Rational Attention and Deliberation
Costs in a Repeated Decision Problem∗
Eugenio Miravete† Ignacio Palacios-Huerta‡
June 2003
Abstract
Deliberation about an economic activity is costly. When information and
deliberation costs are high relative to expected benefits, inertia and inattention may be rational. The deliberation cost theme pervades the discussion in
the literature that postulates deviations from unbounded rationality. Yet, no
empirical evidence on the size of these costs is available in a natural environment. We study this question empirically using data from South Central Bell’s
introduction of local telephone measured tariffs in Louisville, KY. Households
were given the choice to remain in a flat rate scheme, previously the only one
available, or switch to the new measured tariff scheme. Households had to determine each month whether their expected demand the following month would
be above or below a certain threshold. We find that households learn rapidly to
undertake optimal decisions, make no systematic mistakes, and react to potential savings of very small magnitude, typically about $5.00 per month. We find
no support for models where consumers’ responses are determined by inertia,
inattention or impulsiveness. These results have important implications for a
recent literature on the value of information, the optimal degree of experimentation (thinking) in a changing environment, and for bandit problems. From an
econometric methodological viewpoint, the analysis shows how the appropriate
treatment of predetermined endogenous variables and state dependence turns
out to be crucial for interpreting the data.
∗
We thank George Akerlof, Andrew Foster, Giuseppe Moscarini, Ralph Siebert, Dan Silverman,
Johannes Van Biesebroeck, and seminar participants at Berkeley, CEMFI, and the 4th CEPR Conference on Applied Industrial Organization in Leuven for useful comments. We are particularly
grateful to Manuel Arellano and Raquel Carrasco for their help with the estimation of our dynamic
discrete choice panel data model. We alone are responsible for any remaining errors. An earlier
version circulated as CEPR D.P. No. 3604.
†
Department of Economics; University of Pennsylvania; McNeil Building, 3718 Locust Walk;
Philadelphia, PA 19104-6297, and CEPR, London, UK. Email: [email protected]; URL:
http://www.ssc.upenn.edu/˜miravete.
‡
Department of Economics; Brown University, Box B; 64 Waterman Street; Providence, RI 02912.
Email: [email protected]. URL: http://www.econ.brown.edu/˜iph
It is evident that the rational thing to do
is to be irrational where deliberation and
estimation cost more than they are worth.
Frank Knight (1921)
Risk, Uncertainty and Profit.
1
Introduction
There is no question that deliberation about an economic decision is a costly activity.
Many economists, psychologists and other social scientists have expressed the idea
that decision makers try to achieve a balance between the benefits of better decision
making and the effort cost of decision. According to Stigler and Becker (1977), for
instance, “the making of decisions is costly, and not simply because it is an activity
which some people find unpleasant. In order to make a decision one requires information, and the information must be analyzed. The costs of searching for information
and of applying the information to a new situation may be such that habit [and inertia] are sometimes a more efficient way to deal with moderate or temporary changes
in the environment than would be a full, apparently utility-maximizing decision.”
They continue further by discussing how a permanent change in the environment
will tend to induce households to invest in knowledge and skills attuned to the new
environment, and how their incentive to do so will depend on the expected returns
on these investments.
If finding the best action is costly, then the best way to decide on an action
involves trading off decision-making costs with the benefits to improve the choice of
the action. The purpose of this paper is to address this trade-off empirically and to
estimate or infer the size of deliberation and estimation costs that may lead us to
expect, or to separate, rational from irrational behavior. We will study a specific
individual decision making problem in a natural setting where virtually all of the
difficulties that may have impeded previous empirical analyses of this question in
real situations are absent. But before describing the characteristics of the empirical
setting, we shall put into greater perspective the general question that we address in
this paper.
The importance of deliberation and processing costs is an old and significant
question.1 The deliberation cost theme now pervades the discussion in the literature
1
Adam Smith, David Hume and other classical economists already emphasized their decisive role
in their analysis of human preferences.
1
on bounded rationality (e.g., Simon (1996, 1997), Conlisk (1996)). This question is
relevant, either directly or indirectly, for most of the theories that postulate deviations from the assumption of rational, computationally unconstrained agents. These
include the behavioral economics literature (Selten (1978), Simon (1955, 1987)),
the learning literature in macroeconomics (Sargent (1993), Evans and Honkapohja
(2001)), the robust control literature (Hansen and Sargent (2001), Epstein (2001)),
the game theory literature (Rubinstein (1997)), the study of the demand for information in Bayesian decision theory (Moscarini and Smith (2001, 2002)), the determinants
of the adoption of rules of thumb in individual and social learning contexts (Ellison
and Fudenberg (1993)), the study of cognitive dissonance and near-rational theories
(Akerlof and Dickens (1982), Akerlof and Yellen (1985)), and many others. The effects of bounded rationality in game theory are often studied by supposing that agents
choose among distinct behavioral rules, each carrying its own fixed deliberation cost.2
Another approach in this and other areas is to assume limited information processing
capacity, but no limit on agents’ abilities to behave optimally given the constraints
on capacity.3 Also, the predictions of explicit cognitive algorithms of bounded rationality crucially depend on the size of deliberation and active cognition costs that are
assumed within the algorithms (e.g., Gabaix and Laibson (2002), Evans and Ramey
(1998)).
Conlisk (1996) offers a detailed survey of the literature on bounded rationality
which stresses the central role that deliberation and decision costs play in it. The
survey includes many experimental studies by psychologists and economists where
individuals are faced with decisions that have objectively correct answers, but where
they display intransitivity, misunderstand statistical independence, make errors in
updating probabilities, display overconfidence, and violate other assumptions of unbounded rationality. It also includes a number of experiments in which subjects
reason accurately, especially after practice. These results lead him to conclude that
we may “expect that virtually any reasoning error can be made to disappear through
adequate incentives,” and that, ultimately, the important question is “when and why
people get it right or wrong.”
A merit of the deliberation cost idea is that it suggests a discipline for models
of bounded rationality, in that departures from unbounded rationality can then be
systematically related to the deliberation costs involved. This discipline is important for theoretical and empirical progress in this area. Yet, very few theoretical
2
This approach has been used to study the fitness of cheap imitation relative to costly optimization (Abreu and Rubinstein (1988) and Rosenthal (1993)).
3
See, for instance, Rubinstein (1998) in the game theory literature, and Moscarini (2002) and
Sims (2002) in a macroeconomic context.
2
models of deliberation costs have appeared in the literature. The first ones were
developed by Radner and Rothschild (1975), Radner (1975), and Rothschild (1975).
More recently, de Palma et al (1995) and Smith and Walker (1993) have proposed
other models. Perhaps more importantly, to the best of our knowledge, no empirical
evidence from natural settings is available on the size of costs and benefits when individuals deliberate about a problem.4 The importance of estimation and deliberation
costs, and the lack of empirical evidence is what motivates the analysis of this paper.
Thus, in this paper we turn the arguments toward the more constructive questions:
When is bounded rationality likely to be important in a real environment? How large
should the expected benefits from thinking and deliberating about a problem be if
rational attention is to be expected from individuals? In other words, if deliberation
and active cognition costs represent critical physiological limits on human cognition,
when should we expect these limits to be important?
There are a number of problems that can explain the limited amount of theoretical
work and the lack of empirical evidence on these questions:
a. Theoretical Issues. There is a first difficulty with what Savage (1954)
referred to as the “endless” or “infinite regress” problem. The basic idea is that an
optimization problem whose scope covers all considerations including its own costs
raises the question of how imperfect action choices are endogenously generated by
optimal decision procedures. If actions are imperfectly chosen because it is costly to
do better, why aren’t decision procedures chosen imperfectly as well? (Savage (1954,
1967), Winter (1975)).5 The infinite regress problem makes it impossible, even in
principle, to decide how much or how little information to acquire when confronted
with an arbitrary decision problem in which there are costs to information acquisition,
deliberation and processing. Interestingly, Lipman (1991) offers a model representing
the agent’s perceptions of all his actions, including every way to refine his perceptions,
that is not inconsistent with modeling limited rationality by assuming that the agent
uses the “optimal” decision procedure.
A second theoretical problem is concerned with the intrinsic non-concavity in the
value of information. An unresolved problem in Bayesian decision theory is how
4
Smith and Walker (1993) and Gabaix and Laibson (2002) offer some experimental evidence.
Arrow (1987) and Lucas (1987) discuss the limitations of experiments to study bounded rationality
problems.
5
Savage (1954, p. 30) indicates that “it might be stimulating, and it is certainly more realistic,
to think of consideration or calculation as itself an act on which the person must decide. Though
I have not explored the latter possibility carefully, I suspect that any attempt to do so leads to
fruitless and endless regression.” Thirteen years later, Savage (1967, p. 308) asks: “Is it possible to
improve the theory in this respect, making allowance within it for the costs of thinking, or would
that entail paradox, as I am inclined to believe?”
3
to value and price information. Radner and Stiglitz (1984) present conditions under which the marginal value of a small amount of information is zero. Since the
marginal value of costless information is always nonnegative, unless information is
useless and always valueless it must exhibit increasing marginal returns over some
range (see also Chade and Schlee (2002)). Hence, the value of information may not
be globally concave, in which case first order conditions alone do not describe demand. Moscarini and Smith (2002) summarize this problem by indicating that “for
those in the business of buying or selling information, economists have little to say.”
This includes individuals “buying” information by investing costly time and effort
to think and deliberate about an action. This problem, however, turns out to have
a satisfactory solution in some special cases. An important result in Moscarini and
Smith (2002) is that they prove that the demand for information described by first
order conditions is virtually precise if information units are cheap relative to payoffs.
Thus, economists may have something to say in a Bayesian rational learning context
where individuals face explicit costs of active cognition and deliberation if these costs
are small relative to the payoff stakes. But, empirically, when should we expect that
deliberation and cognition activities will be cheap relative to payoffs?
b. Empirical Issues. There is little doubt that theoretical contributions should
feed back to empirical analysis and vice versa. If departures from unbounded rationality are to be systematically related to the deliberation costs involved, it seems
desirable that we have an idea of the size of these costs. Unfortunately, the empirical
problems are typically insurmountable. Estimating and evaluating any theoretical
implication in a real setting has proven extremely difficult in the literature. For instance, in natural settings there are great difficulties in finding individual decision
making situations, in determining individuals’ choice and strategy sets, in observing and characterizing individuals’ choices, and in measuring the incentive structures
that they face. Moreover, detailed data on rewards and the relevant features of the
environment are seldom available. In addition, data on individual choices where the
dynamic effects of learning and incentives can be distinguished from those of unobserved heterogeneity are rarely, if ever, available.
In some cases we may have data on market-level outcomes as opposed to individuallevel outcomes. At the market or other aggregate levels, however, downward-sloping
demand functions can be derived even as consequences of agents’ random choice behavior subject to a budget constraint (Becker (1962) and Gode and Sunder (1995)).
As a result, it may not be possible to distinguish rational from irrational behavior
at any level of aggregation. Similar problems and other empirical difficulties appear
in population settings, especially if individuals may learn not only by doing but also
from others, in settings where preferences are endogenous to the environment, and in
4
strategic settings where agents are involved in game theoretical situations difficult to
characterize empirically.6
Virtually all of the difficulties just mentioned are absent in the natural setting
that will be examined in this paper. In our setting we have an individual decision
making situation where strategy sets can be readily determined, individuals’ choices
can be precisely documented, incentives can be exactly computed, and where a rich
panel data set on choices, rewards, and demographic characteristics is available to
study dynamic learning and incentive effects. There are no market-level outcomes,
no endogenous preferences, no strategic game situations, and no social learning effects that would greatly complicate the empirical analysis. In this sense, this setting
isolates the problem of active cognition and deliberation about an economic decision
in a way that makes it suitable for empirical analysis.
The general structure of the empirical setting that we study can be briefly described as follows. South Central Bell (SCB) implemented a detailed tariff experiment
for the Kentucky Public Service Commission in 1986. SCB collected demographic and
economic information for about 2,500 households in Louisville. In the Spring of 1986,
all households in Kentucky were on mandatory flat rates, paying $18.70 per month
with unlimited local telephone calls. This was the only tariff available. In July of
1986, optional measured services were introduced for the first time in a way that was
unanticipated by consumers. This alternative tariff included a $14.02 monthly fixed
fee, a $5.00 allowance, and a tariff per call that depends on its duration, distance,
and period (time of the day and day of the week). The basic problem that households
faced each month was to determine whether their expected demand for local phone
calls next month would be above or below $19.02, as they would not be billed for
the $5.00 allowance unless their usage level exceeded this limit. A rich panel data set
on all the variables and characteristics of interest is available during the months of
April-June and October-December.
The setting includes a number of desirable characteristics for evaluating whether
or not the costs of attention and learning about their own demand for households
were sufficiently high relative to the expected payoffs so as to induce rational inattention, or whether they were not high enough so that they would think about, and
if necessary react, to the new consumption opportunities. Thus, from the empirical
perspective, we have an individual decision making situation where it is trivial to
6
These difficulties may explain the lack of empirical evidence in natural settings. Foster and
Rosenzweig (1995) is one exception where households learn about the optimal stochastic use of
inputs in the adoption of high-yielding seed varieties (HYVs) in a social, agricultural learning. The
costs of active learning are not modeled in their setting, where the expected payoffs from correct
adoption decisions are very large since households derive all of their income from agriculture.
5
determine strategy sets and to observe individuals’ choices, and where it is easy to
measure the incentive structures and the rewards that they face. There are no selfselection problems since the penetration of telephone service is almost 100 percent
of the population. In this sense, we have a truly representative sample. Moreover,
individuals are as close as possible to a tabula rasa as there was no way that they
could know their demand prior to the introduction of the alternative tariff since phone
calls were not priced at the margin in the flat rate tariff. Data on many economic
and demographic variables for many consumers are available. Interestingly, also data
on consumers’ own expectations are available. The detailed “discreteness” of the
situation is also very useful: the expected consumption level at t + 1 and the tariff
rate to be applied to this consumption are determined at t, consumption takes place
at t + 1. These choices are repeated every month. As indicated above, there are no
market-level data, no endogenous preferences, no strategic setting, and no learning
from others.
Yet, despite all of these desirable characteristics, the empirical analysis is far from
trivial or straightforward. The reason is that we need to estimate a binary choice
panel data model with predetermined variables and unobserved heterogeneity. These
models are hard to estimate. For one, parameter estimates from short panels jointly
estimated with individual fixed effects can be seriously biased and inconsistent when
the explanatory variables are only predetermined as opposed to strictly exogenous
(see Arellano and Honoré (2002) for a review). In linear models with additive effects
the standard response is to consider instrumental-variables estimates that exploit the
lack of correlation between lagged values of the variables and future errors in first
differences. In non-linear models, however, very few results are available.7 For fixed
effects the few methods available are case-specific (logit and Poisson) and, in practice,
√
lead to estimators that do not converge at the usual n-rate. In the case of random
effects, the main difficulty is the so-called “initial conditions problem”: if one begins
to observe individuals after the process in question is already in progress, we need
to isolate the effect of the first lagged dependent variable from the individual-specific
effect and the distribution of the explanatory variables prior to the sample.8 To illustrate the importance of state dependence in panel data modeling, consider the choice
of individual i at time t in the following model: yit = 1 {xit β + γyit−1 + αi + εit ≥ 0}.
In this model there are three sources of persistence: serial correlation in the error
term, ε, unobserved heterogeneity, αi , and true state dependence, γyit−1 . Even if
7
The relevant methods will be discussed in a later section.
Random effects with the strict exogeneity assumption are considered by Chamberlain (1984)
and Newey (1994).
8
6
the error terms are serially independent, the regularity conditions for conditional
maximum likelihood estimation of a fixed effects logit model (or even for the conditional maximum score estimator (Manski (1997)) are not satisfied in the presence of
a lagged dependent variable as εt is not independent of explanatory variables at t − 1.
The distinction between the sources of persistence, however, is very important for
determining the underlying model of behavior. For example, Chiappori and Salanié
(2000) and Chiappori (2000) argue that this distinction is important to differentiate
between moral hazard and adverse selection models in the settings that they examine.
In our case, it will be important to determine whether tariff and consumption choices
are consistent with inertia, rational inattention, or with models where cognition and
deliberation costs do not represent limits to rational behavior.
In order to control for the effect of state dependence appropriately in our setting, we estimate a semi-parametric, dynamic random effects, discrete choice, panel
data model based on the recent work by Arellano and Carrasco (2003). These authors develop a consistent random effects estimator where: (a) explanatory variables
are predetermined but not strictly endogenous, and where (b) individual effects are
allowed to be correlated with explanatory variables. It contains a non-parametric
conditional expectation of the effects given the predetermined variables, but it is
otherwise parametric. This makes the estimation of the model affordable while not
restricting the estimates of the effects by imposing an arbitrary distribution of the
conditional expectation.
The results can be summarized briefly. We find that while consumers facing the
new consumption option may make mistakes initially, they actively engage in tariff
switching in order to reduce the monthly cost of local telephone services. We also
find that mistakes are not systematic. These results are taken to be consistent with
a model of active investments in cognition and deliberation where consumers learn
to undertake the optimal decision. As indicated earlier, incentives and motivation
depend on the size of the costs of cognition and deliberation effort relative to expected
gains. In our case, these incentives are quite low as the magnitude of the differences
between the alternative tariff schemes is very small, typically about $5-$6 dollars
per month. Yet, the observed responses reacting to these potential savings are quite
conclusive. No rational inattention is observed. Despite the small magnitude of the
monetary differences across choices, we find no support for alternative models where
consumers’ responses are determined by habit, inertia, or impulsiveness.
We infer from these results that the costs of attention, cognition, and deliberation
cannot be greater than $5-$6 at the household level and at monthly frequencies when
facing this problem. From the perspective of the literature on bounded rationality,
this imposes a discipline for the study of problems whose complexity is no greater than
7
the one examined in our setting: deliberation and cognitive costs do not represent a
critical limit when a household is tracking every month whether its demand is above
or below a certain threshold and expected benefits are of low magnitude.
From the perspective of the literature on the value of information, the results imply
that Bayesian decision theoretical models may be used to value and price information
even when payoff stakes are as low as $5-$6 at monthly frequencies, as the behavior
we observe is consistent with Bayesian rational learning.
We will also argue in the next section that the problem that households face is
similar to that of a monopolist who has to track his own, unknown demand function
in a possibly changing environment (e.g., Keller and Rady (1999)), and that it also
has a close resemblance to the classic bandit problem in the literature. To the best
of our knowledge, there is no empirical evidence in a natural setting for any of the
implications that arise in either of these problems.
From the empirical methodological viewpoint, we apply for the first time a newly
developed consistent semi-parametric random effects model. Interestingly enough,
the analysis shows how the appropriate treatment of predetermined endogenous variables considered important in the econometrics literature turns out to be crucial for
interpreting the data. The reason is that we also find that when predetermined variables are incorrectly treated as exogenous, consumers do appear to make systematic
mistakes in their choice of tariffs: their ex ante choice of tariff scheme is often not
the one that would have minimized the cost of their local telephone consumption ex
post, and these mistakes are not corrected. In this sense, the appropriate dynamic
analysis of individual learning and investment experiences, which may not have been
adequately appreciated by previous research in the behavioral and experimental economics literature, is important.
The rest of the paper is organized as follows. Section 2 describes the relevant theoretical background and the empirical questions that we will be able to study. Section
3 describes the features of the natural experiment, the data, and reports various descriptive statistics. In Section 4, we first describe the empirical methodology. We
then estimate our random effects binary choice panel data dynamic model, first, to
study the determinants of the choice of tariff option and, second, to analyze whether
ex post mistaken choices are a systematic phenomenon. Section 5 concludes.
8
2
Theoretical Background
This section describes the relevant theoretical background. We begin by describing
the general framework that corresponds to our setting, and then the specific empirical
aspects and implications that arise in this setting that we will be able to study.
A decision maker (DM) must choose an action a from a menu A = {a1 , a2 , ..., aK }.
He has full support prior probability density q (θ) on state θ ∈ Θ = {θ1 , θ2 , ..., θM }.
Action a yields von Neumann-Morgenstern (vNM) utility u(a, θ) in state θ where
u : A × Θ → R. No action is dominated, and there is a unique best action a∗ (θ) =
arg maxa∈A u(a, θ) in each state θ. Let E = hf (· | θ), θ ∈ Θi be an experiment or a
signal, that is a family of state-dependent probability densities on outcomes in X,
each associated with a probability measure µθ .
Before choosing an action, the DM chooses a sample size n, and observes the
outcome of an n-sample xn = (x1 , ..., xn ) ∈ Xn of experiments E. This sample size
may be chosen, for instance, by searching, through explicit information purchases, or
through thinking effort. After observing xn , the DM updates his prior beliefs to the
posterior Pr (θ | xn , q) and takes the action that maximizes his expected utility given
the sample:
a∗ (xn ) = arg max Eθ [u(a, θ) | xn , q] .
a∈A
Thus, the ex-ante payoff from sampling n observations is:
Vq,u (n) = EX n [Eθ [u(a∗ (xn ) , θ) | xn , q]]
Without loss of generality, we consider the binary-state, binary-action case that corresponds to our empirical setting. There are two states Θ = {L, H} and two actions:
A = {F, M }. The two states may be interpreted as low (L) and high (H) demand
respectively, and the two actions as the flat tariff (F ) and the measured tariff (M )
options. Absent a dominated action, we assume:
u (M, L) > u (F, L) ,
u (F, H) > u (M, H) ,
that is, tariff M is preferred if demand is low and tariff F if demand is high. Denote
belief q = Pr(H), so that Vq,u (0) is a piecewise linear, convex function of q. The
DM optimally chooses action F iff q ≥ qb for some qb ∈ (0, 1) . Action M is selected if
beliefs are:
u (M, L) − u (F, L)
.
qn = Pr (H | xn ) < qb =
u (F, H) − u (M, H) + u (M, L) − u (F, L)
9
The expected payoffs in states L and H are:
state L : Pr (qn < qb | L) u(M, L) + Pr (qn ≥ qb | L) u(F, L),
state H : Pr (qn ≥ qb | H) u(F, H) + Pr (qn < qb | L) u(M, H).
Hence,
Vq,u (n) = (1 − q) [(1 − αn ) u (M, L) + αn u(F, L)]+q [(1 − βn ) u (F, H) + βn u(M, H)] ,
where αn and βn denote error probabilities:
αn = Pr (qn ≥ qb | L) ,
βn = Pr (qn < qb | H) .
We next introduce the costs of thinking and gathering information, that is explicit
information purchases. The DM can buy multiple i.i.d. informative signals before
deciding. The intensity level n incurs a flow cost c(n) ≥ 0, where the cost function
c(n) is twice continuously differentiable, increasing, and strictly convex. Thus, the
DM acts as a neoclassical competitive firm, producing information at an increasing
marginal cost and selling it to himself at a price c0 (n).
In general, there are two classes of costs: (i) the costs of buying and searching for
information, and (ii) the cognitive costs of processing this information and thinking
about the problem. In our setting, households are freely provided each month with
a description of their consumption of telephone services. In this sense, the first class
of costs are low, perhaps even zero. Behavior would then seem to be mostly, perhaps
even exclusively, determined by cognitive and deliberation costs. Assume that the
DM chooses n to maximize:
Vq,u (n) − c(n) · n.
His information demand is the number n(p). With a smooth concave payoff function,
demand solves:
V 0 (n (p)) = c0 (n).
If the marginal value V 0 (n) vanishes exponentially, then the demand is logarithmic. But, of course, the demand is discrete and perhaps poorly behaved. Still,
Moscarini and Smith (2002) prove that a log demand formula is almost precise, and
that at “small enough prices” this demand formula is exact to the nearest integer.
“Small enough” means that information units are cheap relative to payoffs. Thus, this
result raises the question of when this type of Bayesian decision formal model may be
used to study the value of information: How low can payoffs be so that deliberation
and cognition activities are still cheap relative to payoffs in a given problem?
10
This is the first empirical aspect that we will examine. Bayesian rational learning
is to be expected in a context with active costs of cognition if these costs are small
relative to the payoff stakes. We do not observe the costs, but we observe in great
detail the payoffs of the actions and we also observe in detail the behavior of DMs.
Thus, from payoffs and from the observed behavior we may broadly infer an upper
bound on the size of the cognitive costs that may be involved iff, as predicted by this
model, individuals behave in a way consistent with rational learning.9
Our second empirical objective comes from Moscarini and Smith (2001) who study
the basic setting that we have just broadly described, where the DM is impatient,
discounts payoffs at a constant rate, and maximizes expected rewards less the costs
incurred. This setting is also similar to that of a two-armed bandit problem, only
that the bandit problems studied in the literature do not have costly information acquisition between switches.10 They derive a number of empirical implications on how
changes in payoffs, costs, or the interest rate affect the level of experimentation.11
One that is suitable for empirical analysis in our setting is that the level of experimentation (intensity of thinking effort) grows with the project’s expected net payoffs:
greater payoffs and/or lower costs raise the experimentation level. In our setting it
will be straightforward to argue that the problem that households face is much less
cognitively costly for households that subscribe to a specific tariff than for households that subscribe to the alternative. This asymmetry provides for a useful source
of testable empirical predictions as households that face the less complex problem
should learn faster and commit fewer mistakes.12
The third empirical goal comes from a related literature on optimal experimentation in a changing environment. Rustichini and Wolinsky (1995), Keller and Rady
(1999) and other authors have examined the problem of a monopolist who does not
know precisely his possibly changing demand and learns about it through experimentation. This setting is very similar to ours in which households learn about their
9
If individuals did not behave in this way, we would be able to infer a lower bound on the size of
these costs.
10
See Bolton and Harris (1999) for a bandit problem that is close to Moscarini and Smith (2001).
11
In Moscarini and Smith (2001) the action that the DM takes is irreversible, while in our empirical
setting consumers may change their minds and switch tariffs multiple times within a month. Only
the last choice that they make is irreversible. This choice determines the tariff that will be applied
to consumption the following month. Thus, the action is irreversible for a month only, as a new
choice is possible every month.
12
Other interesting implications cannot be studied empirically in our setting. For instance, the
implication that one should think hardest when the action to be taken has the highest payoffs
(because if you delay taking it then you pay the discounting cost and have to justify this delay with
a lot of information acquisition) cannot be studied because we do not know when households are
thinking. Moreover, regardless of when the decision about the tariff is made this month, the new
tariff will only start to be applied next month, that is not at the time that the decision is made.
11
possibly changing demand every month. In their models, a given level of noise prevents the household from directly inferring the true state and information gathering
incurs an endogenous opportunity cost but it is otherwise costless. Yet, for a given
intensity of demand switching, more thinking effort is equivalent to less noise and/or
a greater signal. As Moscarini and Smith (2001, Proposition 5) show, a greater signal/noise ratio in turn helps the household track its demand better. Interestingly
enough, Keller and Rady (1999) are able to provide a sharp explicit characterization
of the possible behavior that can be observed in these settings. They show that only
two very different regimes are possible. The first one, when the discount rate on the
future and the intensity of demand switching are low, is characterized by large deviations from myopic behavior: the agent tracks its demand very well and its beliefs
come arbitrarily close to the truth. The second one, when either the discount rate or
the intensity of demand switching are high, is characterized by small deviations from
myopic behavior where the prevailing state of demand is tracked poorly. We will be
able to evaluate empirically this qualitative implication, that is which of these two
regimes arises in our natural setting.
Lastly, the nature of the problem in our setting is such that the null hypothesis of
rational learning has few alternatives. The specific problem that households face is
simply not suitable to study where they display intransitivity, misunderstand statistical independence or violate other assumptions of unbounded rationality. As Stigler
and Becker (1977) and Knight (1921) indicated, the basic alternative to rational attention and learning is that of rational inattention, which is basically equivalent to
the myopic regime characterized by Keller and Rady (1999).
Besides this alternative, there are two other possibilities that can also be studied. A second alternative is impulsiveness, where households’ behavior is random
every period, that is current choices are entirely independent of past choices and
past outcomes. The third alternative is concerned with the temporal discount rate.
The distinct intertemporal nature of the problem whereby households have to plan
one month in advance their consumption level is valuable to study whether or not
households may be time-inconsistent. After the descriptive statistics and the empirical analyses are presented, we will discuss how certain time-inconsistent households
should exhibit a specific pattern of ex post mistakes, and we will see if this pattern
is present in the data.
These three alternatives may be of particular interest in light of the recent theoretical work on the notion that economic agents are often driven not only by inertia
or inattention but also by impulses and are time-inconsistent. We will be able to
assess empirically whether any of them find some support in our setting.
12
3
Empirical Setting and Data
The data come from a tariff experiment held in Louisville, KY, in the mid–eighties.
The experiment was undertaken in order to evaluate the revenue effects of introducing
optional tariffs in fixed local telephony. In this section, we first describe the tariff
experiment and present some descriptive statistics. We also analyze in some detail the
expectations that households have about their own weekly usage levels, and present
a preliminary analysis of the data.
3.1
Features of the Experiment
In the second half of 1986, South Central Bell (SCB) carried out a detailed tariff experiment aimed at providing the Kentucky Public Service Commission (KPSC) with
evidence in favor of authorizing the introduction of optional measured tariffs for local
telephone service. Prior to this tariff experiment, in the Spring of 1986, all households in Kentucky were on mandatory flat rates and SCB collected demographic and
economic information for about 2,500 households in the local exchange of Louisville.
In July of 1986, the tariff was modified in this city. Customers were given the choice
to remain in the previous flat tariff regime—paying $18.70 per month with unlimited
calls—or switch to the new measured service option. The measured service included
a $14.02 monthly fixed fee, a $5.00 allowance,13 and distinguished among setup, duration, peak periods, and distance.14 Choices could be made every month and, unless
a household indicated to SCB otherwise, its choice of tariff was automatically renewed for the following month.15 The regulated monopolist also collected monthly
information on usage (number and duration of calls classified by time of the day, day
of the week, and distance within the local loop), and payments during two periods
of three months, one right before (March–May) and the other (October–December)
three months after the measured tariff option was introduced.
The data set has a number of valuable features. First, local telephony is a basic service and its market penetration is close to 100% in the U.S. Thus, there are
no potential self–selection problems or conspicuous consumption considerations that
13
Consumers on the measured option were not billed for the first $5.00 unless their usage exceeded
that limit. Thus, depending on the accumulated telephone usage over a month, a marginal second
of communication could cost $5.00.
14
The tariff differentiated among three periods: peak was from 8 a.m. to 5 p.m. on weekdays;
shoulder was between 5 p.m. to 11 p.m. on weekdays and Sunday; and off–peak was any other time.
For distance band A, measured charges were 2, 1.3, and 0.8 cents for setup and price per minute
during the peak, shoulder, and off–peak period, respectively. For distance band B, setup charges
were the same but duration was fixed at 4, 2.6, and 1.6 cents, respectively.
15
Switching tariffs simply required a free phone call to request the change of service.
13
may lead to biased estimates because of selection into this market. Second, the low
magnitude of the cost differences between the alternative tariff choices in the data
rules out any risk aversion arguments that could otherwise explain systematic mistakes in the choice of tariffs.16 Third, it is valuable for the purpose of the analysis
that in addition to demographic and economic variables, SCB also collected information on customers’ own telephone usage expectations. During the Spring months,
SCB explicitly requested households to estimate their own average weekly number
of calls. This information can be matched with their actual telephone usage during
the same period. Direct indicators of consumers’ expectations are rarely available in
empirical settings. Furthermore, these estimates are particularly useful because (i)
local calls were never priced before, and (ii) consumers were not aware of the tariff
experiment that was going to be held in the second half of the year. Thus, neither
marginal tariffs nor strategic considerations influence these estimates of customers’
own satiation levels. Even if the formation on individual expectations may be subject
to the effect of unobserved individual heterogeneity, this statistic is perhaps the best
summary available of expected individual usage upon which households may condition their decisions when an alternative tariff becomes available.17 Fourth, given that
the flat tariff regime means that local calls were not priced at the margin, households
might not be aware, at least not perfectly, of their own actual demand for local phone
calls at the time of the experiment.18 In this sense, they are as close as possible to a
tabula rasa and the situation represents a suitable opportunity to test for attention,
learning, and other reactions to a change in consumption options.
Lastly, households receive every month the bill of their consumption. In this sense,
the costs of searching for information are minimal, and thus the costs of deliberation
and cognition, relative to the expected payoffs, would likely be the main, and perhaps
only, determinant of their behavior. Moreover, there is an important asymmetry in
the cognitive costs associated with the problem that households face in the different
tariff options. Households in the measured tariff simply need to compare their actual
bill with the $18.70 cost of the alternative flat tariff in order to ascertain whether
or not they made a mistake. Households in the flat tariff option face a much more
complex problem: they would need to monitor every phone call and compute whether
the total cost of all of their calls in the month would have been above or below $19.02
had they subscribed the measured service, where each call is metered differently
depending on their duration, distance, and periods. Clearly, this task is much more
complex and requires a great deal of monitoring effort. Empirically, we would expect
16
Risk aversion is also ruled out in empirical tests (see Miravete (2000)).
The analysis in Miravete (2003) confirms the importance of this statistic.
18
Measured tariffs were rarely offered in the U.S. before the breakup of AT&T, and local telephone
services typically consisted of just a flat monthly fee as in Louisville (Mitchell and Vogelsang (1991)).
17
14
that these asymmetric cognitive costs are an important driving force of observed
behavior.
3.2
Data and Descriptive Analysis
We begin by presenting some descriptive statistics. Only active consumers were
considered and a small number of observations with missing values for some variables
were excluded.19 Table 1 breaks down the sample into two groups according to their
choice of tariff in October.
[Table 1 here]
Households whose head holds a college degree and those who moved in the past five
years are more likely to choose the measured option. Larger households, those with
teenagers, blacks, and households who receive any kind of federal or state benefits
are more likely to subscribe to the flat tariff option. Households that choose the
measured tariff option are far less intensive users of local telephone service than those
who subscribe to the flat tariff. These households also appear to predict their future
consumption more accurately although, at this level of aggregation, their average
monthly bill exceeds the monthly cost of the flat measured option. This may be
because either (i) most households who subscribed to the measured service made
mistakes in predicting their telephone use, or because (ii) ex post mistakes were not
evenly distributed across subscribers of the measured tariff. The evidence that will
be presented later supports this second interpretation: it is the important mistakes
of only part of these subscribers that explains why the average payment under the
measured service exceeds the $18.70 monthly fee of the flat tariff option. These
descriptive statistics initially suggest that individual heterogeneity in consumption is
important, and that many households were in fact minimizing the cost of their local
telephone service.
As indicated earlier, a valuable feature of the data is that it includes the number
of calls that consumers expect to make per week. These expectations were requested
before the introduction of the new tariff option, that is when all consumers faced an
effective zero–marginal tariff. A comparison of these individual expectations with the
19
The number of observations excluded is very small. Some households did not report their income.
In these cases we recoded the missing observations to the yearly average income of the population in
Louisville and also included a dummy variable, dincome, to control for non–responses (see Miravete
(2002a, Appendix 3, for further details). A feature of the data set is that there is oversampling of
customers that subscribed the optional measured option. While almost 30% of our sample consists of
customers that subscribed to the optional measured service, in the population only 10% subscribed.
All the estimates presented in this paper control for this choice-biased sampling. We use Lerman
and Manski’s (1977) procedure to obtain choice–based, heteroscedastic–consistent, standard errors.
15
actual number of calls that they make during the same period may help identifying
various aspects of the role that expectations may play in choice of tariffs when the
new option is introduced. Figure 1 presents the empirical distribution of the expected
and actual number of weekly calls during the March–May period.
[Figure 1 here]
It is apparent that consumers tend to underestimate their future local telephone
demand. This may explain, at least in part, why some of the households that subscribed to the measured service would have paid less had they subscribed the alternative flat rate option. In order to evaluate whether this underestimation of future
consumption is significant, we computed Anderson’s (1996) non-parametric test of
first order stochastic dominance. Using a uniform 20–fractile division of the empirical support of the number of calls, we obtain that the value of the test statistic
is –5.65. This is strong evidence in favor of first-order stochastic dominance of the
distribution of actual calls over the distribution of expected calls.20
Figure 2 shows the empirical distribution of the expectation bias—the difference
between the actual and the expected number of calls—across individuals.
[Figure 2 here]
Interestingly, the distribution is very symmetric and not overwhelmingly skewed
to the right, as one could have expected from Figure 1. While on average underestimation of future consumption exists at the aggregate level, this is not by all means
a common phenomenon across individuals. Moreover, any systematic discrepancy
between actual and expected consumption would be economically relevant only if it
translates into a systematic erroneous choice of tariff plans.
In an attempt to begin to examine whether households tend to choose the ex
post correct tariff option for their usage levels, we first study the broad pattern of
correlations and decisions using a simple static model of simultaneous choice of tariff
plan and usage level. We estimate the following reduced form model:
yj∗ = XΠj + vj ,
j = 1, 2,
where, conditional on observed demographics, we assume that the tariff choice and
the usage level decisions are not independent:
Ã
(v1 , v2 ) ∼ N (0, Σv ) ;
Σv =
20
1 ρ
ρ 1
!
.
The statistic is distributed as a studentized maximum modulus distribution (Stoline and Ury
(1979)). With 20 multiple comparisons and infinite degrees of freedom the 1% one–tail critical value
is 3.49. Also, stochastic dominance is never rejected for numerous clusters of individuals defined by
the characteristics included in Table 1 (see Miravete (2002b)).
16
These two reduced-form equations are estimated simultaneously as a bivariate
probit model, thereby providing an estimate of ρ (that is, characterizing the distribution of unobservable individual characteristics) and an estimate of the effect of
demographics.21 In this model y1 = 1 if the household subscribes the measured tariff,
and y2 = 1 if the household realizes a low usage level, defined as a consumption
level below $19.02 when metered according to the measured tariff rates. The model
includes the set of demographic variables in both equations to control for the effect of
observable individual heterogeneity over the tariff choice and consumption decisions.
For instance, we would expect that households with more members will tend to make
a more intensive use of the local telephone services and, hence, that they will tend to
subscribe the flat tariff option.22,23 The data also allow us to import household specific information from the Spring months to control, at least in part, for the accuracy
of predictions of individual future usage. We thus include two dummies to indicate
whether consumers significantly over or underestimate future consumption.24 Similarly, we construct an indicator of usage intensity for each household during the Spring
months, low usageSpring , which equals one when the usage level during Spring (at
zero marginal charge) is less than $19.02 had it been metered according to the optional measured tariff that will later be in place during the Fall. We include this
variable in order to account for any systematic effect of demographics not included
in our data on usage. Table 2 reports the reduced form parameters.
[Table 2 here]
21
It is also possible to estimate each equation separately, thereby ignoring any potential correlation
between the disturbances v1 and v2 . Such procedure would have two costly consequences. First,
we would loose the ability to evaluate, using actual data, whether a low usage level is positively
correlated with the subscription to the measured tariff option. Second, if ρ = corr [v1 , v2 ] 6= 0, which
will be our case, the estimates from the individual probit regressions would be inconsistent.
22
For convenience, and in order to follow the same approach of the dynamic discrete–choice model
that will be presented in the next section, all regressors are dummy variables. This includes the
income indicators, for which the original data identifies a household as belonging to one out of nine
income categories. high and low income equal one when the income level of the household exceeds
the mean plus or minus its standard deviation, respectively. The definition of the other dummies is
self–explanatory.
23
It might be argued that if we make use of the expectation indicators, there is no need to include
the demographics in the regression for the choice of tariff (first equation). However, swbias is only
an indicator of usage prediction errors based on the total number of weekly calls. Also, different
households may differ in their time/distance/duration patterns of calling. The combination of all
these other elements contribute to determining whether or not the usage level exceeds $19.02. Thus,
the demographics account for these differences in household behavior in the first equation.
24
The underestimation dummy is equal to one if swcalls exceeds expcalls by more than
50% of the standard deviation of swbias. The overestimation dummy is defined accordingly
when expcalls exceeds swbias.
17
The results show that the effects of income on local telephone usage and tariff
choice are not significant. Only those who do not report their income appear to be
more inclined to subscribe the flat tariff option. Larger households tend to subscribe
the flat tariff option and to realize high usage levels. Households whose head holds a
college degree are inclined to subscribe the measured service option but, conditional
on having subscribed the measured option, they are also more likely to realize a high
demand and, thus, to have (incorrectly) chosen the measured option ex post. A
similar pattern arises for households formed by married couples.25
The exogenous indicators associated with usage and expectations during the
Spring months are revealing of the possible decision process behind the choice of
tariffs. Households with a low usage profile during the Spring months are more likely
to present a low usage pattern in the Fall months as well, and also to choose (correctly) the measured tariff. Consumers that either over or underestimate their future
telephone usage quite significantly are less likely to subscribe the measured option,
and are also less likely to realize a low usage level. This means that given that the
households that make the more important forecast errors in absolute terms are those
with high levels of demand, these households are also more likely to choose the right
option by subscribing to the flat tariff.
The results also show that the parameter ρ accounting for the correlation between
the choice of the measured service and a low demand realization is positive and very
significant. This is perhaps the main result that we would like to emphasize. It
indicates that consumers do not appear to make systematic mistakes when choosing
among optional tariffs. However, from the above results, it seems that when mistakes
are made, it is more likely that they are made when a household subscribes to the
optional measured service.
This model is estimated using the balanced pool sample of the Fall months of
1986, and abstracts from any dynamic and learning issues.26 Yet, these results are
useful in that they characterize, at least initially, the broad patterns of correlations
and decisions in the data, and also in that they qualify the previous discussion on the
underestimation of future usage levels. This descriptive evidence also confirms the
need to account for the existence of state dependence and unobserved heterogeneity
that will be examined in the dynamic analysis of the next section.
25
Consumers are classified as having chosen correctly or incorrectly each tariff option ex post
keeping the usage pattern unchanged, that is independently of price responses. This provides an
approximate upper bound to the gains of switching to a different tariff option.
26
We also estimated the model including monthly dummy indicators but they failed to improve
the regression results. It does not appear to be any important time effect, neither in the estimation
of the model of Table 2 nor in any of the other regressions that will be reported later in the paper.
18
Before concluding the description, and in order to gain further insights into the
basic dynamic patterns of decisions, Table 3 reports the expectation forecast errors,
the percentages of households who made the wrong tariff choice ex post, and the
potential savings of local telephone customers for each possible path of tariff choices
during the three Fall months.
[Table 3 here]
This table illustrates various issues. First, in broad terms, potential savings appear
to drive the choice of tariff options. Second, consumers with more accurate predictions
of future usage tend to subscribe to the measured service, while those who are less
accurate in their predictions tend to remain on the flat tariff option. The latter may
often underestimate their future usage level, but this is not typically costly as the flat
tariff option turns out to be optimal ex post for them very frequently. About 86%
of the population in Louisville always chooses the flat tariff option during the Fall.
Given that, on average, they are saving substantially by doing so, it is not possible
to argue a priori that these consumers are irrational or exhibit rational inattention.
Only from 6.19% to 11.33% of these households could have saved some money ex
post by switching to measured service during these months, while from 55.73% to
66.67% of those who always subscribed to the measured service are mistaken ex post.
In addition to these proportions, the magnitudes are important as well. The average
savings of those always on the flat tariff go from $13.98 to $16.93, while those always
under the measured service spend on average from $0.88 to $2.66 more than they
would have under the flat tariff. Third, there is a small proportion of switchers
during these months. Among those switching from flat to measured service at some
point during the Fall, it appears that many of them make mistakes ex post since
in October they were saving about $2.29 relative to the optional measured service,
but ended up spending about $3.41 in excess of the cost of the flat tariff option in
December. Interestingly, a remarkable 100% of those who were on the measured
service in October and decided to switch in either November or December were losing
money in October and all of them ended up saving money after the switched. They
lost, on average, in excess of $17.00 in October and ended up saving on average $16.33
right after returning to the flat tariff option. Thus, their adjustment basically takes
place in the form of switching tariffs, not in decreasing their level of consumption.
After this descriptive evidence, we turn the arguments toward the more substantive questions: Are the consumption levels, tariff choices, and the switching that we
observe in the data sufficient to provide any significant evidence that consumers respond to potential savings? What is the role of previous tariff choices and demand
realizations on the decision to subscribe to one of the two options? Do consumers
19
simply stay on their previously chosen tariff because of inertia or rational inattention? In order to answer these questions we need more sophisticated econometric
methods that allow us to account for state dependence, unobserved heterogeneity,
and dynamic learning. We study such model next.
4
Econometric Model and Empirical Evidence
In this section we first present a semi–parametric, random effects, discrete choice
model with predetermined variables based on the recent work by Arellano and Carrasco (2003). We also discuss why this model offers useful advantages over the very
few alternative approaches available in the literature. We then implement this model
to study the choices of tariffs and consumption levels.
The basic idea of the model is to define conditional probabilities for every possible sequence of realizations of state variables. In this sense, it is able to deal with
regressors that are predetermined but not exogenous, such as the previous choices
of tariffs and the past realizations of demand in our setting. Then, the estimator
computes the probability of subscribing to a given tariff along every possible path
of past realizations of demand and subscription decisions. The panel data structure
allows us to identify the effect of individual unobserved heterogeneity since consumers
make different decisions even if they share the same history of realizations of state
variables.
4.1
A Dynamic Discrete Choice Panel Data Model
The probability of subscribing to a given tariff option, and hence the probability of
switching tariffs in the future, depends on the particular sequence of past choices and
past realizations of demand for each consumer. As consumers choose differently, they
accumulate different experiences and invest differently in information and deliberation
efforts. These experiences in turn change the information set upon which they decide
in the future. For instance, consumers that have previously chosen the measured
option may have learned that their demand is systematically high, so that in the future
they will be more likely to subscribe to the flat tariff option. Consumers that have
always remained on the flat tariff option have accumulated different experiences and
made different investments, which also affect their conditional probability of renewing
their subscription to the flat tariff option. Given that their consumption was never
priced at the margin in any range, these households may have much less knowledge of
their own demand than those that at some point subscribe to the measured service.
To be more specific, the probability of subscribing to a given tariff option may depend
20
on some intrinsic characteristics of consumers, as well as on their expectation on the
realization of demand. This can be written as follows:
n
³
´
o
yit = 1 βzit + E ηi | wit + εit ≥ 0 ,
³
´
εit | wit ∼ N 0, σt2 ,
where yit = 1 (yit = 0) if the measured (flat) tariff option is subscribed; zit includes
the set of time–invariant characteristics of consumers, xit , plus the past realization of
demand and the previous choices of tariffs, yi(t−1) ; wit = {wi1n, ..., wit } is othe history of
past choices represented by a sequence of realizations: wit = xit , yi(t−1) ; and ηi is an
individual effect whose forecast is revised each period t as the information summarized
by the history wit accumulates.27 In our case ηi is the future individual realization
of demand. The conditional distribution of the sequence of expectations E (ηi | wit )
is left unrestricted, and hence the process of updating expectations as information
accumulates is not explicitly modeled. This is the only aspect that makes the model
semi–parametric. While the assumption of normality of the distribution of errors
is not essential, the assumption that the errors εit are not correlated over time is
necessary for the estimation. Given the history of past decisions, since errors are
normally distributed, the conditional probability of choosing the measured option at
time t for any given history wit is:
³
Pr yit = 1 |
wit
´
#
"
βzit + E (ηi | wit )
.
=Φ
σt
Since all our regressors are dichotomous variables, their support is a lattice defined
by 2J nodes {φ1 , ..., φ2J }. The t × 1–vector of regressors zit = {zi1 , ..., zit } has a
multinomial distribution and may take up to J t different values. Similarly, the vector
wit is defined on (2J)t values, for j = 1, ..., (2J)t . Given that the model has discrete
support, any individual history can be summarized by a cluster of nodes representing
the sequence of tariff choices and demand realizations for each vector of demographics
representing the different combinations of characteristics of individuals in the sample.
Thus, the conditional probability can be rewritten as:
³
´
³
´
pjt = Pr yit = 1 | wit = φtj ≡ ht wit = φtj ,
j = 1, ..., (2J)t .
The estimation relies on a simple intuitive idea. In order to remove the unobserved individual effect we account for the proportion of customers with identical
27
The specification of Arellano and Carrasco (2003) is more general in the sense that it also
includes a time-varying component, γt , common to all individuals. In our case all demographics
are time–invariant. We also included monthly indicators in our empirical analysis but this did not
improve the results of our estimations, even when interacted with past subscription decisions and
past realizations of demand.
21
demographics and history up to time t that subscribe to the measured tariff option
M at each time t. We then repeat this procedure for every cluster of combinations of
demographics and histories that exists in our data. For each cluster we compute the
percentage of consumers that subscribe to M . This provides a simple estimate of the
unrestricted probability p̂tj for each possible history present in the sample. Then, by
taking first differences of the inverse of the equation above we get:
h
³
σt Φ−1 ht wit
´i
h
³
− σt−1 Φ−1 ht−1 wit−1
´i
³
´
− β xit − xi(t−1) = ξit ,
and, by the law of iterated expectations, we have:
h
i
h
³
´
³
´¯
¯
i
E ξit | wit−1 = E E ηi | wit − E ηi | wit−1 ¯ wit−1 = 0.
This conditional moment condition serves as the basis of the GMM estimation of
parameters β and σt (subject to the normalization restriction that σ1 = 1). Arellano
and Carrasco (2003) show that there is no efficiency loss in estimating these parameters by a two–step GMM method where in the first step the conditional probabilities
ptj are replaced by unrestricted estimates p̂tj , such as the proportion of consumers
with a given demographic profile and a given history that subscribe to the measured
service. Then:
t
³
´
ĥt wit =
(2J)
X
n
o
1 wit = φtj · p̂tj ,
j=1
which is used to define the sample orthogonality conditions of the Arellano–Carrasco’s
GMM estimator:28
N
³
´i
³
´o
h
n
h ³ ´i
1 X
dit σt Φ−1 ĥt wit − σt−1 Φ−1 ĥt−1 wit−1 − β xit − xi(t−1) = 0, t = 2, ..., T,
N i=1
n
o
where dit is a vector containing the indicators 1 wit = φtj for j = 1, ..., (2J)t−1 .
Alternative Approaches. There are two problems that we need to deal with.
First, since consumer actions are likely to be conditioned by the individual history
of choices, we need to control for state dependence. In addition, households have
already accumulated different individual experiences through their different choices
during the July–September period. Since these pre–sample individual decision paths
are not observable to us, we also have to deal with the “initial conditions problem”
in the estimation of our econometric model. Had SCB collected data on tariff choices
P
t−1
In practice the number of moment conditions is smaller than t (2J)
because we only consider clusters with at least 4 observations. Also, we use the orthogonal deviations suggested by
Arellano and Bover (1995) instead of first differences among past values of the state variables.
28
22
and usage decisions during the six months of the tariff experiment we would not be
facing this problem because all consumers in Louisville were priced according to the
flat tariff option at the beginning of the experiment. Unfortunately, data are only
available from October to December because the KPSC requested to wait for three
months of adjustment before collecting usage and tariff choice data again.
If we ignored the initial conditions problem our estimates would most likely be
inconsistent since the initial conditions become endogenous if errors are correlated.
A potential solution is to consider that each unobserved individual path of discrete
decisions prior to the initial month of data collection has an effect on the probability
of subscribing to the measured option only through individual fixed effects. Unfortunately, and with few exceptions, discrete choice models with fixed effects cannot
be consistently estimated with finite samples because of the well–known incidental
parameter problem.29
There are very few results in this literature. The logit specification is the only
discrete choice model where the incidental parameter problem is not present. In order
to deal with the issue of state dependence, Honoré and Kyriazidou (2000) include
one lagged dependent variable but require that the remaining explanatory variables
are strictly exogenous, thus excluding the possibility of a lagged dependent regressor.
Also, time dummies are ruled out and, furthermore, their estimator does not converge
√
at the usual n–rate. Honoré and Lewbel (2002) allow for additional predetermined
variables but at the cost of requiring a continuous, strictly exogenous, explanatory
variable that is independent of the individual effects. An alternative to the logit
specification is the maximum score estimator of Manski (1987). However, in addition
to the strict exogeneity of regressors this estimator also requires stationarity in order
to avoid the initial conditions problem. But stationarity should not be expected in
our sample which contains data collected just three months after the tariff experiment
was launched.
In addition to fixed effects models, research has also addressed random effects
models in order to deal with unobserved heterogeneity in discrete choice problems
(e.g., Chamberlain (1980, 1984), Newey (1994)). However, beyond the common requirement of strict exogeneity of regressors, random effects models have the disadvantage that the identification of parameters depends critically on the arbitrary choice
of the conditional distribution of individual effects by the econometrician. This is not
the case in our model because, as pointed out earlier, the conditional distribution of
the individual effect E (ηi | wit ) is not explicitly modeled.
29
On the statistical problems originated by the initial conditions problem, including its relationship
with the incidental parameter problem, see Heckman (1981). On the impossibility of obtaining
consistent fixed-effect estimates with finite samples, see Neyman and Scott (1948) and Lancaster
(2000).
23
Finally, one additional reason in favor of choosing the approach of Arellano and
Carrasco (2003) is that our short panel fits the identification requirements of their
GMM estimator. Alternative fixed-effects approaches such as Honoré and Lewbel
(2002) and Honoré and Kyriazidou (2000) are also far more demanding in terms of
data. In particular, they require variation of the exogenous regressors over time,
something that does not occur in our data, and a minimum of four periods of observations.
4.2
Empirical Evidence
In order to account for the dynamic nature of the learning process where individuals
may invest time, deliberation effort, and other resources to gain knowledge about
their new options and about their own demand for telephone services, we estimate
two dynamic discrete choice panel data models with predetermined variables. These
models control for the existence of state dependence and unobserved individual heterogeneity, as both of these aspects are likely to play a relevant role.
The first model studies whether households tend to remain subscribed to the
same tariff option over time regardless of their past realized usage levels. The study
of whether household choices can be characterized by habit and inertia in a natural
environment is not only of interest per se, but also because it is a necessary condition
for rational inattention.
The second model addresses this issue more closely. It studies the learning process
directly by evaluating whether or not those households that make a mistake are more
likely to continue making systematic mistakes in the future. The estimation of these
two models includes the same demographic and usage expectation regressors already
introduced in Section 3.
In addition to addressing these issues using our dynamic random effects model, we
will also estimate, in each of these two cases, a static probit model and a probit model
where predetermined variables are incorrectly treated as exogenous variables. The
static pool regression will only make use of truly exogenous regressors (households’
demographics) and will ignore the existence of unobserved individual heterogeneity.
It will thus lead to inconsistent estimates if in fact such effects are present. Likewise,
the results of the probit model where predetermined variables are incorrectly treated
as exogenous will also be inconsistent. The reason why we will also report these
estimates is that they may help us evaluate the extent to which ignoring important
aspects of the dynamic learning process may lead to incorrect conclusions with regard
to the classes of models that could be supported by the data.30
30
In some cross–sectional analyses in the literature there is information available on previous
24
4.2.1
Testing for Inertia in Tariff Choices
Table 4 reports the results of these three models in the study of the choice of tariffs.
[Table 4 here]
The results in column 1 correspond to the static case. Larger households, those
with teenagers, those that receive any kind of benefits, and those with greater than
average income tend to subscribe to the flat tariff option. Households formed by married couples and those with a college degree appear to be more inclined to subscribe
to the measured service option. We also observe that those who make important
prediction errors of any sign regarding their own future usage levels are more likely
to subscribe to the flat tariff option.
Column 2 reports the results when predetermined variables are included as exogenous regressors, and thus the estimates are not corrected for possible endogeneity
bias. According to the results of this misspecified model, consumers with low demand
tend to subscribe to the optional measured service and consumers do not significantly
switch tariffs. These results would support the idea that consumers are characterized
by inertia, and that low demand consumers rightly choose the measured option and
tend to stay there. These findings, however, appear to contradict the descriptive
evidence reported in Table 3.
Column 3 reports the results of our dynamic discrete choice model. Intuitively, as
time elapses the effects of accumulated experiences, cognitive efforts, and investments
take over through the updating process embodied in E (ηi | wit ). In this sense, these
effects should become a more important determinant of tariff choices over time. Interestingly enough, when we allow for these effects the results are substantially different.
Households that underestimate their future usage are more likely to subscribe to the
flat tariff option, which is consistent with the initial evidence presented in Tables 1
and 3. Similarly, the paths of past decisions upon which we condition not only capture
the individual effects better over time, but they end up making the time–invariant
demographics much less significant indicators of individual heterogeneity than in the
previous models. In fact, virtually none of them are significant. The most important
result is provided by the parameters of the predetermined variables low usaget−1
and measuret−1 . They are both negative and very significant. The negative effect
decisions of consumers. An incorrect procedure to account for the effects of past experience and
unobserved accumulation of information is to include past choices as lagged variables to explain
current choices (see, for instance, the analysis of automatic renewal of subscriptions to health clubs in
Della Vigna and Malmendier (2001)). We are not aware of any empirical analysis in the experimental
literature that addresses the roles of state dependence and unobserved heterogeneity in a dynamic
setting.
25
of low usaget−1 opens up the possibility of mistakes as it captures the effect of
those consumers that still remain on flat tariff, although their low demand for local
telephone services does not justify such a choice.31 This result would seem to be
consistent with the hypothesis of rational inattention among low demand customers,
which may prefer to pay a small premium, ignore thinking and making any decisions,
and remain subscribed to the flat tariff option. Yet, the evidence in favor of this
hypothesis is not conclusive because consumers may not always remain on the same
tariff. In fact, the negative effect of measuredt−1 indicates that consumers do switch
tariffs significantly and that, contrary to the idea of habit and inertia, automatic renewal of tariff subscription options does not necessarily mean that consumers will
stay in the previously chosen tariff indefinitely.32
We conclude from this dynamic model that individual heterogeneity and state
dependence are crucial to interpret the choice of tariff data, and that the results do
not support the idea that consumers’ responses are determined by inertia or impulsiveness.
4.2.2
Testing for Rational Inattention in the Wrong Choice of Tariff
In Table 5 we study the extent to which ex post mistakes are systematic. The analysis
of whether households make systematic mistakes is important, and perhaps even
more revealing since it allows us to evaluate the extent to which households respond
rationally to very small incentives or whether deliberation, cognition, and processing
costs are exceedingly high relative to expected payoffs so that they do not react to
these incentives. The endogenous variable equals one whenever household i chooses
the wrong tariff option ex post (that is, either the measured tariff and a relatively high
usage level or the flat tariff and a relatively low usage level). As in the previous table,
the first column reports the probit estimates using only exogenous regressors, the
second column gives the pseudo–ML estimates including two predetermined variables
that are incorrectly treated as purely exogenous, and the third column gives the
GMM estimates of our dynamic discrete choice model. The predetermined variables
are whether households made a wrong tariff choice in the previous period and whether
they subscribed to the measured tariff option.33
[Table 5 here]
31
Table 3 indicates that these consumers are far more numerous than those wrongly choosing the
measured tariff. This helps to explain the negative estimate of Low Usage.
32
Impulsiveness or random behavior (e.g., consumers choosing tariffs by flipping a fair coin every
month) would imply a coefficient for measuredt−1 equal to zero.
33
We also estimated specifications of the models in Tables 4 and 5 including interactions among
the state variables, but this did not improve our estimation.
26
The results when controlling for individual heterogeneity and state dependence in
column 3 are quite different from the estimations that ignore such effects in columns
1 and 2. In the first column, most of the demographics play a significant role in
determining whether or not consumers make mistakes, while in the random effects
dynamic model of the third column virtually none of them play a role. Thus, demographics are significant in the first column only because of the limited scope of
that static approach that ignores potential learning and investment effects over time
and unobserved heterogeneity. These very sharp differences again indicate that as
time elapses, accumulated individual experiences, investments, and information associated with past tariff choices and consumption decisions do become more important
determinants of future choices.
With regard to the pseudo–ML estimates of the second column, they indicate that
households that either overestimate or underestimate future usage by a considerable
amount are less likely to make mistakes than those whose predictions are more accurate. This idea is in clear contradiction with the descriptive evidence reported in
Table 3. If these estimates were reliable they would support the idea that individuals
who predict their future consumption best would tend to make systematic mistakes
over time. Interestingly enough, the estimates of these variables are not significant
in the third column. Once we control for the history of past decisions, accumulated
experience, cognitive efforts and information play a more important role in the choice
of tariff over time. The fact that this occurs in such a short period of time is consistent with the idea that consumers are actively engaging in effective thinking, track
their own demand, and learn about the new consumption opportunities provided by
the alternative option.
The estimates of the last two variables are the most important ones. The first
result to note is that both of them are very significant in columns 2 and 3 but with
opposite signs:
(a) The positive sign of measuredt−1 in the second column would be consistent,
for instance, with a model where a household systematically thinks that it is going
to consume below the threshold level but will systematically consume above it. A
naive hyperbolic discounter would exhibit this type of systematic mistake (Strotz
(1956), Laibson (1997)). However, once we control appropriately for the effects of
individual heterogeneity associated to the accumulation of experience, investments,
and information in the third column, the results turn out to be drastically different.
The sign of measuredt−1 becomes negative, and the estimate is much greater in
magnitude and remarkably more significant. The pattern of behavior implied by
this coefficient is also consistent with the broad pattern observed in Table 3. This
result establishes that the switching of tariffs is not symmetric. This asymmetric
27
behavior may be explained by different cognitive and deliberation costs across tariff
choices. In particular, as indicated earlier, it is quite apparent that it is much easier
for households that subscribe to the measured option to monitor whether they have
made the wrong decision: they simply have to compare their actual bill with the
$18.70 flat rate. Households in the flat tariff would have to monitor their phone calls
very carefully and make more complex calculations in order to ascertain whether
or not they are making a mistake. Monitoring and cognitive costs are clearly much
greater for them. The asymmetric switching behavior that we observe is thus perfectly
consistent with these asymmetric differences in complexity and cognitive costs. This
result supports the implication that households that face the less complex problem
learn faster and commit fewer mistakes. Lastly, the negative sign of measuredt−1
also shows that consumers readily react in the very short term to very small monetary
incentives.
(b) The negative sign of wrongt−1 in the third column indicates that mistakes
are not permanent and that the switching between tariff options is aimed at reducing
the cost of local telephone service. This finding is important, and is in sharp contrast
with the positive sign of this variable in the second column which would incorrectly
indicate that households make systematic mistakes. These mistakes, which would be
characteristic of households driven by rational inattention, are not supported by our
random effects dynamic model.
Lastly, we find that no demographic variable other than the dummy indicator for
the older age group, age3, helps to explain mistakes. Thus, the two endogenous
predetermined variables are the ones driving the correctness of households’ decisions,
that is, the relationship between the ex ante choice of tariffs and the ex post realizations of consumption. They do so in the direction predicted by a rational attention
model where cognition, deliberation, and decision costs are not large enough relative
to the expected payoffs to induce inertia and systematic mistakes.
4.2.3
Errare Humanum Est, In Errore Perservare Stultum34
Before concluding, we pursue a bit further the result that mistakes are a transitory
phenomenon, and compute the marginal effects associated with the transition among
different states. Arellano and Carrasco (2003) show that the probability of subscribing
to the wrong tariff plan when we compare two states zit = z 0 and zit = z 1 changes
by the proportion:
N n ³
´
h ³ ´i´o
³
³
´
h ³ ´i´
³
1 X
.
− Φ σ̂t−1 β̂ z 0 − zit + Φ−1 ĥt wit
4̂t =
Φ σ̂t−1 β̂ z 1 − zit + Φ−1 ĥt wit
N i=1
34
“It is human to make a mistake, it is stupid to persist on it” (L. A. Seneca, 4 BC-65 AC).
28
Since the evaluation depends on the history of past choices ωit , these marginal
effects are different for each month of the sample. Table 6 presents four marginal
effects evaluated in October, November, December, as well as the average effect over
the Fall.35
[Table 6 here]
The first two rows show the change in probability of choosing wrongly if consumers
chose wrongly in the previous month. The first row indicates that this probability
decreases on average by 6.91% if consumers subscribed to the flat tariff option while
the second row shows that this probability decreases by 1.22% had they subscribed
to the measured tariff option. Thus, regardless of the choice of tariff, it is less likely,
rather than more likely, that they make another mistake in their choice of tariffs.
Similarly, the last two rows report the change in probability of choosing wrongly
if consumers subscribed to the optional measured service in the previous month. This
probability falls by 15.47% if consumers subscribed correctly to the optional measured
service in the previous month and by 9.78% if they subscribed wrongly to the optional
measured service. Thus, consistent with the asymmetry in the complexity of the
problems discussed earlier, the probability of making a mistake is substantially lower
after subscribing to the measured option. This reduction is more important for those
with low demand for which the measured service is the least expensive option than
for those with an usage pattern above the threshold of $18.70.
In analyzing these marginal effects, wrong equals one when consumers pay any
positive amount above the cost of the alternative option. We repeat the analysis for
different thresholds in increments of 5 cents from $0.00 to $4.00 in order to measure
whether this change in the probability varies significantly with the magnitude of the
mistake. Figure 3 reports the average marginal effects for the Fall.
[Figure 3 here]
Interestingly, these results show that the effects experience an abrupt jump in the
first 25-30 cents and are extremely constant once consumers realize a mistake above
these 25–30 cents. Recall that under the measured service option consumers are not
billed for the $5 allowance unless their usage is above $19.02. This is 32 cents more
than the $18.70 cost of the flat tariff option. We find it remarkable that this amount
is almost identical to 25-30 cents.
35
These four transitions exhaust the relevant effects to be reported. To compute the marginal
effects of going in the opposite direction, just reverse the sign of the corresponding effect in Table 6.
29
4.3
Discussion and Implications
To our knowledge, the effects of unobserved heterogeneity and unobserved investments in information, active cognition and deliberation costs in determining current
choices have not been addressed before, neither in the experimental nor in the empirical learning and behavioral economics literature. The fact that the evidence turns
out to be drastically different when predetermined variables and unobserved heterogeneity are appropriately treated indicates that they play an important role in the
dynamic learning process.
Our first empirical objective was quantitative in nature: When should we expect
that deliberation and cognition activities represent critical limits to rationality or,
alternatively, that they are cheap relative to payoffs? Since we can measure the
magnitude of the cost difference between potential choices, the finding that households
display rational attention lead us to infer an upper bound on the costs of active
cognition and deliberation that are involved. This upper bound is low, and would
even be lower if search and other costs were likely to be large. However, search costs
are unlikely to be high since households receive every month a bill specifying their
consumption behavior.
The second empirical goal was a specific behavioral implication. Households who
face the less complex, cognitively cheaper problem behave as predicted: they learn
faster and are much less likely to make mistakes.
The third empirical goal was qualitative in nature. The results support the regime
characterized by high experimentation (thinking effort) and good tracking of demand,
that is by large deviations from myopic behavior (Keller and Rady (1999)). They are
not consistent with moderate experimentation and poor tracking of demand, that is
with small deviations from myopic behavior.
Our findings allow us to discard not only models of inertia and inattention, but
also models where households are driven by impulsiveness or are time-inconsistent.
Random choice of tariffs would simply imply no effect of lagged dependent variables
(lagged choices and lagged outcomes), something that we do not observe in the data.
Thus, our households are not impulsive. Neither they seem to be time-inconsistent.
Households with non-constant temporal preferences, like those in models of myopic
hyperbolic discounting (Strotz (1956), Laibson (1997)), would display a systematic
overvaluation of the future: their actual consumption would systematically be above
their planned consumption. Iff the optimal tariffs associated with these two consumption levels is different, then systematic ex post mistakes should be observed in the
data.36 Empirically, we see no evidence that households who systematically choose
36
We would not be able to detect myopic hyperbolic discounters if this is not the case, that is if
30
the measured service, expecting a low consumption level, systematically consume
above the level that makes the measured tariff optimal ex post. Thus, the evidence
does not support the idea of time-inconsistent households.
5
Concluding Remarks
The systematic analysis of individual responses to changes in the environment has
important implications not only for the study of many kinds of economic and social
phenomena but also for understanding the extent and formation of rationality. The
natural scenario we have examined and the panel structure of the data offer a number
of valuable advantages that allow us to avoid many of the crucial difficulties that may
explain the lack of empirical studies that asses the extent of individual rationality
in natural environments. We are able to uncover households’ responses in isolation
from a number of other conflicting considerations which almost always exist in other
circumstances.
We find that forward–looking households recognize that choices today affect their
utilities in the future and that they actively react to a new option despite potential
savings of very small magnitude. It is at least conceivable that households would
not have reacted to potential savings of much smaller magnitude, say 10-15 cents,
perhaps even up to 50 cents-1 dollar per month. In this sense, the results allow us
to infer an upper bound, rather than a lower bound, on the size of deliberation costs
that may be involved in the problem that we study.
The costs of deliberation, decision and active cognition pervade the discussion on
bounded rationality. They are important for models in this literature because departures from rationality may be systematically related to the deliberation costs involved.
If we take the Gödelian statement that “the rational thing to do is to be irrational
where deliberation and estimation cost more than they are worth” (Knight, 1921),
then, for the level of complexity of the problem examined in this paper, individuals’
deliberation and cognitive costs must be less than $5-$6 at monthly frequencies. In
Bayesian decision theory it is typically not possible to value and price information
since the value of information is not globally concave (Radner and Stiglitz (1984)),
unless information units are cheap relative to payoffs (Moscarini and Smith (2002)).
Thus, an implication of our results is that formal models may be used to price information and to derive empirical implications even if expected payoffs are of low
magnitude for problems whose complexity is no greater than the one we have examined.
we do not have that the optimal tariff for the planned consumption is the measured tariff and the
optimal tariff for the actual consumption is the flat tariff.
31
Lastly, the generality of our results need not extend to problems that are more
complex than the one we have studied. Yet, we have no way of assessing the degree of
complexity of this problem relative to other problems that households may typically
face. It is possible to argue, however, that the demand for many consumption goods
and services is similar to this problem in that the optimal action often requires households, at the very least, to determine whether they expect their demand at a future
date to be below or above a certain threshold. To the extent that this conjecture is
correct, these results may gain a substantial deal of generality.
32
REFERENCES
Abreu, Dilip and Rubinstein, Ariel. “The Structure of Nash Equilibrium in Repeated Games with Finite Automata,” Econometrica 56 (December 1988): 1259-1271.
Akerlof, George A. and Dickens, William T. “The Economic Consequences of
Cognitive Dissonance,” American Economic Review 72 (June 1982): 307-319.
Akerlof, George A. and Yellen, Janet L. “Rational Models of Irrational Behavior,”
American Economic Review 77 (May 1987): 137-142.
Anderson, Gordon. “Tests of Stochastic Dominance in Income Distribution,”
Econometrica 64 (September 1996): 1183-1194.
Arellano, Manuel and Bover, Olympia. “Another Look at the Instrumental Variable Estimation of Error–Components Models,” Journal of Econometrics 68 (July
1995): 29–51.
Arellano, Manuel and Carrasco, Raquel. “Binary Choice Panel Data Models with
Predetermined Variable,” Journal of Econometrics, 115 (July 2003): 125-157.
Arellano, Manuel and Honoré, Bo E. “Panel Data Models: Some Recent Developments.” In Handbook of Econometrics, vol. 5, chapter 53. Edited by J. E. Heckman
and E. Leamer, North-Holland, 2001.
Arrow, Kenneth J. “Rationality of Self and Others in an Economic System.” In
Robin M. Hogarth and Melvin W. Reder (eds.) Rational Choice. The Contrast
between Economics and Psychology. Chicago: University of Chicago Press, 1987:
201-15.
Becker, Gary S. “Irrational Behavior and Economic Theory,” Journal of Political
Economy 70 (February 1962): 1-13.
Bolton, Patrick and Harris, Christopher. “Strategic Experimentation,” Econometrica 67 (1999): 349-374.
Chade, Hector and Schlee, Edward E. “Another Look at the Radner-Stiglitz Nonconcavity in the Value of Information,” Journal of Economic Theory 107 (December
2002): 421-452.
Chamberlain, G. “Analysis of Covariance with Qualitative Data,” Review of Economic Studies 47 (April 1980): 225-238.
Chamberlain, G. “Panel Data.” In Griliches, Z. and M. D. Intriligator (eds.)
Handbook of Econometrics, vol. 2, Elsevier Science, Amsterdam, 1984.
33
Chiappori, Pierre-André. “Econometric Models of Insurance under Asymmetric
Information.” In Georges Dionne (ed.), Handbook of Insurance, chapter 12, Kluwer
Academic Publishers, 2000.
Chiappori, Pierre-André and Salanié, Bernard. “Testing for Adverse Selection in
Insurance Markets,” Journal of Political Economy 108 (February 2000): 56-78.
Conlisk, John. “Why Bounded Rationality?” Journal of Economic Literature 34
(June 1996): 669-700.
De Palma, André; Myers, Gordon M. and Papageorgiou, Yorgos Y. “Rational
Choice Under an Imperfect Ability to Choose,” American Economic Review 84 (June
1994): 419-440.
Della Vigna, S. and U. Malmendier. “Self–Control in the Market: Evidence from
the Health Club Industry.” Mimeo, Harvard University, 2001.
Ellison, Glenn and Fudenberg, Drew. “Rules of Thumb for Social Learning,”
Journal of Political Economy 101 (August 1993): 612-643.
Epstein, Larry. “Sharing Ambiguity,” American Economic Review 91 (May 2001):
45-50.
Evans, George W. and Honkapohja, Seppo. “Adaptative Learning and Expectational Stability: An Introduction,” In Learning and Rationality in Economics. Eds.:
Alan Kirman and Mark Salmon. Oxford: Basil Blackwell, 1995.
Evans, George W. and Ramey, Garey. “Calculation, Adaptation, and Rational
Expectations,” Macroeconomic Dynamics 2 (June 1998): 156-182.
Foster, Andrew D. and Rosenzweig, Mark R. “Learning by Doing and Learning from Others: Human Capital and Technical Change in Agriculture,” Journal of
Political Economy 103 (December 1995): 1176-1209.
Gabaix, Xavier and Laibson, David. “Bounded Rationality and Directed Cognition,” Mimeo, MIT, 2002.
Gode, Dhananjay K. and Sunder, Shyam. “Allocative Efficiency of Markets with
Zero-Intelligence Traders: Market as a Partial Substitute for Individual Rationality,”
Journal of Political Economy 101 (February 1993): 119-137.
Hansen, Lars P. and Sargent, Thomas J. “Robust Control and Model Uncertainty,” American Economic Review 91 (May 2001): 60-66.
Heckman, James J. “The Incidental Parameters Problem and the Problem of
Initial Conditions in Estimating a Discrete Time–Discrete Data Stochastic Process.”
34
In C.F. Manski and D.L. McFadden (eds.): Structural Analysis of Discrete Data with
Applications. Cambridge, MA: MIT Press, 1981b.
Honoré, Bo E. and Lewbel, Arthur. “Semiparametric Binary Choice Panel Data
Models without Strictly Exogenous Regressors,” Econometrica 70 (November 2002):
2053-2063.
Honoré, Bo E. and Kyriazidou, Ekaterini. “Panel Data Discrete Choice Models
with Lagged Dependent Variables,” Econometrica 68 (July 2000): 839–874.
Keller, Godfrey and Rady, Sven. “Optimal Experimentation in a Changing Environment,” Review of Economic Studies 66 (July 1999): 475-507.
Knight, Frank. Risk, Uncertainty and Profit, University of Chicago Press, 1921.
Laibson, David I. “Golden Eggs and Hyperbolic Discounting,” Quarterly Journal
of Economics 112 (May 1997): 443–77.
Lancaster, Tony. “The Incidental Parameter Problem since 1948.” Journal of
Econometrics 95 (April 2000): 391–413.
Lerman, S. R. and Manski, Charles F. “The Estimation of Choice Probabilities
from Choice Based Samples,” Econometrica 45 (November 1977): 1977–1988.
Lipman, Barton L. “How to Decide How to Decide How to ...: Modeling Limited
Rationality,” Econometrica 59 (July 1991): 1105-1125.
Lucas, Robert E. Jr. “Adaptive Behavior and Economic Theory.” In Robin
M. Hogarth and Melvin W. Reder (eds.) Rational Choice. The Contrast between
Economics and Psychology. Chicago: University of Chicago Press, 1987: 217-42.
Manski, Charles F. “Semiparametric Analysis of Random Effects Linear Models
from Binary Panel Data,” Econometrica 55 (1987): 357-362.
Miravete, Eugenio J. “Choosing the Wrong Calling Plan? Ignorance and Learning,” CEPR DP 2562 (September 2000).
. “Estimating Demand for Local Telephone Service with Asymmetric
Information and Optional Calling Plans,” Review of Economic Studies 69 (October
2002a): 943-971.
. “The Welfare Performance of Sequential Pricing Mechanisms.” Mimeo,
University of Pennsylvania, 2002b.
. “Choosing the Wrong Calling Plan? Ignorance and Learning,” American Economic Review 93 (March 2003): 297-310.
Mitchell, Bridget M. and Vogelsang, Ingo. Telecommunications Pricing. Theory
and Practice. Cambridge, U.K.: Cambridge University Press, 1991.
35
Moscarini, Giuseppe. “Limited Information Capacity as a Source of Inertia,”
Mimeo, Yale University, 2002.
Moscarini, Giuseppe and Smith, Lones. “The Law of Large Demand for Information,” Econometrica 70 (November 2002): 2351-2366.
. “The Optimal Level of Experimentation,” Econometrica 69 (November
2001): 1629-1644.
Newey, Whitney K. “The Asymptotic Variance of Semiparametric Estimators,”
Econometrica 62 (1994): 1349-1382.
Neyman, J. and Scott, E. L. “Consistent Estimation from Partially Consistent
Observations.” Econometrica 16 (1948): 1–32.
Radner, Roy. “Satisficing,” Journal of Mathematical Economics 2 (June-September
1975): 253-262.
Radner, Roy and Rothschild, Michael. “On the Allocation of Effort,” Journal of
Economic Theory 10 (June 1975): 358-376.
Radner, Roy and Stiglitz, Joseph E. “A Nonconcavity in the Value of Information.” In Bayesian Models in Economic Theory, edited by Marcel Boyer and Richard
E. Kihlstrom. NY: Elsevier Science Publishers, 1984, pp. 33-52.
Rothschild, Michael. “Further Notes on the Allocation of Effort,” in Adaptive
Economic Models. Edited by Richard H. Day and Theodore Groves. New York:
Academic Press, 1975, pp. 195-220.
Rosenthal, Robert W. “Rules of Thumb in Games,” Journal of Economic Behavior
and Organization 22 (September 1993): 1-14.
Rubinstein, Ariel. Modeling Bounded Rationality. Cambridge, Mass.: MIT Press,
1998.
Rustichini, Aldo and Wolinsky, Asher. “Learning About Variable Demand in
the Long Run,” Journal of Economic Dynamics and Control 19 (July-Sept. 1995):
1283-92.
Sargent, Thomas J. Bounded Rationality in Macroeconomics. Oxford: Oxford
University Press, 1993.
Savage, Leonard J. The Foundations of Statistics. New York: Wiley, 1954.
. “Difficulties in the Theory of Personal Probability,” Philosophy of Science 34 (1967): 305-310.
Selten, Reinhard. “The Chain Store Paradox,” Theory and Decision 9 (April
1978): 127-159.
36
Simon, Herbert A. “A Behavioral Model of Rational Choice,” Quarterly Journal
of Economics 69 (February 1955): 99-118.
. “Rationality in Psychology and Economics,” in Robin M. Hogarth and
Melvin W. Reder (eds.) Rational Choice. The Contrast Between Economics and
Psychology. Chicago: University of Chicago Press, 1987.
. The Sciences of the Artificial. Cambridge, Mass.: MIT Press, 1996.
. An Empirically Based Microeconomics. Cambridge, UK: Cambridge
University Press, 1997.
Sims, Christopher A. “Implications of Rational Inattention,” Mimeo, Princeton
University, 2002.
Smith, Vernon L. and Walker, James M. “Monetary Rewards and Decision Costs
in Experimental Economics,” Economic Inquiry 31 (April 1993): 245-261.
Stigler, George J. and Gary S. Becker. “De Gustibus Non Est Disputandum,”
American Economic Review 67 (March 1977): 76–90.
Stoline, Michael R. and Ury, Hans K. “Tables of Studentized Maximum Modulus
Distribution and an Application to Multiple Comparisons Among Means.” Technometrics 21 (February 1979): 87-93.
Strotz, Robert H. “Myopia and Inconsistency in Dynamic Utility Maximization,”
Review of Economic Studies 23 (1956): 165–80.
Winter, S. “Optimization and Evolution in the Theory of the Firm,” in Adaptive
Economic Models. Edited by Richard H. Day and Theodore Groves. New York:
Academic Press, 1975, pp. 73-118.
37
Table 1. Descriptive Statistics
Variables
Description
ALL
0.2971
FLAT
(0.46)
MEASURED
Optional measured service chosen this month
EXPCALLS
Household own estimate of weekly number of calls
26.8884 (31.34)
30.1341 (35.05)
19.2104 (17.78)
CALLS
Current weekly number of calls
37.6093 (38.48)
44.4898 (42.62)
21.3326 (17.64)
BIAS
CALLS — EXPCALLS
10.7209 (39.92)
14.3558 (45.67)
2.1223 (18.04)
SWCALLS
Household average number of calls during Spring
37.9434 (37.16)
44.0499 (40.80)
23.4980 (20.32)
SWBIAS
SWCALLS — EXPCALLS
11.0550 (39.37)
13.9158 (44.55)
BILL
Monthly expenditure in local telephone service
19.4303
18.7000
SAVINGS
Potential savings of switching tariff options
SAVINGS-SPR
Potential savings of subscribing the measured option
SAVINGS-OCT
SAVINGS-NOV
1.0000
4.2876 (21.39)
(0.00)
21.1578
–9.9223 (16.53)
–15.1557 (16.45)
2.4578
(7.82)
–15.4206 (15.27)
–18.7859 (16.21)
–7.4596
(8.56)
Potential savings in October
–9.4898 (16.99)
–14.2444 (17.61)
1.7578
(7.60)
Potential savings in November
–9.2864 (15.03)
–13.6444 (15.30)
1.0230
(7.47)
SAVINGS-DEC
Potential savings in December
–10.9908 (17.41)
–16.4967 (17.22)
2.0340
(8.83)
INCOME
Monthly income of the household
7.0999
(0.81)
7.0767
(0.84)
7.1547
(0.74)
HHSIZE
Number of people who live in the household
2.6168
(1.51)
2.7858
(1.56)
2.2170
(1.28)
TEENS
Number of teenagers (13–19 years)
0.2440
(0.63)
0.2908
(0.68)
0.1336
(0.49)
DINCOME
Household did not provide income information
0.1577
(0.36)
0.1831
(0.39)
0.0977
(0.30)
AGE1
Head of household is between 15 and 34 years old
0.0632
(0.24)
0.0614
(0.24)
0.0676
(0.25)
AGE2
Head of household is between 35 and 54 years old
0.2686
(0.44)
0.2604
(0.44)
0.2880
(0.45)
AGE3
Head of household is above 54 years old
0.6682
(0.47)
0.6782
(0.47)
0.6444
(0.48)
COLLEGE
Head of household is at least a college graduate
0.2240
(0.42)
0.1821
(0.39)
0.3230
(0.47)
MARRIED
Head of household is married
0.5253
(0.50)
0.5342
(0.50)
0.5042
(0.50)
RETIRED
Head of household is retired
0.2433
(0.43)
0.2417
(0.43)
0.2471
(0.43)
BLACK
Head of household is black
0.1161
(0.32)
0.1295
(0.34)
0.0843
(0.28)
CHURCH
Telephone is used for charity and church purposes
0.1711
(0.38)
0.1785
(0.38)
0.1536
(0.36)
BENEFITS
Household receives some federal or state benefits
0.3095
(0.46)
0.3282
(0.47)
0.2654
(0.44)
MOVED
Head of household moved in the past five years
0.4025
(0.49)
0.3899
(0.49)
0.4324
(0.50)
Observations
1,344
(4.41)
0.0000
MEASURED
949
(7.82)
395
Mean and standard deviation of demographics and usage variables. This balanced sample contains 1,344 household observations. Income is
measured in logarithms of thousands of 1986 dollars.
–i–
Table 2. Choice of Tariff and Usage Level
MEASURED
LOW USAGE
Constant
–0.6763 (5.56)
–0.8099 (7.06)
LOW INCOME
–0.0604 (0.57)
0.0418 (0.46)
HIGH INCOME
–0.2317 (1.79)
–0.0320 (0.32)
DINCOME
–0.4846 (4.23)
–0.1144 (1.43)
HHSIZE = 2
–0.3548 (3.32)
–0.3128 (3.46)
HHSIZE = 3
–0.5645 (4.29)
–0.3979 (3.81)
HHSIZE = 4
–0.4854 (3.17)
–0.3866 (2.97)
HHSIZE > 5
–0.7187 (4.04)
–0.6709 (4.22)
TEENS
–0.1768 (1.27)
0.0115 (0.11)
AGE1
–0.0216 (0.14)
0.1761 (1.38)
AGE3
–0.0491 (0.53)
0.1707 (2.03)
COLLEGE
0.2910 (3.42)
0.0709 (0.93)
MARRIED
0.2301 (2.47)
–0.0509 (0.66)
RETIRED
0.0497 (0.43)
–0.1967 (2.24)
BLACK
0.0287 (0.26)
–0.1845 (1.72)
CHURCH
–0.0274 (0.30)
–0.0084 (0.11)
BENEFITS
–0.2189 (2.03)
–0.0360 (0.42)
MOVED
–0.0542 (0.64)
0.0915 (1.24)
UNDERESTIMATION
–0.4164 (4.14)
–1.1597 (9.70)
OVERESTIMATION
–0.3548 (2.42)
–0.7881 (5.17)
0.6418 (4.87)
1.4125 (11.26)
LOW USAGE
ρ
Observations
Spring
0.8408 (7.46)
4,032
The endogenous variable MEASURED equals one if the household subscribes the optional measured service during the current month. The UNDERESTIMATION dummy indicates that
SWCALLS exceeds EXPCALLS by more than 50% of the standard deviation of SWBIAS. The OVERESTIMATION dummy
is defined accordingly when EXPCALLS exceeds SWBIAS. The
LOW USAGE dummy indicates whether the monthly consumption during the Spring months would have exceeded the $18.70
threshold if billed according to the optional measured tariff
available during the second half of 1986. Estimates are obtained
by weighted ML (bivariate probit). Absolute, choice–biased
sampling, heteroscedastic consistent, t–statistics are reported
in parentheses.
– ii –
Table 3. Potential Savings and Tariff Switching
PATH
FFF
FFM
FMF
FMM
MFF
MMF
SAMPLE OBSERVATIONS
POPULATION SHARE
953
0.8603
5
0.0045
1
0.0009
38
0.0343
28
0.0067
13
0.0031
375
0.0901
12.5845
-0.1954
9.3826
-0.1631
-51.0870
-1.7669
3.3370
-0.0027
15.1761
0.0593
0.5246
-0.2019
3.0600
-0.2598
1.0000
16.7640
1.0000
17.5189
0.5733
1.1859
0.0000
-14.4198
1.0000
14.8302
0.5573
0.8899
SWCALLS–EXPCALLS
PERCENT
MMM
OCTOBER
WRONG
POTENTIAL SAVINGS
0.1070
-15.2358
0.6000
-2.7268
0.0000
-7.6810
0.4211
-2.0849
NOVEMBER
WRONG
POTENTIAL SAVINGS
0.1133
-13.9896
0.6000
-1.8204
1.0000
5.6830
0.5789
2.5909
DECEMBER
WRONG
POTENTIAL SAVINGS
0.0619
0.4000
0.0000
0.6842
0.0000
0.0000
0.6667
-16.9373
2.6848
-4.0760
3.7008
-15.3860
-18.3705
2.6647
PATH denotes the sequence of tariff choices (F=Flat, M=Measured) by households during the October–December
period. No household followed the MFM sequence. POPULATION SHARE is the proportion of households in the
sample after correcting for choice-biased sampling in October. WRONG denotes the proportion of sample households
that each month would have saved a positive amount in their telephone bill had they chosen the alternative tariff
plan and had they kept their local telephone usage level unchanged. POTENTIAL SAVINGS indicates the average
magnitude of the mistake of all the houselholds in that path. The mistake is positive for WRONG households and
negative for all others.
– iii –
Table 4. Testing for Attention and Inertia in Tariff Suscription
STATIC
PSEUDO–DYNAMIC
RANDOM EFFECTS
POOL
PANEL
DYNAMIC PANEL
Constant
–0.6275 (10.83)
–1.2448
(16.49)
–1.8180
(6.95)
LOW INCOME
–0.0406
(0.78)
–0.0625
(0.91)
–0.1105
(0.42)
HIGH INCOME
–0.2180
(4.06)
–0.2092
(2.89)
–0.1082
(0.41)
DINCOME
–0.4654
(9.19)
–0.3965
(6.20)
–1.2911
(4.94)
HHSIZE = 2
–0.3885
(7.91)
–0.2932
(4.66)
–0.2421
(0.93)
HHSIZE = 3
–0.6375 (10.15)
–0.4636
(5.77)
0.1631
(0.62)
HHSIZE = 4
–0.5488
(7.70)
–0.4251
(4.68)
0.4255
(1.63)
HHSIZE > 5
–0.7721
(8.92)
–0.5657
(5.35)
0.2058
(0.79)
TEENS
–0.1905
(3.49)
–0.1602
(2.37)
–0.0641
(0.24)
AGE1
–0.0210
(0.29)
–0.0252
(0.26)
0.1313
(0.50)
AGE3
–0.0288
(0.67)
–0.0385
(0.68)
–1.2077
(4.62)
COLLEGE
0.2963
(7.82)
0.2242
(4.47)
–0.2865
(1.10)
MARRIED
0.2366
(5.08)
0.1882
(3.19)
0.5212
(1.99)
RETIRED
0.0433
(0.86)
0.0330
(0.52)
–0.5431
(2.08)
BLACK
0.0144
(0.26)
0.0764
(1.09)
–0.1452
(0.56)
CHURCH
–0.0334
(0.76)
–0.0208
(0.37)
–0.1421
(0.54)
BENEFITS
–0.2332
(4.78)
–0.1750
(2.86)
–0.3390
(1.30)
MOVED
–0.0541
(1.37)
–0.0476
(0.92)
–0.1958
(0.75)
UNDERESTIMATION
–0.4478 (10.15)
–0.3282
(5.64)
–0.5730
(2.19)
OVERESTIMATION
–0.3538
–0.2926
(3.26)
–0.1294
(0.49)
LOW USAGEt−1
0.4034
(7.21)
–3.9039
(14.93)
MEASUREDt−1
3.1919
(41.30)
–6.1359
(23.46)
(5.43)
The endogenous variable equals one if the household subscribes to optional measured service at time
t. Sample includes 1,344 individual observations over a three-month period. Absolute, choice–biased
sampling, heteroscedastic–consistent, t–statistics are reported in parentheses. The models of the
first and second column are estimated by weighted ML (probit). The random effects dynamic model
of the third column is estimated by GMM.
– iv –
Table 5. Testing for Persistence in the Wrong Choice of Tariff
Constant
STATIC
PSEUDO–DYNAMIC
RANDOM EFFECTS
POOL
PANEL
DYNAMIC PANEL
–0.5114
(9.71)
–1.0033
(16.69)
–1.4118
(6.30)
LOW INCOME
0.0065
(0.14)
–0.0013
(0.02)
–0.1166
(0.52)
HIGH INCOME
–0.0788
(1.56)
–0.0267
(0.50)
–0.0729
(0.33)
DINCOME
–0.1975
(4.63)
–0.1014
(2.17)
–0.1238
(0.55)
HHSIZE = 2
–0.2682
(6.03)
–0.1446
(2.92)
–0.2172
(0.97)
HHSIZE = 3
–0.3800
(6.80)
–0.1884
(3.18)
–0.1589
(0.71)
HHSIZE = 4
–0.3317
(4.96)
–0.1786
(2.54)
–0.1152
(0.51)
HHSIZE > 5
–0.5214
(6.65)
–0.3188
(3.87)
–0.0922
(0.41)
TEENS
–0.1236
(2.50)
–0.0866
(1.69)
–0.1582
(0.71)
AGE1
0.1227
(1.84)
0.1486
(2.09)
–0.0370
(0.17)
AGE3
0.0869
(2.20)
0.0745
(1.74)
–0.4698
(2.10)
COLLEGE
0.1767
(4.83)
0.0948
(2.40)
–0.1226
(0.55)
MARRIED
–0.0105
(0.25)
–0.0539
(1.25)
–0.3837
(1.71)
RETIRED
–0.1533
(3.13)
–0.1390
(2.62)
–0.1689
(0.75)
BLACK
–0.1205
(2.29)
–0.0879
(1.57)
–0.0992
(0.44)
CHURCH
–0.0235
(0.59)
–0.0113
(0.26)
–0.1233
(0.55)
BENEFITS
–0.1213
(2.72)
–0.0692
(1.44)
–0.2260
(1.01)
0.0425
(1.21)
0.0335
(0.87)
–0.2657
(1.19)
UNDERESTIMATION
–0.7510 (17.47)
–0.6278
(13.65)
–0.2452
(1.09)
OVERESTIMATION
–0.5773
–0.5000
(7.77)
–0.0724
(0.32)
MEASUREDt−1
0.8087
(15.40)
–6.0301
(26.92)
WRONGt−1
1.2331
(29.80)
–1.2128
(5.41)
MOVED
(9.51)
The endogenous variable equals one whenever the household subscribes to the wrong tariff choice
for their realized consumption. Sample includes 1,344 individual observations over a three-month
period. Absolute, choice–biased sampling, heteroscedastic–consistent, t–statistics are reported in
parentheses. The models of the first and second column are estimated by weighted ML (probit).
The random effects dynamic model of the third column is estimated by GMM.
Table 6. Marginal Effects
October
November
December
FALL
From (0,0) to (0,1)
–10.76
–6.10
–3.87
–6.91
From (1,0) to (1,1)
–0.06
–1.63
–1.97
–1.22
From (0,0) to (1,0)
–17.59
–17.52
–11.29
–15.47
From (0,1) to (1,1)
–6.89
–13.05
–9.39
–9.78
These marginal effects represent the percent change in the probability of
choosing wrongly the tariff option conditional on each transition among
states. The pair (i, j) represents the state, where i denotes the choice of
tariff (i = 0 if Flat and i = 1 if Measured) and j denotes if the tariff is
chosen correctly (j = 0) or incorrectly (j = 1).
–v–
1
0.9
0.8
Cumulative Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
Actual Calls (continuous) and Expected Calls (dotted)
Figure 1. Distribution of Actual and Expected Calls
0.030
Frequency
0.025
0.020
0.015
0.010
0.005
0.000
-110 -100 -90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
60
70
80
90
100 110 120 130 140
Bias=Calls-Expected Calls
Figure 2. Empirical Density of Expectation Errors
Percentage Change of Probability
Percentage Change of Probability
1.00
1.00
1.25
1.25
1.50
1.50
2.00
2.25
2.00
2.25
From (0,0) to (1,0)
1.75
From (0,0) to (0,1)
1.75
2.50
2.50
2.75
2.75
3.00
3.00
3.25
3.25
3.50
3.50
3.75
3.75
4.00
4.00
-12.0
-11.5
-11.0
-10.5
-10.0
0.25
0.25
0.50
0.50
0.75
0.75
Figure 3. Marginal Effects
0.00
0.00
-9.5
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
-13.0
0.75
0.75
-15.6
0.50
0.50
-12.5
0.25
0.25
-15.4
-15.2
-15.0
-14.8
-14.6
-14.4
0.00
0.00
-14.2
-7.0
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
Percentage Change of Probability
Percentage Change of Probability
1.00
1.00
1.25
1.25
1.50
1.50
2.00
2.25
2.00
2.25
From (0,1) to (1,1)
1.75
From (1,0) to (1,1)
1.75
2.50
2.50
2.75
2.75
3.00
3.00
3.25
3.25
3.50
3.50
3.75
3.75
4.00
4.00