Forms of Bounded Rationality in Economic Theory

be generated, i.e. �... by relaxing one or more of the assumptions of subjective
utility theory (SEU)�. In particular he identi�es three neoclassical economic
postulates on which SEU theory is based:
SEU1 Choices are made among a given, �xed set of alternatives;
SEU2 The (subjective) probability distributions of outcomes for each choice
are known;
SEU3 Decisions makers maximize the expected value of a given utility func-
tion.
Postulates SEU1 and SEU2 imply that economic agents operate in an
environment that is �xed and completely known by them (at least in the
features that are relevant for their decisions). The agents must know all the
alternative choices available and must be able to predict the consequences of
each possible choice, according to the most accurate theories. In a context of
strategic interdependence this means that agents can also predict the relevant
choices of other agents. In the case of variables whose value changes over
time (i.e. prices) the agents have to know their laws of motion, as given by
economic theory. As a consequence the only kind of expectations admitted
are the rational expectations. All other expectations (adaptive, static, etc...)
make the agent a boundedly rational agent since s/he necessarily has a lack
of knowledge and their forecasts are usually wrong1.
Condition SEU3 is a postulate that concerns the mechanism through which
an agent makes her/his decisions. In order to be fully rational, or simply ra-
tional, agents must be able to use the available information optimally. Among
all the alternative choices the rational agent selects the one that maximizes
the expected value of his utility function. On the contrary, even if the subject
makes the right (i.e. optimizing) choice, they are not necessarily boundedly
rational in the sense of Simon. In fact, according to the postulates of Simon�s
SEU theory, the �nal choice is not important but the mechanism through which
the choice is made.
Maximizing choice ; Rationality (according to SEU)
but
Rational agent ) Maximizing Choice
In short, the rational agent is a good economist (he knows all the relevant
economic theories) and an excellent mathematician (he solves continuously
1The same statement concerning expectations and rationality is present in [63] and [104].
6
optimum problems, even when it is not possible to obtain analytically the
best choice!2).
It appears clear that the image of economic agent that emerges is quite
unrealistic. The literature concerning the features of a more realistic economic
agent is huge. The literature, recently surveyed in [27], presents a subject that
does not maximize her/his objective function but makes satisfying choices. A
lot of experiments have shown that people quite often adopt heuristic deci-
sions, following rules of thumb that lead to systematic errors.
It seems to me that H.Simon�s de�nition is quite exhaustive and includes
all the situations where the decision maker is not assumed to be the homo
economicus of the neoclassical theory. It is a residual de�nition. I do not
agree with those authors who think that Simon�s idea is rarely followed by
decision theorists of bounded rationality (see, among others, [43] and [49]).
I share the Conlisk�s opinion [27] that �... there are many models (in the
literature) which allow for bounded rationality...� even if �...they are only a
small fraction of the total literature...the models spread in all directions...�.
Sometimes bounded rationality is even included in works whose authors do
not intend to speak about it at all. At the end of his de�nition in [111] Simon
himself gives the example of the Lucas�s rational expectationist theory of the
cycle, based on the hypothesis that businessmen are unable to discriminate
between industry prices�movements and general prices�s movements. It is
a clear cognitive limitation of the agents, a variant of the Keynesian money
illusion.
1.2 Thesis Structure
In this thesis I propose examples of boundedly rational behavior. The repre-
sentative consumer of chapter 2 does not completely know her/his own utility
function. S/He also adopts a rule of thumb to make his consumption deci-
sions. The heterogeneous consumers of chapter 3 are connected to each other
through the in�uence that their choices exercise on each others�preferences.
They don�t know (or don�t take into consideration) this network structure.
A rational consumer would know that in this situation it is fundamental to
generate a (rational) expectation on the consumption choice of the other. The
boundedly rational consumers of the model presented here simply ignore it.
The duopolists of the models presented in chapters 4 and 5 violate the pos-
tulates mentioned by Simon because they adopt static expectations3 about the
2Think, for instance, of the case of nonlinear functions that correlate variables.
3Remember here that expectations are called static when the agent thinks that the vari-
able does not change from one period to the next one.
7
choices of the concurrent �rms so they know neither the mechanism through
which the others take their decisions nor (as a consequence) the market price
corresponding to each production choice. Moreover, the �rms of chapter 5 do
not possess a complete knowledge of the demand function they face, so they
approximate it in a boundedly rational way.
Finally, the speculators presented in chapter 6 do not know the law of
motion of the stock prices and exchange rate. In this case we consider two
kinds of speculators that have di�erent (and almost opposite) views about
price trends.
8
Part I
Boundedly Rational
Consumers
9
Chapter 2
A model with unknown
(endogenous) preferences
2.1 Introduction
In his seminal paper, Alchian [3] does not assume that rationality is a given
characteristic of economic agents, but rather considers it the asymptotic out-
come of di�erent and intermingled dynamic processes (i.e. evolutionary, adap-
tive or learning processes). Evolutionary processes are those that permit fully
rational agents to emerge from a population formed by agents endowed with
di�erent degrees of rationality. It recalls Darwinian selection processes in ecol-
ogy. Adaptive and learning processes refer to models where boundedly rational
agents try to reconstruct elements of the environment where they operate on
the basis of data issuing from past choices and from the operations of the mar-
ket mechanism. These agents, in their decisional process, could even follow
simple rules of thumb that do not require excessive informative and computa-
tional capacities and permit them to �nd the best direction in terms of utility,
pro�ts, payo�s, etc. We can say that the methodological approach of this
branch of research consists in analyzing contexts, conditions and decisional
processes of boundedly rational agents where the rational choice emerges as a
�nal outcome through di�erent dynamic processes (see [27]).
In more recent economic analysis and game theory literature, the original
approach (the purpose of which was the research of a more solid knowledge ba-
sis for the rational choice), has been replaced by a more instrumental approach
concerning the multiplicity of equilibria: the research of evolutionary, learn-
ing and adaptive dynamical processes able to select among that embarrassing
multiplicity of equilibria (see [72] and [109]).
This huge and varied literature has been and is applied to various contexts
11
of strategic interaction both in macro and microeconomics. In the last case,
it seems that applications in the theory of �rms prevail over the consump-
tion theory ones. Among them is the contribution of D�Orlando and Rodano
[39]. Their aim is to evaluate if and under which circumstances the maxi-
mizing behavior could really emerge in consumption theory as an equilibrium
result of a dynamic process that begins with choices that are not maximizing
by assumption. In their model, preferences are given but they are not fully
known ex-ante. Only experience permits the consumer to undestand his/her
own preferences more deeply , through comparison between the utility reached
ex-post and the ex-ante expected utility, that led the consumer to buy and con-
sume that quantity of goods. Discrepancy between ex-ante and ex-post utility
causes a change in the consumption choice, initiating a dynamic process. The
primary methodological feature of the model is that the maximizing choice
could be one possible asymptotic outcome, analytically de�ned as a stationary
point of the dynamic system. For particular sets of parameters�constellations,
the D�Orlando and Rodano results suggest that the rational choice is not the
only possible �nal outcome of the dynamic process, but cyclical and chaotic
dynamics may be �nal outcomes as well. Moreover, they show that the en-
dogenization of the preferences reduces the sets of parameters that lead to a
rational choice.
This study assumes a methodological approach similar to D�Orlando and
Rodano. Alternative hypotheses regarding the knowledge capacity and the
decisional process concerning the consumption choice are taken into consid-
eration. In an initial model, preferences are supposed to be given but the
representative consumer only possesses a local knowledge of them. In par-
ticular, he/she is able to evaluate the utility corresponding to choices quite
similar to the actual ones. Moreover, the consumer modi�es the quantities of
goods chosen according to an adaptive mechanism in the direction that locally
permits increased utility. This behavioral rule is not very expensive for the
consumer because it only requires a local knowledge of the slope of the utility
function. This information could be acquired through the experience derived
from consumption choices made in previous perdiods.
The assumed hypothesis gives rise to a dynamic system whose only sta-
tionary point is the rational choice and the dynamic behaviors of the system
are analyzed in this work. Another purpose of the present work is to show the
limits of the local linear analysis that permits an evaluation of the attractive-
ness of a stationary point only within a relatively small neighborhood of it and
does not consider all possible initial conditions. At the same time, I propose
the global analysis approach for nonlinear dynamic systems. This approach
is formed by a suitable combination of analytical, numerical and computer-
12
graphical techniques that allow us to detect coexistence among di�erent kinds
of attractors (stationary points, periodic and chaotic trajectories) and their
basins of attraction.
The decisional mechanism proposed present a strong analogy to the gra-
dient algorithm, widely utilized in numerical analysis as an iterative method
for �nding local maxima of functions (see [64]). In game theory, the gradient
method is also used for the computability of the Nash equilibrium in strategic
form games, starting from [107]. The purpose of these studies still consists in
the analysis of contexts and conditions that permit the numerical detection of
particular points (optimum points or Nash equilibria). The decisional mecha-
nism proposed has also been interpreted as the choice mechanism of boundedly
rational economic agents. This branch of research only considers the supply
side and the �rms. Most of these studies analyze the dynamic behavior of
oligopolistic markets where �rms operate having only a local knowledge of
the market demand function and decide the production level for the next pe-
riod following a gradient rule (see, among the others, [5]; [28]; [38]; [45]; [66];
[84]; [118]). In particular, these studies search for the conditions on the para-
meters that favour convergence towards the Cournot-Nash equilibrium. The
Cournotian game in [17], however, demonstrates the emergence of di�erent
kinds of attractors (periodic and chaotic orbits), analyzing the di�erent basins
of attraction, i.e. the set of initial conditions that converge asymptotically to
the various (coexisting) attractors. Only two studies adopt a di�erent context
for the application of that decisional mechanism. In [10], that "optimally im-
perfect decision process" is inserted into a continuous time monopoly model
and the authors, through the Lyapunov functions technique, demonstrate the
global convergence to the canonical equilibrium of the static game with global
knowledge of the demand function. Taking into consideration the same model
but in a discrete time framework, in [85] the authors show the emergence of
chaotic dynamics.
A second model introduces the endogenization of the preferences as in
D�Orlando and Rodano [39].
2.2 The model
Let us consider a consumer whose preferences with respect to two goods (x
and y) are de�ned by a Cobb-Douglas utility function
U(x; y) = xy1 (2.1)
with 0 <  < 1. The consumer�s choices have to satisfy the budget constraint
px + y = m (2.2)
13
where the price of the good y is normalized to 1 and the income of the consumer
is the positive constant m. It is well known that in this case the rational choice
is given by the vector
(x; y) = (m=p; (1 )m) (2.3)
The consumer knows prices and income but, di�erently from the canoni-
cal approach, knowledge of his own preferences is the outcome of a dynamic
process that involves consumption in previous periods.
Concerning available information from the agent, we use two hypoteses
that act together at the same time. We can distinguish them for the sake
of expositive clarity: a) knowledge about preferences is local and not global,
and b) such knowledge becomes concrete only ex-post through the consump-
tion experience. The �rst assumption permit us to point out the fact that
consumers are not actually able to evaluate utility levels corresponding to the
in�nitly possible consumption bundle but they can still determine the util-
ity levels reachable with combinations of goods quantitatively similar to the
current one.
Through the second assumption I want to highlight the material nature
of the process producing information about preferences, which is based on
consumption activity.
From the point of view of the decisional process, I introduce a mechanism
that permit the consumer to move locally towards combinations of goods char-
acterized by a higher level (with respect to the actual one) of utility.
The auxiliary function V (x)
V (x) = x(m px)1 (2.4)
de�ning the utility in terms of the good x through the budget constraint,
permit us to de�ne the informative set and the decisional mechanism of the
boundedly rational consumer.
It is quite easy to prove that V (x) is generally a unimodal concave function,
that intersects the x-axis in the origin and in the point m=p, and that it has
its only maximum point in x = m=p, i.e. the rational choice for the good x.
From a geometrical point of view, it is possible to show how V (x) derives
from the utility function. In �g.(2.1)a it is su� cient to perform the orthog-
onal projection on the plane x=U , after the determination of the section of
the surface that represents the utility function through the plane the budget
constraint belongs to. It could be interpreted as follows: variations in V (x)
that are the consequences of a variation in x, represent variations in the util-
ity derived from a di�erent level of consumption of the good x (and y) and
so they are a measure of changing utility as the consumption bundle varies.
14
Figure 2.1: Derivation of V (x)
Two general observations can be made concerning the sign and the intensity
of the utility variation. First of all, this variation is positive before the max-
imum point and is negative after it. Moreover, the intensity of the variation
changes with the variation of the consumption bundle. Adopting the notion
of derivative as a good approximation for utility variations, these aspects of
V (x) can be observed in �g.(2.1)b.
Of course V 0(x) is a decreasing function and it intersects the x-axis at the
rational choice point. V 0(x) di�ers from the notion of marginal utility because
the latter is obtained from the utility function keeping �xed the level of the
other good y, whereas the �rst is obtained by respecting the budget constraint,
which implies a negative variation in the consumption of the good y.
The qualitative behavior of V (x) depends on the con�guration of the para-
meters�set. Figure (2.2) shows to show the in�uence of preference coe� cient
 on V (x). The general properties of V (x) are preserved for three di�erent
values of , but it is possible to note that in the extreme cases (a and c) char-
acterized by a preference parameter particularly favorable to one of the goods,
the branches that form the graph of the function are qualitatively steeper with
respect to the intermediate case (b) with more balanced preferences.
Whenever the preference for the good x is relatively low (case a), starting
from the origin the graph of V (x) quickly increases because the consumption
of x is at low levels. It suddenly reaches its maximum because, given the low
preference for x, the saturation level is also low. At the opposite end, whenever
15
Figure 2.2: V (x) for di�erent values of 
the preference for x is relatively high (case c), the graph of V (x) slowly reaches
its maximum point and then quickly decreases because the saturation point is
high.
In terms of knowledge capacity, at each time period t, the agent (due to
the current consumption experience) is supposedly able to evaluate correctly
the e�ect of a relatively small increase in the consumption of x (and then a
corresponding decrease in the consumption of y) on his utility. In other words,
he can determine the consequences of a passage to a basket made up of a higher
level of x and a lower level of y. This cognitive hypothesis implies that at each
time period t the consumer has a local knowledge of the derivative V 0(x) (in
a neighborhood of the current consumption).
Moving from this minimal cognitive hypothesis, it is possible to draw the
following decisional mechanism with respect to the consumption choices. At
each time period t, if the current consumption experience indicates an increase
(resp. decrease) in the utility passing to a basket containing a higher quantity
of x, then the consumption of x in the following period t+1 will increase (resp.
decrease), xt+1 >(<) xt, and the intensity of such a change in the consumption
bundle will be directly proportional to the foreseen variation of the utility and
to a subjective reactivity factor. With this decisional mechanism, at each time
period t consumption bundles satisfy the budget constraint but not necessarily
the optimum conditions. Moreover, in the Cobb-Douglas preferences case, the
intensity of the utility variation deriving from the consumption of the two
goods is stronger when preferences are unbalanced in favour of one of the
goods.
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2.3 Dynamics
The possible analytical speci�cations of the previously described decisional
mechanism, involving the value of x in di�erent time periods, permit the
construction of a dynamic model.
In the discrete-time case, if at time t the agent consumes xt, the dynamics
of the decisional process under analysis can be described by the following �rst
order di�erence equation
xt+1 = xt + '

V 0(xt)

(2.5)
with '[0] = 0 and '0[0] > 0.
In particular, a linear speci�cation of eq. (2.5) is proposed
xt+1 = f(xt) = xt + 
V 0(xt) (2.6)
where the parameter 
 > 0 measures the speed of reaction of subjective reac-
tivity of the consumer with respect to possible utility variations in a neighbor-
hood of the point representing actual consumption. Analogously, it is possible
to de�ne the decisional dynamics in continuous time with the following �rst
order (and generally nonlinear) di�erential equation1
:
x = g(x) = 
V 0(x) (2.7)
The application of the stationarity condition both in the discrete time case
(xt+1 = xt = x) and in the continuous time case ( :x = 0) permits us to obtain
the optimal basket (x; y) = (m=p; (1)m). Then, if the adaptive process
of revision of the consumption bundle converges, it will converge to the rational
choice, according to the canonical theory and similarly to the case described
in [39].
From a dynamic point of view, economic conditions that determine the
emergence of the rational choice in the long run or the emergence of other
kinds of asymptotical behaviors (periodic cycles, chaotic dynamics, etc.) must
be analyzed.
2.3.1 Dynamics in the continuous time case
The �rst proposition shows how continuous time evolution provides a strong
dynamical and adaptive basis to the adoption of the rationality hypothesis in
economic models. In fact, in this context, it is easy to demonstrate the global
stability property of the rational choice:
1
In the continuous case the equation corresponding to (2.5) is


x =  [V 0(xt)] with  [0] = 0
and 0 [0] > 0.
17