**Authors:**
Dockner E., Feichtinger G. and Novak A.J.

**Abstract:**
A non-linear version of Parlar's production and marketing
model is analysed. The optimal production and advertising
policies are derived using the Maximum Principle. A stability
analysis of the canonical system is carried out to determine the
behaviour of production and marketing decisions over time. As
the main result we show that the model might exhibit stable
limit cycles as optimal policies. Using the Hopf Bifurcation
Theorem the existence of a stable limit cycle is shown in a
numerical example.

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
As discussed in Rauscher (1990) the optimal policies of consumption and
exploitation of a renewable resource by an indebted country may be
complex; cyclical and unstable paths cannot be excluded. Nevertheless,
until now the existence of stable cycles has still been in question. In
this note a numerical example leading to such stable cycles is presented.

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
In an article by Ngo Van Long it is shown that the optimal policy in
exploiting the fish stocks of a predator-prey system by a country may
be rather complex: closed orbits, oscillating convergence or monotonic
convergence are possible. Nevertheless, the question about the
existence of stable cycles is still unresolved. In this note, a
numerical example leading to stable cycles is presented.

**Authors:**
Feichtinger G., Kaitala V. and Novak A.J.

**Abstract:**
We consider a resource management problem for which the management
objective is to sustain both employment and stock level in an
open-access common property fishery. The socially optimal policy
includes utilities derived from the employment level, the resource
level, and from the regulation actions. The optimal management policy
includes stable limit cycles in the resource level, employment, and the
regulation policy.

**Authors:**
Novak A.J. and Feichtinger G.

**Abstract:**
In this paper we analyze the joint determination of optimal
consumption and allocation of time between learning (accumulation
of human capital), working for wages (used for consumption and
accumulation of financial capital)
and leisure. Using Hopf bifurcation theory we are able to show that
* cyclical * training, working, leisuring and consumption
is optimal under certain constellations of parameters.

**Authors:**
Hartl R.F., Mehlmann A. and Novak A.J.

**Abstract:**
In this paper, we present a new approach for
modelling the dynamic intertemporal confrontation between
vampires and humans. It is assumed that the change of the
vampiristic consumption rate induces costs and that the vampire
community also derives some utility from possessing humans and
not only from consuming them. Using the Hopf bifurcation
theorem, it can be shown that cyclical bloodsucking strategies
are optimal. These results are in accordance with empirical
evidence.

**Keywords:** Maximum principle; limit cycles; economics of human
resources; vampire myths.

**Authors:**
Novak A.J. and Feichtinger G.

**Abstract:**
A new optimal control model of chemotherapeutic cancer treatment
is presented including both the number of cancer cells and the
level of toxicity as state variables. It is shown that under certain
conditions periodic drug administration is optimal.
Although the model is highly stylized and too simple to describe
realistic chemotherapeutic treatment patterns it may be seen as a
first step to prove periodic treatment strategies to be superior
to therapies with monotonic decreasing or increasing dosage rates.

**Authors:**
Feichtinger G., Novak A.J. and Wirl F.

**Abstract:**
This paper starts from well-behaved (i.e. concave) one-state-variable
optimal control models. The crucial feature is that indigenous growth
is present. The replacement of the control by an adjustment process
and presumably penalizing these adjustments may convert stable (and
only stable) fixed point equilibria into limit cycles. Indeed, given
positive growth and proper externalities (such that discounting exceeds
growth), one can generate stable limit cycles with ease, e.g., for a
separable framework. Examples from such diverse fields as renewable
resources, optimal saving and public choice highlight the economic
applicability.

**Keywords:** Stable limit cycles; Hopf bifurcation; Renewable
resources; Optimal saving; Subsidies.

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
Empirical evidence indicates that a given moderate number of
ads per year may achieve higher effect when concentrated in
flights than when spread equally. In the control -theoretic
literature a few approaches have been developed for which the
optimal policy is a pulsing schedule. The present communication
analyses a two-state diffusion advertising model for repeat
purchasing. Recognizing the interaction between customers and
potential buyers, the optimal advertising policy turns out to be
a persistent periodic oscillation. This provides a further
interesting example in marketing for the existence of a stable
limit cycle.

**Keywords:**
Optimal control; Advertising by word-of-mouth; Stable
limit cycle; Hopf bifurcation; Diffusion model;

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
Ancient Chinese history reveals many examples of a cyclical
pattern of social development connected with the rise and the decline of
dynasties. In this paper, a possible explanation of the periodic alternation
between despotism and anarchy by a dynamic game between the rulers
and the bandits is offered. The third part of the society, the farmers, are
dealt with as a renewable resource which is exploited by both players in a
different manner. It is shown that the Nash solution of this one-state
differential game may be a persistent cycle. Although we restrict the
analysis to open-loop solutions, this result is of interest for at least two
reasons. First, it provides one of the few existing dynamic economic
games with periodic solutions. Second, and more important, the model is
an example of a three-dimensional canonical system (one state, two
costates) with a stable limit cycle as solution. As far as we see, our model
provides up to now the simplest (i.e., lowest dimensional) case of a
persistent periodic solution of an intertemporal decision problem.

**Keywords:**
Differential games,open-loop Nash solutions,
Hopfbifurcations, stable limit cycles.

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
The aim of the present paper is to illustrate how extremely complex patterns
may be generated in a simple model of educational planning.
In particular, we will show
that certain dependencies of the flow rates on the teacher/student ratio
imply nonlinearities which are substantial enough to generate erratic
behaviour of the time paths. The main message is that chaos in educational
planning may result
from assumptions which are indeed *qualitatively* realistic but
which are quantitatively exaggerated.

**Authors:**
Novak A.J., Kaitala V. and Feichtinger G.

**Abstract:**
In this paper we analyse a fishery resource
exploitation model in which a single firm or a cartel has leased
the rights to manage the resources independently. Two variables,
resource level and the capital level, determine the dynamics of
the resource system. The leasing contract includes an incentive
for the agent to maintain the resource level high. The main
result is that sole-agent resource management and efficiency of
the resource use do not necessarily imply that the fishery is
stabilized at a unique steady state level. Instead, the optimal
resource exploitation may lead to periodic capital investments in
fishing vessels and gear which in turn causes cycles in the
resource economy. We show analytically that nonzero discount rate
and low capital depreciation rate both favor the conditions under
which periodic optimal solutions may occur. Simulation results
related to a Baltic herring fishery are used to illustrate the
results.

**Authors:**
Feichtinger G. and Novak A.J.

**Abstract:**
In this paper we present a simple time-continuous behavioural model
of habit formation. Addictive behaviour is damped by a
threshold which adapts itself to the habit.
This adaptive behaviour of the threshold may lead to periodic
fluctuations of the consumption rate, the habit and the
threshold. It turns out that both a low adjustment rate of the threshold
as well as a steep consumption function favour oscillatory patterns.

**Keywords:**
nonlinear dynamical systems, limit cycles, threshold models, habit formation,
addiction.

**Authors:**
Chand S., Moskowitz H., Novak A.J., Rekhi I., Sorger G.

**Abstract:**
Management of process improvement activities is an essential part of
the manufacturing strategy of a firm to remain globally competitive in
the long run. This paper considers a manufacturing environment where
process improvement activities require use of the productive capacity
of the firm in addition to other investments. Thus the firm must
allocate its productive capacity between production activities and
improvement activities. The output of production activities is used to
meet customer demand. Process improvement activities improve the
quality of the output, which in turn leads to lower quality related
costs (both internal and external) and possibly lower per-unit
production cost. It is assumed that the demand function is downward
sloping and that revenue is a concave function of output. A
continuous-time, finite-horizon, profit maximization, resource
allocation model is developed to find an optimal time path for process
improvement activities and production activities. Computational
results are provided to study the effect of various problem parameters
on the optimal decisions.

**Authors:**
Wirl F., Novak A.J., Feichtinger G. and Dawid H.

**Abstract:**
This paper introduces a differential game that captures
the strategic interactions between the ruling class,
their hidden misbehavior (corruption, adultery,
drinking, drugs, etc.), and the uncovering pursued by
a tabloid press with market power. Aside from
investigating these topical issues, the purpose of this
paper is to draw attention to two formal aspects that
are so far by and large neglected in the literature on
differential games: First, the open loop Nash equilibrium
need not be unique, i.e. an entire family of solutions
istead of a single pair of strategies may exist,
similar to nonlinear Markov strategies in linear
quadratic differential games; yet contrary to nonlinear
Markov strategies, all different Nash equilibrium strategies
converge, if they converge at all, to the same and
unique long run strategies. Second, the long run strategies
need not be constants but may follow a (stable)
limit cycle. These two results are established for a
simple (but nonlinear), one state variable differential game
so that this potential complexity holds already in the
lowest conceivable dimension of one state and two costates.
Therefore, a limit cycle, which is derived using the
Hopf bifurcation theorem and then numerically calculated,
is only possible, if the open loop Nash equilibrium is
not unique. ln fact, indeterminacy is a generic
property of the investigated game.

**Authors:**
Pötscher B.M. and A.J. Novak

**Abstract:**
Frequently, parameter estimation is preceded by a data-driven model
selection procedure. In such a case traditional distribution theory
based on an a priori given model is no longer valid. In Pötscher
(1991) the asymptotic distribution of estimators that are preceded by
model selection has been studied. In this paper we study the small
sample distribution and, in particular, evaluate the accuracy of the
approximation provided by the asymptotic distribution in small samples.
We also show, how one of the assumptions in Pötscher (1991) can be
weakened substantially.

**Keywords:**
Model selection, post-model-selection estimator, AIC.

**Authors:**
Asada T., Semmler W. and Novak A.J.

**Abstract:**
The paper shows that the Romer (1990) model of endogenous growth,
its version for the social planning problem, has a unique steady state and
is saddle path stable. The steady state can be monotonically reached by an
appropriate choice of the initial values of the control variables if the
initial values of the state variables lie in the vicinity of the steady
state. For the original version as well as for generalizations of it we
demonstrate that Hopf bifurcation can be excluded. The technique employed
here may be useful in studying other variants of endogenous growth models.
We also demonstrate in the paper that the market variant of the Romer model
as, for example, presented in Benhabib, Perli and Xie (1994), does admit
Hopf-bifurcation and stable periodic solutions.

**Authors:**
Feichtinger G., Jorgensen S. and Novak A.J.

**Abstract:**
The paper is concerned with a celebrated collection of love poems, the
14th century Italian poet Francis Petrarch's Canzoniere. A striking
feature of these poems is the emotional ups and downs experienced by
Petrarch and his platonic mistress Laura. Recently, attempts have been
made to model these emotional swings by catastrophe theory or nonlinear
differential equations. This paper takes a different approach.
Starting with a pair of differential equations that model the dynamics
of the emotions of the two individuals, we formulate an optimal control
problem. A key hypothesis of this problem is that Petrarch was a
rational addict of his desire for Laura. With specific functional
forms and parameter values we identify a stable limit cycle that gives
a representation of the oscillating emotions of Laura and Petrarch.

**Keywords:**
Differential equations; Optimal control theory; Addictive
behavior; Limit cycle; Love affairs.

**Authors:**
Hartl R. F., Kort P.M. and Novak A.J.

**Abstract:**
This paper studies the optimal behavior of a firm over time that faces
the probability of causing an environmental disaster by its activities.
Here, we can think of explosions in the chemical industry, oil tankers
that lose oil, etc. In the static literature, a distinction is made
between small firms and large firms. Small firms are calles
undercapitalized in the sense that the firm cannot pay for the accident
because the accidental damage exceeds the value of the firm. As soon as
this happens, the firm is declared bankrupt. This gives an incentive to
spend part of the safety budget on distributing dividends to the
shareholders under the motto: pay dividends as long as it is still possible.

In this paper, we extend the static framework to a dynamic one. The major reason is to find out under what circumstances it can be optimal for a small firm to expand in order to become a large firm that has the means to fully pay for the damage caused by an environmental accident. Admittedly, this paper is only a first step in reaching this goal. Here, we determine the optimal safety expenditures and the convergence behavior of long-run investment policies.

**Author:**
Novak A.J.

**Abstract:**
In a recent paper Faria and Andrade (1998) present a model of two
different representative agents, borrowers and lenders, and investigate
conditions such that the borrower's problem results in a cyclical
relationship between capital and loans by applying Hopf bifurcation
theory. Nevertheless, the question about the existence of stable cycles
is still unresolved. In this note a numerical example leading to stable
cycles is presented.

**Keywords:**
Hopf bifurcation theory; Stable limit cycles; Investment.

**Author:**
Novak A.J.

**Abstract:**
In many manufacturing applications the regulation of a quality process
to attain a given target is of central interest. In a recent paper Liu
and Nam (1999) present a model of optimal quality regulation.
The underlying quality process evolves due to
regulation actions and superimposed random disturbances.
Optimal regulation is sought as to minimize the regulation costs and
the mean squared deviation from the desired target over a finite time
horizon.
Unfortunately the model is incorrectly analyzed in Liu and Nam (1999)
and we therefore present the correct results in the following paper.

**Keywords:**
Quality improvement; Optimal control; Diffusion models

**Authors:**
G. Feichtinger, Hartl R.F., Kort P.M. and Novak A.J.

**Abstract:**
In some countries, for instance Egypt, terrorists try to
hurt the country's income from the tourism industry by violent actions
against tourists. Another example are actions of the Kurds to bring tourism
down in the east of Turkey. This paper is a first attempt to model some
relevant aspects in that prey-predator relation. The country tries to
maximize profits from the tourism industry, where profit is defined by the
difference between revenue from the tourism industry and the sum of
expenditures on tourism industry investments and expenditures on enforcement
associated with reducing terrorism. It turns out that for reasonable
parameter values the optimal trajectory exhibits a cyclical strategy. The
interpretation is that, after starting out with a low number of tourists and
terrorists, tourism investments are undertaken to increase tourism. This
attracts terrorists which reduces the effect of tourism investments.
Therefore investment declines and so does the number of tourists. This makes
it less attractive for terrorists to act so we are back in the original
situation, where the whole thing starts again.

**Keywords:**
Hopf bifurcation, Limit cycles, Tourism industry, Law enforcement.

**Authors:**
Liski M., Kort P., Novak A.J.

**Abstract:**
We consider optimal fishery management under the assumption of increasing
returns that is supported by previous empirical evidence. We improve the
tractability and realism of the previous approaches by introducing flow
adjustment costs on changes in harvest rate. Our framework is the first to
provide a link between stable limit cycle policies and increasing returns in
harvesting. The type of the harvest policy depends on flow adjustment costs:
for relatively costly adjustments the usual steady state harvest policy is
conceivable, whereas for relatively cheap adjustments the harvest policy is
cyclical. We also show a connection between chattering control policies and
limit cycles, which helps us to develop a clear economic meaning for
cyclical harvesting.

**Keywords:**
Nonconvexities; Adjustment costs; Hopf bifurcation; Renewable resources.

**Authors:**
Dawid H., Feichtinger G., Novak A.J.

**Abstract:**
In this paper we use a differential game analysis to study the dynamic
strategic interaction between a criminal gang extorting money from
local shop-owners and the local police force. In particular, we are
interested in characterizing which factors are important in determining
whether the capital stock of local shop-owners keeps growing in spite
of extortion or the criminal activity leads to a phase of stagnation of
the local economy. A Markov perfect equilibrium of the game is
characterized in order to address this question and several policy
implications are derived to facilitate growth in regions affected by
extortion.

**JEL Classification:**
C73, D92, K42.

**Keywords:**
Extortion, Differential Games, Capital Accumulation.

**Authors:**
Kort P.M., Greiner A., Feichtinger G., Haunschmied J.L.,
Novak A.J. and Hartl R.F.

**Abstract:**
Efficient investment programs in touristic infrastructure have to take
into consideration that any kind of tourism reduces the environmental
quality. Since pollution shows negative repercussions as concerns the
attractiveness of a touristic region, tourism planners have to
determine a trade-off between adequate services for tourists and a
clean environment. To deal with this problem in a dynamic context, a
three-state optimal control model is formulated. It turns out that,
even if pollution reduction is not a goal in itself, the
profit-maximizing tourism industry should care for ecological
conservation. The paper further shows that persistent periodic
investment policies are optimal for realistic parameter sets, and
provides an economic intuition for such behavior. From an economic
point of view, this result implies that expansionary periods with high
investment are followed by periods of stagnation with low investment.

**Keywords:**
Tourism Industry, Environment, Optimal Control, Pontryagin's Maximum
Principle, Limit Cycles.

**Authors:**
Hartl Richard F., Kort Peter M. and Novak A.J.

**Abstract:**
This paper considers a dynamic model of the firm in which the firm has
debt financing as an additional means to provide funds for investments.
It is assumed that it is more difficult for the firm to get a loan when
the debt-equity ratio is large. Consequently it is imposed that the
interest rate increases with this ratio. The model under consideration
has been studied before by Steigum (1983, Econometrica). He obtained
that the problem reduces to a strictly concave optimal control problem
with a unique saddle point. In this paper we prove this result in a
different way and provide a generalization to his model by extending
the objective functional such that positive utility is assigned to
dividends as well as equity. Despite the fully concave setting, we show
that the optimal solution of a mainly equity oriented firm has history
dependent equilibria.

**JEL Classification:**

**Keywords:**

**Authors:**
Hartl R.F., Novak A.J., Rao A.G. and Sethi S.P.

**Abstract:**
We consider a market consisting of two populations, termed rich and
poor for convenience. If a product is priced such that it is very
expensive for the poor, but affordable to the rich, then it becomes a
status symbol for the poor and this makes it more desirable for the
poor. At a lower price the product is affordable by both populations.
However, as more of the poor buy the product, it ceases to be a status
symbol, and becomes less appealing to the rich. We present a two-state
non-linear optimal control problem that aims to obtain
profit-maximizing prices over time in this environment. We find that
there are three categories of optimal price paths. One is status
symbol pricing with high initial price, declining over time. The other
two are mass market pricing, with price declining in one, and
increasing and then decreasing in the other.

**Keywords:**
Optimal control, marketing, pricing, market diffusion, aspirational
group.

**Authors:**
Faria Joao R and Novak A.J.

**Abstract:**
A naive model of cannibalism assumes that it decreases
intra-species competition, increasing per capita supply of resources.
As a result, there is a cycle between prey population and carrying
capacity of the environment, which oscillates around cannibals'
population. The model is applied to examine the behaviour of firms in
an oligopolistic industry. From this exercise emerges a general model
of cannibalism which shows the economic rationale for the existence of
cannibalistic strategies and has stronger properties than the naive
model.

**JEL Classification:**
L11, L19, L29, J51.

**Keywords:**
Cannibalism, Entry and exit, Predation, Monopolization strategies.

**Authors:**
Feichtinger G., Novak A.J.

**Abstract:**
The purpose of this paper is to study efficient measures to combat
terror. To choose efficient actions against terror organizations, the
response of terrorists as well as the public
has to be taken into
consideration. Thus, an appropriate framework to study the intertemporal
strategic interactions of Western governments, and terror organizations
is dynamic game theory.

Using the open-loop Nash solution concept it turns out that the system may exhibit long-run persistent oscillations. However, transitory behavior is nonunique.

**Keywords:**
Differential games, Counterterror measures, Limit cycles,
Indeterminacy.

**JEL Classification:**
C73, H56.

**Authors:**
Wirl F., Novak A.J., Hof F.X.

**Abstract:**
This paper departs from the standard open-economy Ramsey model and
introduces additional concerns for wealth, status and Easterlin's (2001)
hypothesis that consumption changes, in particular increases, are important
and not only the level. These extensions induce first of all interior steady
states, which are lacking in the standard model, multiple steady states
(separated by thresholds leading to history dependence) and limit cycles.
The existence of cyclical consumption patterns could provide a so far
ignored source for real business cycles. Surprisingly, introducing status
conferred by private wealth or conspicuous consumption has no effect despite
the involved externalities (the outcomes are observationally equivalent) as
long as the social influence associated with the status externalities
remains moderate.

**Keywords:**
wealth, valuing consumption increases, social influence,
stability, limit cycles.

**JEL Classification:**
E21, C61, D91.

**Authors:**
Novak A.J., Feichtinger G., Leitmann G.

**Abstract:**
The question of how best to prosecute the 'war on terror' leads to
strategic interaction in an intertemporal setting. We consider a
non-zero sum differential game between a government and a terrorist
organisation. Due to the state-separability of the game we are able to
determine Nash and Stackelberg solutions in analytic form.
Their comparison as well as the sensitivity analysis deliver
interesting insight into the design of efficient measures to combat
terror.

**Keywords:**
Differential games, Counterterror measures, Nash solution,
Stackelberg solution, State-separable games.

**JEL Classification:**
C73, H56.

**Authors:**
Feichtinger G., Novak A.J., Veliov V.M.

**Abstract:**
The main result in this short note is that the integral form of the Leitmann-Stalford sufficiency conditions can be verified for a class of optimal control problems whose Hamiltonian is not concave with respect to the state variable. The main requirement for this class of problems is that the dynamics is sufficiently dissipative. An application to a Stackelberg differential game between a producer and a developer is exemplified. Using our result we show that the necessary conditions implied by Pontryagin's maximum principle are also sufficient. This allows a complete characterization of the solution.

**Keywords:**
Optimal control, sufficient optimality conditions, differential games,
Stackelberg equilibrium.

**Authors:**
João Ricardo Faria, Damien Besancenot,
Andreas J. Novak

**Abstract:**
This paper deals with two elements of Thomas Kuhn (1962) ideas regarding paradigm: Depletion and resiliency. The possibility of paradigm depletion taking resilience into account, given the hierarchy among scientists, is modeled as a Stackelberg differential game between editors [leaders] and authors [followers]. A number of results emerge from the model: i) Paradigm depletion can be optimal; ii) The optimal editor’s shadow price of potential knowledge must be non-positive, if it is positive, the editor is just a keeper of the orthodoxy rather than a scientist; iii) Editor’s and/or researcher’s impatience is always bad for science; iv) In equilibrium editor’s behavior does not matter for optimal research effort, while only editor’s behavior matter for the paradigm.

**Keywords:**
Paradigm; Economics of science; Research effort.

**JEL Classification:**
C79; J39; Q39.

**Authors:**
Jonathan P. Caulkins, Gustav Feichtinger, Richard F. Hartl, Peter M. Kort,. Andreas J. Novak, Andrea Seidl.

**Abstract:**
Becker and Murphy (J Polit Econ 96(4):675-700,1988) have established the existence of unstable steady states leading to threshold behavior for optimal consumption rates in intertemporal rational addiction models. In the present paper a simple linear-quadratic optimal control model is used to illustrate how their approach fits into the framework of multiple equilibria and indifference-threshold points. By changing the degree of addiction and the level of harmfulness we obtain a variety of behavioral patterns. In particular we show that when the good is harmful as well as very addictive, an indifference-threshold point, also known in the literature as a Skiba point, seperates patterns converging to either zero or maximal consumption, where the latter occurs in the case of a high level of past consumption. This implicitly shows that an individual needs to be aware in time of these characteristics of the good. Otherwise, he/she may start consuming so much that in the end he/she is totally addicted.

**Keywords:**
Optimal control, indifference points, history dependence, rational addiction.

**Authors:**
Jonathan P. Caulkins, Gustav Feichtinger, Dieter Grass, Richard F. Hartl, Peter M. Kort,
Andreas J. Novak, Andrea Seidl.

**Abstract:**
We present a novel model of corruption dynamics in the form of a nonlinear optimal dynamic control problem. It has a tipping point, but one whose origins and character are dinstinct from that in the classic Schelling (1978) model. The decision maker choosing a level of corruption is the chief or leader who presides over a bureaucracy whose state of corruption is influenced by the leader's actions, and whose state in turn influences the pay-off for the leader. The policy interpretation is somewhat more optimistic than in other tipping models, and there are some surprising implications, notable that reforming the bureaucracy may be of limited value if it takes its cues from a corrupt leader.

**Keywords:**
Corruption, tipping point, Skiba point, optimal control.

**JEL Classification:**
C79; J39; Q39.

**Authors:**
Wirl F., Novak A.J.

**Abstract:**
This paper provides a new and surprising reason for growth, namely costs.
More precisely, adding adjustment costs of the control to a one-dimensional,
strictly concave optimal control problem does not affect the steady
state(s). Then, sufficiently high adjustment costs turn an interior and
saddlepoint stable steady state of the original, one-state variable model
into a source that can lead to unbounded growth. Given a version of the open
economy Ramsey model, the initial conditions determine whether unbounded
growth or impoverishment results. Related to this threshold property, the
strict concave two-state variable control model allows for thresholds even
if it has a unique and stable steady state.

**Keywords:**
growth, adjustment costs, thresholds, strict concave
dynamic optimization, thresholds.

**JEL Classification:**
E21, C61.

**Authors:**
Jonathan P. Caulkins, Gustav Feichtinger, Richard F. Hartl, Peter M. Kort,
Andreas J. Novak, Andrea Seidl.

**Abstract:**
We consider a semi-rational addiction model in which the user has perfect foresight over all things within the user's control, but not necessarily with respect to exogenous parameter shocks, e.g., those stemming from changes in national policy. We show that addictive substances are more likely to have state-dependent solution trajectories, and that in turn can create path dependence at the macro-policy level; in particular, legalization may be an irreversible experiment. Also, in this model, shifting from a nuanced policy that differentiates between high and low intensity users, to a tougher one where the government makes life hard for every user reduces initiation considerably. However, it also may have perverse effects. In particular, we show that making the policy tougher in this way could drive some people from a "happy" stable saddle point equilibrium with moderate consumption into increasing rather than reducing their consumption and addiction stock. So implementing zero tolerance policies may increase rather than reduce aggregate drug use, depending on the population's distribution of parameter values and initial consumption stocks. Further, we consider the impact of announcing a policy change.

Last modified: April 15