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Certain major topics on the subject of interregional and international migration of population stocks, as well as of the interregional and international flows of the two central input factors in the economic production process – namely capital and labor – are addressed in this paper. The key findings are that: first, uncontrolled global migration inevitably leads to a uniform utility level equal to the globally prevailing average: that of the average Indian and Chinese peasants. Second, that uncontrolled flows of capital and labor result in dynamically chaotic urban, regional and national economic structures. This paper revisits, summarizes and extends a number of papers (with a focus on three of them) by this author, written and published in the period 1975 – 1985.
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Dimitrios S. Dendrinos, Ph.D.
Professor Emeritus, University of Kansas, Lawrence, Kansas, US
In residence, at Ormond Beach, Florida, US
May 11, 2018
The 1979 “Equilibrium x2*, and Optimum x* City Size” Model by D. S. Dendrinos
Table of Contents
Brief Summary
Part I. Spatial Population Flows
The Alonso-Henderson-Dendrinos Model
Borders, Transport Costs inhibit Free Flow and Optimum Size
Part II. The Incongruous Spatial Free Flows of Labor and Capital
The Cusp Catastrophe Model
The Spatially Incongruous Labor and Capital Flows Model
Dynamics in Archeological Time and Space
Copyright Statement
Certain major topics on the subject of interregional and international migration of population
stocks, as well as of the interregional and international flows of the two central input factors in
the economic production process – namely capital and labor – are addressed in this paper. The
key findings are that: first, uncontrolled global migration inevitably leads to a uniform utility level
equal to the globally prevailing average: that of the average Indian and Chinese peasants. Second,
that uncontrolled flows of capital and labor result in dynamically chaotic urban, regional and
national economic structures. This paper revisits, summarizes and extends a number of papers
(with a focus on three of them) by this author, written and published in the period 1975 – 1985.
Much is being debated, and from a variety of viewpoints, in the current political arena about
countries with open or closed borders to population migration flows, be those legal or illegal. The
theoretical underpinnings of such heatedly debated issues, however, were set at a much earlier
and more innocuous time period, the late 1970s and early 1980s in the US, and drew from what
is now considered Classical Spatial (Urban and Regional) Dynamics and Economics literature. This
paper revisits some work this author undertook and published during the late 1970s and the first
half of the 1980s, specifically in the 1975 – 1985 time period, while a Professor of Urban, Regional
and Transportation Planning at the University of Kansas (since 1975), and a Visiting Scholar with
the US Department of Transportation, Research and Special Programs Administration, in
Washington DC (in the one-year period, from August 1979 till August 1980, while on Leave from
the University of Kansas). Specifically, three papers are revisited, which can be considered as
containing material that is found at the core of the current political debate on international
population migration, as well as the interregional and international flows of capital and labor.
The first paper (presented in Part I) contains an inter-spatial free flow of population model, which
the author stated in the late 1970s and early 1980s. This model is now being referred to in the
Urban Economics literature as the “Alonso-Henderson-Dendrinos model”. The crux of this paper
is its finding that uncontrolled borders result in a stable, globally uniform utility level, as high as
that of an average background utility, specifically that of an average Indian and Chinese peasant,
given the current population sizes of China and India and their spatial demographic conditions.
This is graphically shown in the preamble to this paper diagram.
In Part II of this paper, two models are presented which depict the spatial counter-flows of two
input factors in production, namely labor and capital, freely moving in space-time. The author
proposed and published these two papers in the first half of the 1980s. Their key finding is that
capital and labor move in opposite directions; and that their combined dynamics, in a nonlinear
mathematical phase space, create chaotic outcomes, for most cases and scenarios envisioned.
These two sets of spatial dynamical flow models (found in Parts I and II) were stated under an
assumption of free flow, i.e., in absence of either transport costs (implying no impedance and no
congestion externalities were incorporated into these models); or enforced borders (that is,
under an “open borders” policy). Here, transport impedance restrictions and border-related
policies are considered in revisiting these two sets of models. These restrictions allow for
different model outcomes, and for differences in market equilibrium and optimal flows’ sizes.
In addition, and in view of work in Archeology this author has undertaken in the time period from
August 2014 till today, as an Emeritus Professor, some insights from prior millennia of the Human
Existence, under what this author has labeled “archeological time” and “archeological space”,
are now incorporated into the Spatial Economics, Economic Geography, Regional Science, and
Urban/Regional Analysis/Planning models, enhancing our understanding of population migration
and flows of labor and capital in space-time over the span of millennia and continentally.
In this paper, some prior work by this author is revisited, enhanced and, in view of some more
recent work by the author, it is framed within a broader perspective. A number of papers are
cited that contain prior analysis that is now being revised and enhanced.
The first set of papers contain the spatial flow of population from some hinterland (presumably
containing rural or hinterland population involved in agriculture) into an urban area - presumably
a city - that exhibits the Urban Form found in the Spatial Economics literature, as it is stated by
(and it is based on) the Alonso model of urban population density and urban land prices, see
reference [1]. The 1964 W. Alonso model is fundamentally an extension of the original 1826 J. H.
von Thunen model of agricultural rents and land uses, see reference [2].
Work, undertaken by the author of this paper in 1979, which contains all the necessary for the
reader references to trace its pedigree, was first published in a 1980 US Department of
Transportation Report cited in reference [3]. More elaboration on the 1979 Dendrinos model, a
model that is now being referred to in the Urban Geography and Economics literature as the
“Alonso-Henderson-Dendrinos” model, is found in the author’s paper cited in reference [4], and
in the book published by this author (with Henry Mullally) cited in reference [5]. The model is
further discussed, elaborated and analyzed in Part I below.
In Part II, a set of papers, see references [6] and [7], are discussed which present the flows in
space-time of two input factors in the economic production process, namely labor and capital. It
was argued that these two spatial-temporal flows are counter to each other, and in their
interactive flow dynamics potentially generate turbulent outcomes. The work by the author, as
undertaken in the 1980s, contained spatial-temporal flows that were modeled under the
simplifying assumptions of absence of both transport costs and open regional/national borders.
Now the work is expanded incorporating both restrictions and the associated spatial impedances.
Their concomitant policies’ effects are analyzed as to the ensuing urban and regional structures.
Finally, in the concluding remarks of this paper, some insights are incorporated emerging from
the latest work by this author on subjects of Archeology, where the concepts of “archeological
time” and “archeological space” are involved (and on these notions, see references [8] and [9]).
These two key notions (of archeological time and space) allow one to extend the spatial and
temporal horizons usually encountered in standard Neoclassical Economics. The limited space-
time horizons of standard economic analysis (which incorporate either static or dynamic
equilibria, quite short in duration though when compared with archeological, millennia long, time
horizons) is now extended to allow for longer in duration dynamic conditions that do not, in
general, involve dynamic equilibria (but rather a series of dynamic disequilibria).
Hence, this paper is an extension of that prior work by this author, which was carried out in the
late 1970s and the first half of the 1980s. In revisiting these papers, a deeper look into the
complex spatial dynamics of flows involving population stocks, capital and labor is obtained.
Part I. Spatial Population Flows
The Alonso-Henderson-Dendrinos Model
This aforementioned model can be viewed as the fundamental model of inter-spatial population
movements. It generates the major finding emerging from the free flow of population. Stated in
summary and in simple terms, as the underlying Mathematics and Economics are quite involved,
a uniform and stable outcome is expected to result, if no restrains exist to in- or out-migration of
population. This outcome consists of relatively large settings with an average utility level equal
to a background (agricultural) one. Put differently, and in vivid terms, if population flows among
urban or regional or national settings is unrestricted, then the likely dynamically stable outcome
will be a globally prevailing utility level equal to that enjoyed by the average Chinese and Indian
(by far the currently biggest national population stocks on Earth) rural resident.
This model was first elaborated in the late 1970s by the author and published in ref. [3]. The
model is based on two previous models. First, the model by W. Alonso found in ref. [1], which is
in turn based on the model by J. von Thunen, found in ref. [2]. And then on the model by J. V.
Henderson, see ref. [10]. The basics of this model are set in the Report by the author of ref. [3],
and ref. [4]. A more detailed nonlinear dynamical analysis of this 1979 Dendrinos model is
presented by the author in ref. [5], Appendix IV, pp: 156-157, in a book written in collaboration
with Henry Mullally and published by Oxford University Press in 1985.
Figure 1 below captures with a simple diagram the inter-spatial dynamics and the essence of this
model. On the vertical y-axis of this diagram, the average expected utility level of a setting’s (be
that a city, a region or a nation) resident is plotted. That utility level is derived by the von Thunen
– Alonso – Henderson Neoclassical Economics based set of models. Along the horizontal x-axis of
the diagram, the scale (in terms of population size) of the setting is plotted. Whereas the von
Thunen based Alonso model proclaimed a linear positive relationship between utility level and
population size, the Henderson model demonstrated that when externalities of agglomeration
are incorporated into the analysis, the utility level is a nonlinear (in this case a second degree)
function of population size (scale).
The author added a dynamic component to this long run static von Thunen, Alonso, Henderson
model(s) by explicitly introducing the background utility level. This background utility level can
be a rurally prevailing utility level for the case of a nation’s city or region; or an internationally
prevailing background average utility level, in case of a Nation existing within a global economic
system with freely interacting settings. The average and uniformly prevailing background utility
is depicted by the horizontal line (or isocline of a mathematical nonlinear phase portrait)
intersecting the second-degree utility level depicting isocline. The two isoclines intersect at two
dynamic equilibrium points, x1* and x2*, so that x1*<x2*. Of the two equilibria, the smaller
population size x1* is an unstable one; whereas the x2* is a stable equilibrium. Furthermore, the
author’s extension of the Alonso and Henderson models allowed for the derivation of dynamic
equilibrium and optimum city (or any other setting’s) sizes.
Figure 1. The 1979 Dendrinos model. The author’s work, shown in the above diagram, extended
the Neoclassical so-called “New Urban Economics” type models by William Alonso and J. Vernon
Henderson. On the vertical y-axis the expected utility level U is plotted of an average resident of
a setting (a city, a region or a Nation); whereas, along the horizontal x-axis the population size of
the setting (again, a city, region or Nation) is plotted. Utility level U* identifies the uniformly
prevailing at the setting’s background (agricultural, in case the setting is a city or a region within
a national economy, or average international, in case the setting is a Nation). Arrows depict a
combined motion in both utility level and population size, at a setting enjoying specific locational
comparative advantages responsible for the specific form of the 2nd degree curve shown above.
Points x1* and x2* are dynamic market equilibrium size points, although x1* is unstable,
whereas x2* is stable. Instability implies that slight perturbations from equilibrium will lead the
setting’s population size to either a continuous increase (if the perturbation is to the right of x1*)
and to extinction (if the perturbation is to the left of x1*). The point on the utility-size path that
corresponds to a maximum possible utility level, designated as Umax in the above diagram, and
a corresponding population size x* (the point of an optimum city size) is a saddle in the general
case, meaning the dynamics there are unstable, approaching it from its left-hand size, and leaving
it on its right-hand side. The only exception is when the U*-line becomes tangent to the 2nd
degree curve (i.e., when U*=Umax, and x1*=x*=x2*) for some specific locational comparative
advantages within a National or International context, at some point in time. Point x2* is
dynamically stable, meaning that any slight perturbation (to its right or left) will lead the
combined dynamics to converge back to it. Source of diagram: the author, from ref. [3].
Hence, the author’s model concludes that the only dynamically stable solution is a relatively large
(as x1*<x2*) market equilibrium population size, enjoying an average utility level equal to a
background one. It is noted that in the above model, the locational comparative advantages
enjoyed by the setting within the National (or international) environment define the exact shape
of the second-degree utility function, i.e., shifts in local comparative advantages result in shifting
that curve, and hence in different U*max and x1* and x2* points. No matter however where
these three distinct points are on the x-axis and on the y-axis, the setting’s prevailing average
residents’ U* is always the average background utility level. This model points out that the
optimum setting’s size, x*, is not dynamically stable under market conditions in the general case.
The 1979 Dendrinos model is a confirmation of the dire consequences involved in a set of “open
borders” policies, as it demonstrated (almost half a century ago, and when such issues were not
at the forefront of the public debate, and when international migration flows were not as strong
as they are at present) how these policies will significantly and negatively impact the welfare of
settings which adopt such free population flow (mostly uncontrolled in-migration) public policies.
And especially so, for settings within which residents enjoy relatively high levels of (as defined by
Neoclassical Economics) utility levels. Even though the precise mathematical and economic
assumptions and specifications, namely variables parameters and model formulations, may seem
to be restrictive, the conclusions are robust, general, and quite palpable.
Borders, Transport Costs inhibit Free Flow and Optimum Size
Now, the assumptions regarding free flow in the above-mentioned model will be relaxed, and
the implications of relaxing these assumptions will be briefly and qualitatively examined. At the
outset it must be noted that this model, shown in Figure 1 above (and described in its Economics
and Mathematics in references [1] – [5], that contain extensive analysis that will not be repeated
here), population growth is assumed to be both of the endogenous demographic growth type, as
well due to (in- or out-) migration. If one wishes to restrict migration, for example by the
imposition of barriers or transport costs, then this new set of assumptions will result in either of
the following two effects.
If transport (of entering or leaving the setting) costs are now introduced into the calculus of the
average (existing or potential) resident, then the 2nd degree in shape combined utility level and
population size curve will shift (downwardly, lowering the Umax level, and shrinking the market
equilibrium population size x2*, by pushing it towards the origin of the Cartesian diagram in
Figure 1), hence resulting in a decrease of the stable equilibrium setting’s population size.
Further, if enforced restrictions on entry public policies are introduced (that may involve the
establishment of even physical barriers, in the form of strict borders enforcement policies
regulating in-migration), then this policy will obviously alter the market (no longer the outcome
from free flow) equilibrium point. The closed border policy will force it to the left of x2* in the
diagram of Figure 1.
Potentially, this type of closed borders public policy may lead the setting to its true optimum size,
point x* in Figure 1. This is another powerful and insightful outcome of the model, and indicative
of the model’s capability to incorporate instruments that produce policy-rich insights.
Part II. The Incongruous Spatial Free Flows of Labor and Capital
The Cusp Catastrophe Model
In 1982, the author published a paper (see ref. [7]) in which for the first time in the Spatial
Economics, Regional Science and Economic Geography literature the inter-urban/regional (or
more broadly, spatial) free market flow dynamics of two key input factors in the economic
production process, namely capital and labor, were modeled. Although the focus then was on
the Urban areas and Regions of a National Economy, the analysis could be as applicable to inter-
national capital and labor flows under unrestricted, free flow, market conditions.
That paper considered spatial-temporal productivity differentials in these two input factors of
economic production to be the governing elements in their free markets’ flows. Increasing,
constant and decreasing returns to scale conditions for these two input factors were considered,
under some applicable in space-time economic production technology.
A Cusp-type Catastrophe Theory mathematical model was shown to describe these spatio-
temporal dynamics. In general, it was uncovered that spatial economies (be those urban, regional
or National) exhibit a number of dynamic equilibria. Relatively low or high wages (reflective of
low or high labor productivity) differentials (no matter the various returns to scale in labor
conditions) characterized these market equilibria allocations of labor. But all regions, under free
flow in capital and absence of any type of capital controls or capital flows service costs/fees, and
under perfect spatial arbitrage conditions experience equal in capital productivity market
equilibrium points.
It was discovered that a variety of inter-spatial market equilibrium allocations of capital and labor
were possible. However, under free flow of both capital and labor, only under decreasing returns
to scale, the low labor productivity market equilibrium point was dynamically stable. Whereas,
relatively high labor productivity market equilibrium points were dynamically unstable under
increasing returns to scale, see figures 1, cases A and B, in ref. [7]. It was shown that when
constant returns to scale are present in the economic production process, the market equilibrium
is undefined meaning that any number of market equilibria can be the outcome, with their
stability conditions being undeterminable. In fact, the analysis implied in a straight forward
manner that a dynamically chaotic regime could be possible under those spatio-temporal flows.
That 1982 Dendrinos model examined also the cases where limits to labor and/or capital growth
were imposed. These limits could be the end result of any, among many possible, restrictions.
Some of these restrictions reflect endogenously produced (due to concentrations of both capital
and labor) externalities, such as congestion effects due to labor agglomerations. Other
restrictions could be those of public policy linked restrains on labor supply (such as border
restrictions). Limits to either labor or capital growth could be enforced by public policy
instruments directly affecting the free flow of these two input factors in production. Hence,
imposition of transport costs (fees to regulate congestion for instance) or high stock density
related taxes, or the presence of borders, can be some examples in the rich menu of such “limits
to growth” type public policies. This is where the Cusp-type Catastrophe Theory model appears,
see figures 2, 3, and 4 of ref. [7]. It is the case where issues of “optimal” inter-spatial (Urban,
Regional, National) resource (labor, capital) allocations can be addressed and determined.
What the 1982 Dendrinos paper uncovered is that under limits to growth, through any public
policy conditions, favor only those relatively low labor productivity locations (no matter the
returns to scale in production assumptions). Only these low labor wages locations exhibit (or are
capable of) dynamically stable equilibria. Under inter-spatially free flow of capital (meaning
uniformly, over space, equal interest rates – reflective of equal marginal capital productivity and
perfect spatial arbitrage conditions) labor is forced into (or is attracted to) regions of low wage
rates. This is the counter-intuitive result of the Cusp Catastrophe model of the paper in ref. [7],
which will be further analyzed in the subsection below, when another paper by this author, that
in ref. [6] will be discussed in juxtaposition.
Before doing so however, a few more remarks are needed on the Cusp catastrophe model of ref.
[7]. This economic-ecological model, as stated in section #3 of ref. [7], is of some interest because
it demonstrates the extreme sensitivity of the spatial dynamics to even slight changes in public
policy aiming at imposing (through a variety of means in either capital or labor) restrictions in
their spatio-temporal flows. As technology in production switches economics, from increasing to
constant or decreasing returns to scale, violent changes in the allocation of resources may result.
The nature of the ensuing dynamic equilibria can quickly, and at times drastically, reverse course
in the phase space of their motion, creating chaotic dynamics. These differences can be detected
in the variations of dynamical paths shown in the diagrams (phase portraits) of these various
models, in figures 2, 3, and 4 of ref. [7]. They can also be detected in the reshaping of the two
basic arcs, which define the two isoclines, corresponding to the paths delineating the temporal
and spatial course of capital and labor respectively.
The Spatially Incongruous Labor and Capital Flows Model
The counter intuitive result of the 1982 Dendrinos model presented earlier, namely that labor
can be attracted to low wage settings (be those Urban, Regional or National), was further viewed
and explored in the author’s 1985 paper, see ref. [6]. Whereas, paper [7] dealt with market
clearing allocations (the end result of demand and supply interactions), the paper in [6]
addressed in specific excess supply of labor and excess demand for capital conditions at settings
in the then underdeveloped or developing countries. It did so, due to the then (early 1980s)
obvious (the beginning of what is now globally prevalent) and large in scale movements of capital
(from developed to underdeveloped countries, regions or cities) and labor (from economically
underdeveloped to what was then considered economically developed spatial settings).
It was becoming increasingly obvious back then that the beginnings of large in scale movements
of potentially low wage labor was flowing from underdeveloped economies (origins/settings of
thick in volume excess labor out-migration) into developed economies, destinations/settings (of
potential predominantly low skilled labor in-migration) where the average prevailing wage rates
were much higher than those at the origins of out-migrating labor. Hence, flow of labor
(expressed as both surplus labor at the low wage origins of migration, and excess demand for
labor at the high wage destination of the migration) was determined by significant wage
differentials between origins and destinations of labor flows. It was so that, the greater the wage
differential between the origin and the destination, the greater in strength (volume) the flow of
potential labor recorded. Excess supply of labor at the origin was moving towards settings where
it was perceived that an excess demand for low skilled labor demand existed (at the destination
of the movement). Perceptions on expected differences in average prevailing wages (over an
economic time horizon) between undeveloped or developing economies and developed
economies were fueling international migration (of population in general, and labor in specific)
flows, mainly at the relatively low skills end of labor markets.
In effect, and in conclusion, population (potential labor) was flowing from low average wage rates
prevailing Regions/Continents (Africa, Asia, and Latin America) of the World into high average
wage prevailing Nations (the US and Western Europe, what is currently the European Union).
And it did primarily, albeit not exclusively, concern unskilled or low skilled labor sectors.
On the other hand, capital was flowing the other way: from the developed World (especially the
US and the European Union, that is the relatively high average wage Regions of the global
economy, where capital productivity was relatively high) into the either under-developed or
developing Regions (i.e., the relatively then low average prevailing wage Regions, and where
capital productivity was relatively low as well). That was the incongruous interspatial/temporal
labor and capital movement that the paper in ref. [6] sought to identify, explain and model for
the first time in the Spatial Economics, Economic Geography and Regional Science literature.
Simply put, and in restating the model in ref. [6], this author put forward the basic model of the
incongruous labor and capital spatial flows, in simple differential equations, stated as follows:
dx/dt = F[w(t) – w’(t)] > 0, dy/dt = [w’(t) – w(t)] > 0, w(t) > w’(t)
dx’/dt = F’[w’(t) – w(t)] > 0, dy’/dt = ’[w(t) – w’(t)] > 0, w’(t) > w(t)
where x and y are labor and capital stocks respectively, and w stands for prevailing wage rate at
a setting, with the prime symbol above being indicative of a developing setting (nation, region,
city). Index t stands for time. It is implicitly assumed in the above, incongruous flows, model that
x and y are fast moving variables; whereas, w is a slow moving (changing) state variable – acting
as a parameter in the short run. Symbols F, F’, and indicate functions locally specific,
meaning defined by “developed” and “developing” (or “undeveloped”) settings, the latter
category indicated by a prime.
The paper in ref. [6] identified these flows as examples of classical analysis regarding convection
in fluid dynamics. In essence, the 1985 Dendrinos “incongruous labor flow dynamics” model
established the chaotic inter-regional dynamical movements in the spatial flows of capital and
labor. In that sense, it enriched and expanded on the 1982 Dendrinos paper on the “labor and
capital mobility” Cusp catastrophe model. As it was the case with the model in ref. [7], the author
introduced in ref. [6] limits to growth which could be either due to endogenous in a region factors
or exogenous, the partial result of public policies exerted on either end of the migration of labor
movement. In specific, conditions were also explored with the 1985 Dendrinos model, as they
were in the 1982 model, that brought into the spatial analysis forces and factors from the field
of Mathematical Ecology. The chaotic dynamics involved were analyzed through computer
simulations. These public policy restrictions to (labor, population) growth could be the result of
border enforcement, that is to policies inhibiting the free spatial flow of labor. These policies
were shown to limit the range and mitigate the effects of chaotic dynamics.
At this stage, one must ask why the Cusp catastrophe type model of the Dendrinos paper in ref.
[7] produce the counter-intuitive result that low skilled labor is expected under conditions of
dynamic equilibrium to be in effect stuck to relatively low wage settings (be them cities, regions
or nations); whereas, the Dendrinos paper in ref. [6] and as further elaborated in the above
presentation, clearly identifies flows of low skilled labor into relatively high average wage
settings. A quest for a possible answer to this seemingly contradictory set of findings must be
sought for by analyzing the very core of these two models and their structures. And this leads to
the informative conclusion that under the specifications of ref. [7] the free flow of unskilled labor
among regions has as an end result the lowering of the prevailing wages at the receiving end of
such spatial labor flow, and the deterioration of the setting’s wage conditions.
Dynamics in Archeological Time and Space
Now, some possibly basic additions to the above models will be indicated, by incorporating
certain insights obtained by the author from his recent work on various subjects in the field of
Archeology. The above analysis looked at “economic horizons”, possibly time horizons that could
involve a limited length of time, certainly no longer than one or at most two generations in the
case of labor, far shorter in the case of capital – consider the almost instantaneous flows of capital
in Stock Exchanges for instance. In this short and last section of the paper, the author will attempt
in brief to explore the effects of much longer time horizons, what this author has labeled
“archeological time” unfolding in “archeological space”, as for example in references [8] and [9].
This term, “archeological time”, was defined in these two references in terms of centuries, and
specifically it covered a span of about a couple of centuries. The reader is directed to these two
references to obtain a full gamut of the implied assumptions of the notion of “archeological time”
and its associated components in terms of economic equilibrium and disequilibrium conditions
and their implied Nonlinear Dynamics.
The resulting Economics, Geography, and Mathematics of both notions (archeological time and
archeological space) necessitate a profound re-statement of the underlying Theory, a task far
beyond and in excess of this short paper. Hence, only brief and as succinct as possible
foundational statements will be afforded here and at present. Neoclassical Economics, in short,
are ill-prepared and wrongly structured to address such topics. New work is badly (one might
lament as woefully) needed. Again, for a slightly more extensive coverage, along these lines of
inquiry the reader is directed to ref. [8] and ref. [9] for more elaboration and exposition.
The guiding principles towards formulating this New Theory and work are simple. First and
foremost, in archeological time dynamic equilibrium never exists. Archeological settings are
marred by states of successive (at times violent and differing in nature) dynamic disequilibria.
The notion of convergence (or divergence) from steady states (dynamic equilibria) is largely
meaningless. Environmental shocks (events which are to a large extent exogenous,
unpredictable, extremely difficult to endogenously incorporate within any Neoclassical
Economics based model) play a significant role in the dynamics of cultures appearing and
disappearing in archeological space, over archeological time. New time constants and related
dynamical specifications are needed. Therefore, one can’t count on the roles that relatively fast
(in reference to archeological time) changing variables, such as technology, prevailing wage rates
or capital productivities, natural resources availability, etc., play in the large in scale migration of
population stocks and cultures in archeological space-time. Clearly, a new theoretical framework
is needed, and the one suggested in ref. [8] and [9], which was based in part on the Universal
Map of Discrete Relative Dynamics, see ref. [11], could supply such a means to address these
issues. That was attempted by an application to the subject of spatial-temporal mobility of
ancient artifacts model put forward in ref. [8] and [9].
In this “retro” type paper, an effort was made by this author to revisit and partially revise and
extend some early models of Spatial Dynamics proposed by the author back in the late 1970s and
in the decade of the 1980s, and to reconcile some of their differences at some specific findings
they entailed. In view of the current debates on subjects of migration (in both capital and labor,
as well as population flows) in our contemporary World, these models acquire new interest and
offer some fundamental insights into this discussion and on the ensuing public policy issues, such
as the consequences of an “open borders” public policy.
The Economics-defined “long term” dynamic equilibrium spatial configurations, resulting from
such policies, were analyzed under various contexts. At the end, a call was made towards
restating the fundamental theory of Spatial Dynamics in a new and post-Economics (Neoclassical
Economics, that is) framework involving two new notions, and specifically what this author
defined as “archeological time” and “archeological space” in order to address some issues that
transcend the limited temporal and spatial scope of Neoclassical Economics. Some brief
comments were added at the end of the paper, extending some recent work by this author along
these lines. Further work is obviously (and rather urgently) needed, of course.
[1] William Alonso, 1964, Location and Land Use, Harvard University Press, Cambridge, Mass.
[2] Johann H. von Thunen, 1826, Der isolierte Staat in Beziehung auf Landwirtschaft und
Nationalekonomie, Hamburg.
[3] Dimitrios S. Dendrinos, 1979, “A Basic Model of Urban Dynamics as a set of Volterra-Lotka
Equations”, in Dimitrios S. Dendrinos, June 1980, Catastrophe Theory in Urban and Transport
Analysis, Report #DOT/RSPA/DPB-25/80/20, US Department of Transportation, Research and
Special Programs Administration, Office of Systems Engineering, Washington, DC, pp: 79-103.
[4] Dimitrios S. Dendrinos, July 1980, “Dynamics of City Size and Structural Stability: The Case of
a Single City”, Geographical Analysis, Vol. 12, No. 3, pp: 236-244.
[5] Dimitrios S. Dendrinos (with Henry Mullally), 1985, Urban Evolution: Studies in the
Mathematical Ecology of Cities, Oxford University Press, Oxford.
[6] Dimitrios S. Dendrinos, 1985, “On the Incongruous Spatial Employment Dynamics”, in Peter
Nijkamp (ed.), 1986, Technological Change, Employment and Spatial Dynamics; Proceedings of
an International Symposium on Technological Change and Employment: Urban and Regional
Dimensions held at Zandvoort, the Netherlands in April 1-3 1985, Volume 270 in the Series
(managing eds. Martin Beckmann and W. Krelle), Lecture Notes in Economics and Mathematical
Systems, Springer-Verlag, Berlin, pp: 321-339.
[7] Dimitrios S. Dendrinos, 1982, “On the Dynamic Stability of Interurban/regional Labor and
Capital Movements”, Journal of Regional Science, Vol. 22, No. 4, pp: 529-540.
[8] Dimitrios S. Dendrinos, January 31, 2018, “On Ancient Artifacts I: Their Nonlinear Dynamic
Paths in Space-Time”, The paper is found here:
[9] Dimitrios S. Dendrinos, February 10, 2018, “On Ancient Artifacts II: An Application of the
Universal Map of Discrete Spatial Relative Dynamics in Archeological Time”, The
paper is found here:
[10] J. Vernon Henderson, 1977, Economic Theory and the Cities, Academic Press, New York.
[11] Dimitrios S. Dendrinos, Michael Sonis, 1990, Chaos and Socio-Spatial Dynamics, Springer-
Verlag, Berlin.
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Full-text available
This is the second paper in a series of two papers on ancient artifacts movement in space-time. The paper presents the theoretical mathematical model, which is an extension of the Universal Map of Discrete Relative Spatial Dynamics developed by D. Dendrinos and M. Sonis in the 1980s. The paper expands on this model and offers a comprehensive approach to both population stocks and artifacts' accumulations and flows in space-time.
Full-text available
This is the first paper of a two-paper series, dealing with ancient artifacts and their movement in space-time. It provides the background to the theoretical mathematical model supplied in the second paper.
We open the presentation of a universal mapping in discrete-time, discrete-space relative stock dynamics, the main subject of this book, by supplying a brief survey of the recent developments in the field of mathematics, the natural sciences, biology, mathematical ecology, and the spatial socioeconomic sciences. These developments provide the background for presenting the geographical applications of our universal map.
A Basic Model of Urban Dynamics as a set of Volterra-Lotka Equations
  • Dimitrios S Dendrinos
Dimitrios S. Dendrinos, 1979, "A Basic Model of Urban Dynamics as a set of Volterra-Lotka Equations", in Dimitrios S. Dendrinos, June 1980, Catastrophe Theory in Urban and Transport Analysis, Report #DOT/RSPA/DPB-25/80/20, US Department of Transportation, Research and Special Programs Administration, Office of Systems Engineering, Washington, DC, pp: 79-103.
Technological Change, Employment and Spatial Dynamics
  • Dimitrios S Dendrinos
Dimitrios S. Dendrinos, 1985, "On the Incongruous Spatial Employment Dynamics", in Peter Nijkamp (ed.), 1986, Technological Change, Employment and Spatial Dynamics; Proceedings of an International Symposium on Technological Change and Employment: Urban and Regional Dimensions held at Zandvoort, the Netherlands in April 1-3 1985, Volume 270 in the Series (managing eds. Martin Beckmann and W. Krelle), Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, pp: 321-339.