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}<ffy SERIE RESEARCH mEmORRRDR
LOGIT MODELS AND CHAOTIC BEHAVIOUR
P. Ni j kamp
A. Reggiani
Researchmemorandum 1989-54 augustus 1989
VRIJE UNIVERSITEIT
FACULTEIT DER ECONOMISCHE WETENSCHAPPEN
EN ECONOMETRIE
A M S T E R D A M
ABSTRACT
Chaos theory and discrete choice theory have been developed as two
separate analytical tools from various different disciplines.
This paper aims to link chaos theory (emerging mainly from physics)
to discrete choice theory (emerging mainly from geography and economics)
by showing the formal conditions under which a dynamic logit model can
exhibit chaotic behaviour.
Furthermore, the analysis will be illustrated and extended by
developing a time-delayed logit model related to a modal (or route)
choice problem incorporating congestion effects in a dynamic setting. It
will be illustrated by means of simulation experiments, that different
types of behaviour can emerge depending on critical values of the
utility function and initial conditions.
1
1. Introduction
An important scientific event in the past decade has been the dis-
covery of chaos theory and its relevance for the social sciences. The
theory of chaos deals with deterministic, non-linear systems which are
able to produce complex motions of such a nature that they sometimes
seem to be completely random. In particular, small uncertainties may
grow exponentially (though all time paths are bounded), leading to very
different trajectories in the long run, so that precise predictions are
- under certain conditions - almost impossible.
After an interesting series of studies on chaotic phenomena in the
field of physics, chemistry, biology, metereology and ecology (see for a
historical review e.g., Nijkamp and Reggiani, 1988d), chaos theory has
been introduced and investigated also in economics and geography with
the purpose of getting a better understanding of the potential useful-
ness of this new dynamical tooi regarding both theory and application in
the social sciences.
The possibility of chaotic behaviour was mainly identified in growth
models or business cycle models (see, among others, the pioneering
studies by Benhabib and Day, 1982; Day, 1982; Guckenheimer et al., 1977,
Grandmont, 1985, and the special issue of Journal of Economie Theory
edited by Grandmont, 1986), in cobweb models (Chiarella, 1988) and in
the economie analysis of long waves (Nijkamp, 1987a, Sterman, 1985).
Chaotic models have also been explored in regional economics and geog
raphy, for instance in the field of industrial systems (White, 1985),
urban macro dynamics (Dendrinos 1984), spatial employment dynamics
(Dendrinos, 1985), relative population dynamics (Dendrinos and Sonis,
1987; Sonis and Dendrinos, 1987), migration systems (Reiner et al.,
1986) and urban evolution (Nijkamp and Reggiani, 1988d).
Interesting surveys of chaos theory can be found in various con-
tributions (see, for example Boldrin, 1988, Kelsey, 1988, Lung, 1988,
and the special issue of System Dynamics Review, edited by Andersen
1988), giving some further interesting insights into chaotic analysis
related to economics and geography. However, so far the contributions
concerning chaos theory have been developed mainly at a macro level. In
this paper an attempt will be made at linking micro- and macro-dynamic
behaviour by demonstrating how a dynamic logit model (emerging from
micro-economie theory) can exhibit a macro-chaotic behaviour, under
certain conditions of lts dynamic utility function.
2
It is noteworthy that contributions to dynamic logit models are up
till now also very scarce in the literature, despite some few interest-
ing examples (see Ben-Akiva and De Palma, 1986; De Palma and Lefevre,
1983; Fischer and Nijkamp, 1987; Haag, 1986; Leonardi, 1986). In this
respect the present study aims to show also such new developments in
dynamic logit models in order to fill this gap. The first part of this
paper will be mainly theoretical in nature and is devoted to show the
chaotic properties of dynamic logit models: the field of applications
can be manifold, since it is well known that logit models can be applied
to various fields of research, such as migration analysis, transporta-
tion aanalysis, residential choice analysis, industrial location
analysis, etc. (see the special issue of Regional Science and ürban
Economics, edited by Nijkamp, 1987b, and Golledge and Timmermans, 1987).
Furthermore, it will also be shown in this paper how the introduction of
time lags in a dynamic logit model may exhibit a more complex behaviour
under specific conditions of the utility function. Finally the stability
of a dynamic logit model in case of a constrained optimization process
will be examined.
It should be noticed that in the sequel we assume some elementary
knowledge on the mathematics of chaos and non-linearity, so that exten-
sive mathematical definitions can be dropped. For further details we
recommend some Standard references: Collet and Eckmann (1980), Devaney
(1986), Guckenheimer and Holmes (1983), loos (1979) and Poston and
Stewart (1986).
2. Existence of Chaos in Dynamic Logit Models
In this section it will be shown that a discrete choice model of the
logit type can exhibit chaotic behaviour under some conditions of the
utility function.
Let us consider the following dynamic logit model:
P j t = exp(ujjt)/S exp(uJjt) (2.1)
where P. represents the share of the population choosing a given dis-
tinct alternative j (j=l,...,J) at time t and u. the utility of
choosing j at time t.
Model (2.1) is the temporal version of a static logit model (see
Heekman, 1981). It is noteworthy that it can also emerge as a solution
3
of an optimal control spatial interaction model, whose objective func-
tion is a cumulative entropy function (see Nijkamp and Reggiani, 1988a;
1988b) .
Let us first consider the rate of change of P. with respect to time
(i.e., d P ./dt): J ' u
^ d F ^ " *j,t " Ht I ^ ^ j . t ^ exP<uJ,t^ (2-2)
Expression (2.2) leads after some simple computational exercises (see
also Nijkamp and Reggiani, 1988c) to:
P. = u. _ P. _(1-P. _) (2.3) j,t J,t j,t' jft'
where u. = du. /dt represents the time rate of change of u. Now it J i J ) - J > *-
is interesting to explore in more detail possible trajectories of u.
By assuming, for instance, that the utility of choosing alternative j is
increasing linearly with time, through a fixed a., we get:
u. _ = cc. (2.4)
so that the final expression (2.3) becomes:
P. = a. P. ,_ (1-P. J (2.5) J.t J J,t j,t'
Expression (2.5) is the well-known continuous time logistic equation
depicting logistic growth (based on the Verhulst equation).
Let us now approximate equation (2.5) in discrete time by considering a
unit time period:
P. «_,,-P. - a. P. _ (1-P. J (2.6)
j,t+l j,t j j,t j.t'
which can be re-arranged as follows:
a. p- ^i - (a.+l) P. _ (1 - TT P- J ' (2-7) j,t+l J j,t a.+l j,t/
It is now interesting to compare eq. (2.7) with the Standard discrete
logistic growth model for a biological population X (X <1):
4
X t + 1 - N Xt (1-Xt) (2.8)
where N is a parameter reflecting the growth rate (0<N<4).
The equivalent of N in our equation (2.7) is a.+l, so we can rewrite
(2.7) as follows:
P. f . - N P. f (1 • p. ) (2.9) j,t+l j,t N j,t'
N-l It is evident that for an increasing value of N we have -rr— -*• 1, and
consequently (2.9) coincides with (2.8). The model (2.8) has been
thoroughly investigated by May (1976). This author showed that if:
K N O (2.10)
the fixed point (1-^ , 1~ ) i-s a n attractor and the system will settle
at the stable point . For N=3 the system bifurcates to give a cycle of
period 2. Next for:
3<N<4 (2.11)
successive bifurcations give rise to a cascade of period doublings in
the range 3<N<3.8... In particular for the value N = 3.8... a fixed
point of period three appears giving rise to a chaotic regime, according
to the famous statement of Li and Yorke (1975) "Period three implies
chaos". The sequence of events of this process is shown in Fig. 1;
details can be found in May (1976), Collet and Eckmann (1980) and
others.
It is now interesting to show that also the dynamic (discrete) logit
model as specified in (2.9) belongs to the same family of May-models.
In fact, if we map in the plane (P. N) eq. (2.9), we get exactly the J >
same type of bifurcations as in the May model leading to a fixed period
of period three (and hence to a chaotic behaviour), the only difference
being the upper limit greater than 1 (see Fig. 2). However, this diagram
shows that for N — 3 the system bifurcates and gives rise to unfeasible
For the definition of a fixed point, an attractor and other terms re-lated to non-linear analysis see also Nijkamp and Reggiani (1988d).
5
Fig. 1 Bifurcation Diagrams of a May-Logistic Map
(2.95<N<4) Y-axis = X ; X-axis = N
Fig. 2 Bifurcation Diagrams of a Logit Map
(2.95<N<4). Y-axis = P. ; X-axis = N
6
values of P. (in particular P.>1). In this case one would have to switch
to values of P.<1, thus causing sudden jumps in the system's trajectory.
The straightforward conclusion is that when -3<N<4, the discrete
logit map (2.9) enters an unstable regime: in other words when the slope
a. of the utility function u. with respect to time (related to the
dynamic logit model (2.2)) is less than » 2 (note: a. = N-l), we have
stable or periodic solutions in choosing alternative j; when 2<Q.<3
chaotic behaviour emerges, characterized by unpredictable movements
which are hardly foreseeable by modellers and planners.
3. Chaotic Dynamics in Delayed Logit Models
In this section the dynamic behaviour of a delayed logit model will
be examined. In other words, we will model the growth of the population
share P. (choosing option j) whose ability to grow in any given time
span is govemed by the population in the previous time span; i.e., we
transform equation (2.9) in the following way:
p- -j.1 = N P . f l - ^ P . .) (3.1) j,t+l j,t N j.t-l'
Clearly, in population biology terms, the new discrete logit model
(3.1) contains a non-linear term regulating the population size with a
time delay of one generation.
Equation (3.1) can be transformed into a two-dimensional system by
introducing the auxiliary variable:
Q- • . - * • - i (3.2) J.t J,t-1
so that eq. (3.1) becomes:
^ t + l - ^ j . t ^ ^ Q j . t ) (3.3)
Qj,t+1 = Pj,t
System (3.3) is a special case of more general classes of well-known
prey-predator relationships. Q. may here be interp reted as an expected
(cumulative) lack of availability of alternative j (e.g., lack of road
capacity in travel behaviour) (predator) whose influence will be the
reduction of population P. (prey) through the parameter N-l.
7
The stability analysis shows a fixed point at (0,0), which is stable
for 0<N<1, but which becomes unstable for N>1.
The second fixed point (1,1) is unstable for 0<N<1; it is stable for
1<N<2 but it is subject to a Hopf bifurcation for N=2 (see Annex A).
These results are indeed most interesting if they are compared with
those emerging from the Standard delayed logistic equation for a
biological population P (see Maynard Smith, 1968):
pt + i = N pt u-*t-i> (3.4)
th where P represents the population density in the t generation and N is
a parameter reflecting the growth rate. Model (3.4) has been inves-
tigated by several authors (see for example Aronson et al., 1982;
Lauwerier, 1986; Pounder and Rogers, 1980).
In particular it has been shown that the fixed point ( N-l N-l N ' N '
related to (3.4) is stable for 1<N<2. As N passes through the value 2,
this fixed point loses stability via a Hopf bifurcation, giving rise to
a chaotic regime. Consequently the parameter values showing a chaotic
attractor are N>2 for both the delayed logit model (3.1) and the delayed
logistic model (3.4). the only difference being the value of the non-
trivial fixed point. It is also interesting to note that equation (3.4)
spawns for N = 2.27 a point of homoclinic tangency (see Pounder and
Rogers, 1980), leaving a strange attractor whose shape is quite similar
to the pictures obtained by Henon (1976) (See Fig. 3).
Fig. 3 The Strange Attractor for the Delayed Logistic
Map (3.4)
Source: Aronson et al., (1982, p. 309)
Yaxis = P t+1 Xaxis
8
If we observe now the bifurcation diagrams (see Fig. 4 and F
emerging from both cases, we may conclude that also in the delaye
model (3.1) a chaotic regime appears for 2<N<2.27.
Fig. 4 Bifurcation Diagrams for a Delayed Logistic
Map (1.9 < N < 2.27) Y-axis - P ; X-axis = N
Fig. 5 Bifurcation Diagrams for a Delayed Logit Map
(1.9 < N < 2.27) Y-axis = P ; X-axis = N
9
It should also be noted that - in our specific case of a delayed
logit model - the chaotic regime means that negative (or greater than 1)
values of P. appear thus leading to unfeasible system's behaviour (i.e.,
extinction) (see Fig. 6).
1 ? 2 0
, b
i 4 K
1 ! 21
ï 1 '
11 -0 471
Jfl. Ü i I' I !
i
l!
I
lil! Il' f U M
ij ! i j l i! I! i' i' !l !l' I I i il I t l ! I
1 h
I II
I i 1
1!
Ii ff
L (f !! iJ ff
! l
/ l i l
ii
t'
u.u 25.00 » • i • ' i
50.00 75.00 100.00
Tirn>?
Fig. 6 Delayed Logit Model for N = 2
In conclusion for the parameter values 0<N<1, system (3.3) embodies
stability in its fixed point (0,0); while for 1<N<2 system (3.3) repre-
sents stability in its fixed point (1.1). Consequently, in a delayed
logit model more restrictive conditions for the parameter N emerge (in
comparison to the non-delayed logit model (2.9)) in order to have a
stable behaviour.
Once again (see also Nijkamp and Reggiani, 1988c) the introduction
of time lags in dynamic equations shows a richer spectrum of complex
behaviour which has to be taken into account, mostly in planning and
forecasting problems.
4. An optimal Gontrol Approach for a Dynamic Logit Model
In this section we will present an optimal control approach in order
to evaluate the stability of a dynamic logit model in case of a con-
strained optimization process. In particular we will use here as the
objective function a cumulative entropy function whose use in dynamic
10
spatial interaction analysis has been suggested elsewhere (see Nijkamp
and Reggiani, 1988a and Sonis, 1986).
If we assume - in line with the previous sections - a spatial inter
action system (e.g., a transportation system of a residential choice
system) , we may assume the following objective function:
max U = I [-0 E P. (inP.-l) - j8« S o. (ina.-l)] dt (4.1) JQ ^ - J J J l -j J J
where the first term in brackets represents the cumulative entropy of
the actors concerned (i.e., the maximum probability of choosing the
alternative j for a given time horizon T ) , while the second term in
brackets represents the maximum interaction between the marginal
Utilities of the individuals (see also, e.g., Wilson, 1981). The coeffi-
cients /L and /3„ represent the relative weights attached to the two
utility components.
The state variable is P. while we assume as control variable the J
marginal utility a. (depending on the attributes of the Utilities). Then
we get the following optimal control problem
U = [ [-B, S P. (inP.-l) - 0 O E a. (ina.-l)] J 0
P l j J J 2 j J J
s. t.
P. = cc. (l-P.)P. J J J J
EP. = 1 j J
(4.2)
Clearly, the evolution of the state variable P. in (4.2) should obey
assumption (2.4) defining a..
Owing to the static constraint in (4.2), we have to use the follow
ing Lagrangean L as a first-order condition for a maximum:
L = -p. S P. (in P.-l) - /30 S a. (in a.-l) + A. a. P. (1-P.) F l j J J ^2 j j 2 J J J J
+M (S P.-l) (4.3) j J
where A. is the costate variable related to the P. constraint and u is J J
the Lagrangean multiplier related to the static constraint.
11
The necessary first-order conditions for a global maximum are:
3L 8a.
3 u
8L SP.
J =
• -A.
J
3L 3A. =
• P.
J
Vi
Vi
Vi
(4.4)
The conditions (4.4) are also sufficiënt for optimality since the objec-
tive function is a concave function. Next by using the condition:
f^- - -/90 in a. + X. P. (1-P.) - 0 (4.5) 5«j H2 j 3 3 3
we can derive:
A. in a. = -^— P. (1-P.) (4.6)
From equation (4.6) we can see that the shadow (dynamic) price A. is
positive, when a.>l and negative when a.<l. The expression for the op-
timal value a. is therefore the following (see also Fig. 7):
a. - exp [X.?. (1-P )//92] (4.7)
Aj Pj d-Pj)/^
Fig. 7 Relationship between marginal Utilities
and costate variables
12
Front expression (4.7) and Fig. 7 it is clear that a.=l is an equilibrium
point for the values P.=0 and P.=l. It is also evident that a.=l is a J J J
bifurcation point, from which two paths emerge.
In that case a planner - who wants to avoid chaotic behaviour in the
state variable - may manipulate the attributes of the utility function
or try to identify the optimal value of X. from the second condition in
(4.4) and to check whether this value is negative (and hence low mar-
ginal Utilities) in order to have a value of a.<l for stable behaviour.
A further remark is that from (4.6) the following relationship can also
be derived:
a. £n a. = X. P./fl0 (4.8)
In other words, the entropie form of a. is directly related to the
dynamic evolution of a logit model through the shadow price A..
Consequently, we may expect that if we also want to control a
delayed logit model no drastic change will occur. In particular when the
expected mobility Q. (see Annex B) is equal to zero, the marginal
utility Q. is decreasing or increasing if the costate variable A. is
negative or positive.
In conclusion, it is clear from our optimal control analysis, that
stable behaviour of the actors (following dynamic logit model) emerges
when their marginal Utilities take low values, and in particular for a.
less than 1.
5. Conclusions
In this paper the possibility of chaotic behaviour for both a
dynamic logit and a delayed logit model has been shown.
In this respect some further remarks are in order. Firstly since
chaos appears in both these models for high values of the marginal
utility a., we have to examine the economie or physical conditions
producing high a.'s and hence impossible predictions on the actor's
behaviour.
Secondly, the chaotic possibility emerging from dynamic logit models
evokes the need to raise the question whether other types of discrete
choice models (such as probit, nested logit, etc.) can exhibit, under
13
certain condltions, chaotic behaviour; in other words if chaos can ap-
pear not only for actors whose behaviour can be represented by
multinomial logit models suffering from the well-known IIA assumption.
Thirdly, an optimal control entropy model applied to a dynamic logit
model shows once more the importance of the marginal utility a., from
which stable behaviour can be controlled. Thus this parameter appears to
be of critical importance for the emergence of chaos.
Acknowledgements
The second author would like to thank the computer assistance by
Andrea Gerali and Luca Schionato (University of Bergamo). Also the
grants C.N.R. n. 88.00409.11 and M.P.I. 60% 1988 are gratefully acknow-
ledged.
14
Annex A Stability Analysis for the Delayed Logit Model
In this Annex, we will analyse the stability properties of the
delayed logit model. For the sake of complicity we rewrite system (3.3)
as follows:
V i - ^ J . t ^ - T V (A.1)
Qj , t+1 Pj , t
It is easy to see that the fixed points (i.e. the values that do not
change when the mapping is iterated) are Ff.(0,0) and F, (1,1).
The local behaviour of the map (A.1) at a fixed point F is governed
by its local linearization for which the Jacobian J taken at F can be
calculated from the corresponding matrix. Since:
J =
N - (N-l) Q (N-l) P. J
(A.2)
we get:
J(0,0)
N
1
0
0 (A.3)
the characteristic equation of the matrix (A.3) is:
K - NK = 0 (A.4)
with eigenvalues K, = 0 , K„ = N. It is evident that |K1|<1 and [K„|<1
for N<1, leading to the stability of the fixed point F~.
15
Then if we consider:
J(1,D 1 1-N
(A.5)
we get the following characteristic equation:
K - K + (N-l) = 0 (A.6)
with eigenvalues:
K± 2 - (1 ± V 5 - 4N/2) (A.7)
The characteristic equation (A.6) is the same as the one emerging
from the linearization of the logistic delayed model (3.1) around its N-l N-l f ixed point (—TJ— , —zr~) , which was thoroughly investigated by Pounder
and Rogers (1980).
Consequently we have a stable solution for 1<N<2. For N=2, we enter
in a Hopf bifurcation.
As a synthesis, following Pounder and Rogers (1980), we can sum-
marize the results in a compact final form (see table 1):
0<N<1
1<N<1.25
1.25<N<2
2<N
stable node
unstable node
unstable node
unstable node
unstable node
stable node
stable focus
unstable focus
Table 1. Stability Solutions for the Delayed
Logit Model (3.1)
16
Annex B. An Optimal Control Approach for a Dynamic Delayed
Logit Model
In this Annex we will analyze an optimal control problem whose ob-
jective function is a cumulative entropy function as defined in (4.1),
subject to a dynamic delayed logit model, as defined in (3.3).
For the sake of simplicity we will approximate system (3.3) in con-
tinuous terms (by supposing a unit time period) as follows:
P. = P. ,_,, - P. „ = (N-l) P. _ - (N-l) Q. ^ P. _ j j , t+1 j , t v J . t v XJ , t j , t
(B.l)
Q. = Q. ., - Q. = P . - Q. vj vj,t+1 vj , t j , t vj , t
By recalling the condition a. — N-l , system (B.l) results:
P. = Q. P. (1-Q.) J J J J
Q. - P. - Q. J J J
(B.2)
System (B.2) is a special type of prey-predator relationship. P.,
representing the prey, is the share of the poptilation choosing alterna-
tive j (e.g., the form or mode of travel in a transport system) and Q.
(predator) can be integrated as the expected mobility or the lack of
capacity in the mode of transport j (see also Nijkamp and Reggiani,
1988c).
The optimal control problem is therefore the following:
17
rT
max U = [-/3, S P. (inP.-l) - p0 S a. (ina.-l)] dt Jg i ^ ^ i ^ ^
s.t.
P. = a. P. (1-Q.) J J J J
Q.= P. - Q. 3 J J
(B.3)
E P. = 1 j J
S Q. = 1
j J
where a. represents the control variable, and P. and Q. are the state J J J
variables.
The first-order condition for a maximum can be expressed by intro -
ducing the following Lagrangean:
L = -P1 S P. (in P.-l) - 02 E a. (in a.-l) +
+ A. a. P. (1-Q.) + V- (P- - Q-) + M (S P.-l) + J J J J J J 1 j J
+ f (S Q.-D (B.4) j J
where A. and il), ave the costate variables and u, and £ the static shadow J J
prices. Then the optimal value a. results from:
§£-- -/?2 in Q j + Aj Pj (1-Qj) = 0 (B.5)
From (B.5) we can derive the condition:
in a = X P (1-Q ) / (B.6)
which shows that the costate variable A. can be positive or negative,
depending on whether or not a.>l.
18
Analogously to section 4, we can derive the optimal value for a.
a. = exp [AjPj (1-Qj)/^] (B.7)
From (B.7) it is clear that when P.=0 or Q.=l, we get the equi-
librium value a.=l (see also Fig. 8).
»j Pj d-Qj)/-32
Fig. 8 Relationships between optimal control
values and shadow prices related to a
dynamic delayed logit model
It is clear that a. =1 is a bifurcation point. Depending on the sign
of A., the optimal value can be decreasing or increasing and hence lead-
ing to stable or unstable behaviour.
19
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Haag, G., A Stochastic Theory for Residential and Labour Mobility in-cluding Travel Networks, Technological Change. Emplovment and Spatial Dynamics (P. Nijkamp ed.), Springer-Verlag, Berlin, 1986, pp. 340-357.
Heekman, J.J. Statistical Models for Discrete Panel Data, Structural Analysis of Discrete Data with Econometrie Applications (C.F. Manski and D. McFadden, eds.), MIT Press, Cambridge, Mass., 1981. pp. 114-178.
Hénon, M., A Two-dimensional Mapping with a Strange Attractor, Communications in Mathematical Physics, vol. 50, 1976, pp. 69-70.
Iooss, G., Bifurcations of Maps and Applications. North-Holland, New York, 1989.
Kelsey, D., The Economics of Chaos or the Chaos of Economics, Oxford Economie Papers. 40, 1988, pp. 1-31.
Lauwerier, H.A., Two-Dimensional Iterative Maps, Chaos (A.V. Holden, ed.), Manchester University Press, Manchester, 1986, pp. 58-95.
Leonardi, G., An Optimal Control Representation of a Stochastic Multistage-Multiactor Choice Process, Evolving Geographical Structures (D. Griffith and A. Lea, eds.), Martinus Nijhoff, The Hague, 1983, pp. 62-72.
Li, T.Y. and J.A. Yorke, Period Three Implies Chaos, American Mathematical Monthly. vol. 82, no. 10, 1975, pp. 985-992.
Lung, Y., Complexity and Spatial Dynamics Modelling: From Catastrophic Theory to Self-Organizing Process: a Review of Literature, The Annals of Regional Science, vol. 22, no. 2, 1988, pp. 81-111.
May, R., Simple Mathematical Models with Very Complicated Dynamics, Nature. vol. 261, 1976, pp. 459-467.
Maynard Smith, J., Mathematical Ideas in Biology. Cambridge University Press, Cambridge, 1968.
Nijkamp, P., Long-Term Economie Fluctuations: A Spatial View, Socio-Economic Planning Sciences, vol. 21, no. 3, 1987a, pp. 189-197.
Nijkamp, P. (ed.), Discrete Spatial Choice Analysis: Special Issue Regional Science and Urban Economics. vol. 17, no.1, 1987b.
Nijkamp, P., and A. Reggiani, Entropy, Spatial Interaction Models and Discrete Choice Analysis: Static and Dynamic Analogies, European Journal of Operational Research, vol. 36, no. 2, 1988a, pp. 186-196.
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Nijkamp, P., and A. Reggiani, Dynamic Spatial Interaction Models: New Directions, Environment and Planning A. vol. 20, 1988b, pp. 1449-1460.
Nijkamp, P. and A. Reggiani, Chaos Theory and Spatial Dynamics, Proceedings IX Annual Conference of Italian Regional Science Association. Torino, vol. 2, 1988c, pp. 583-602.
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1986-1 Peter Nijkamp New Technology and Regional Development
1986-2 Floor Brouwer Aspect5 and Application of an Integrated Peter Nijkamp Environmental Model with a Satellite Design
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1986-3
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1986-5
1986-6
19B6-7
1986-8
1986-9
1986-10
1986-11
1986-IE
1986-13
1986-14
1986-15
1986-16
Peter Nijkamp
Peter Nijkamp
Peter Nijkamp
Peter Nijkamp Jacques Poot
Henk Folmer Peter Nijkamp
Floor Brouwer Peter Nijkamp
Han Dieperink Peter Nijkamp
Peter Nijkamp Aura Reggiani
E.R.K. Spoor
V. Kouwenhoven A. Twijnstra
F.C. Palm E. Vogel vang
M. Wortel A. Twijnstra
A. de Grip
F.C. Palm C.C.A. Winder
25 Vear* of Regional Science: Retrospect and Prospect
Information Centre Policy in a Spatial Perspeetive
Structural Dynamics in Cities
Dynamics of Generalised Spatial Interaction Models
Methodological Aspects of Impact Analysis of Regional Economie Policy
Mixed Oualitative Calculus as a Tool in Policy Model ing
Spatial Dispersion of Industrial Innova-tion: a Case Study for the Netherlands
A Synthesis between Macro and Micro Models in Spatial Interaction Analysis, with Special Reference to Dynamics
De fundamenten van LINC
Overheidsbetrekkingen In de strategie en organisatie van ondernemingen
A short run econometrie analysis of the international coffee market
Flexibele Pensioenering
Causes of Labour Market Imperfections in the Dutch Construction Industry
The Stochastic life cycle consumption model: theoretical results and empirical evidence
1986-17 Guus Holtgrefe DSS for Strategie Planning Purposes: a Future Source of Management Suspicion and Disappointment?
1986-18 H. Visser H.G. Eijgenhuijsen J. Koelewijn
The financing of industry in The Netherlands
1986-19 T. Wolters Onderhandeling en bemiddeling in het be-roepsgoederenvervoer over de weg
1986-20
1986-ei
1986-82
1986-23
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1986-26
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1986-31
1986-32
1986-33
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1986-36
S.C.M. Eijffinger J.W. in 't Veld
E.R.K. Spoor
J.T.C. Kool A.H.O.M. Herkies
W. van Lierop L. Braat
R.M. Buitelaar J.P. de Groot H.J.W. Uijland
E.R.K. Spoor & S.J.L. Kramer
Herman J. Bierens
Jan Rouwend*1 Piet Rietveld
W. Keizer
1986-29 Max Spoor
1986-30 L.J.J. van Eekelen
F.A. Rooien
H.J. Bierens
R. Huiskamp
W.J.B. Smits
P. Nijkamp
H. Blommestein P. Ni jkamp
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