance for the unreliability and incompleteness of information is by no means a minor step in such enquiries. It was this belief that prompted the submission of this methodological entry.
Author information
Authors and Affiliations
Editor information
Copyright information
© 2018 Macmillan Publishers Ltd.
About this entry
Cite this entry
Champernowne, D.G. (2018). Divided Populations and Stochastic Models. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_673
Download citation
DOI: https://doi.org/10.1057/978-1-349-95189-5_673
Published:
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-349-95188-8
Online ISBN: 978-1-349-95189-5
eBook Packages: Economics and FinanceReference Module Humanities and Social SciencesReference Module Business, Economics and Social Sciences