Abstract
There is a long standing discussion in the literature,
whether expected utility theory (EU) or prospect theory (PT) explains best the
behavior with respect to risky choices. Often these two approaches are compared
by putting questions to students in laboratory situations. Here we try to
investigate stated preferences of farmers which are functioning under high
levels of risk in real life. As part of a larger survey, four binary choices
were offered during two successive years. The experimental test was done for 199
farmers in two different districts in Tigray,
In order to identify the factors that
affect farmer’s preferences a binary choice model was used. Household income were found to be positive and significant, while value of livestock
had the expected negative sign and directly related to a decrease in risk
aversion. This result has important implications for the characterization of
risk attitudes in policy applications.
Keywords: risky choice, risk attitude, expected utility, prospect
theory, Tigray.
1 Introduction
In
semi-subsistence agriculture, farm households face numerous natural, market and
institutional risks in generating means of survival. Yield risk, crop price
risk, risk of illness and injuries are important risks that prevail in
developing economies. Households have developed various mechanisms for coping
with risk. These mechanisms offer short-term protection at long-term cost (e.g.
diversification versus specialization). Their attitude towards risk, therefore,
tends to display an explanation for the many observed economic decisions.
In
measuring attitude towards risk, two approaches are identified: econometric and
experimental. The econometric approach is based on farmers’ actual behavioral
data, which typically assumes that farmers maximize the expected utility of
income. Given a production technology, the risk associated with production and
market conditions, the observed level of input use can reveal the underlying
degree of farmers risk aversion. Examples of this line of research include
(Bar-Shira et al., 1997; Kumbhakar,
2002). The experimental approach is based on questionnaires regarding
hypothetical risky alternatives with or without real payments. Here,
respondents are asked to choose between lotteries that differ in payoffs and
probabilities or both. The experimental approach is further classified into
expected utility and non-expected utility approaches. For example Binswanger
(1981) measured attitude towards risk in rural
Using
the experimental approach without real payments, this paper will identify which
choice model best describes risk attitude of Northern Ethiopian subsistence
farmers.[1]
The objective of this paper is to measure farmer’s attitude toward risk and to
see how an individual’s attitude toward risk relates to observed
characteristics. Specifically this study seeks to answer: (1) does the expected
utility theory explains risk attitude of north Ethiopian farmers better than
the non-expected utility approach (such as prospect theory)? (2) Are farmers
risk-averse to gains and risk seeking to losses? And are there concave utility
shapes for gains and convex utility shapes for losses? (3) Are there systematic
differences in attitudes amongst farmers? (4)Is there any evidence to suggest
that farmer’s socio-economic variables determine aversion to risk? From the
research question it is attempted to test whether the farmers made decisions
according to the expected utility theory or the non-expected utility theory
(prospect theory). Empirical studies on how risk varies across individuals can
be useful in predicting households’ technology adoption, participation on off-farm
work and in crop portfolio selection, since risk and risk aversion behavior
plays an important role in these decisions.
In the next
section a data set containing farmers’ choices of hypothetical binary lotteries
are presented. Experimental results on the shape of the utility function and a
test of the independence axiom are discussed in section 3. In section 4 factors
affecting risk behavior are econometrically determined. Section 5 concludes.
2 Expected Utility versus Non-Expected Utility:
Literature Overview
2.1 Expected utility: background
In
general the expected utility (EU) model has been the dominant model for the
last decades in modeling behavior under risk. Von Neumann and Morgenstern (vNM)
are the major contributors to a large body of work that provides the
justification for the use of the expected utility model by a rational decision
maker. This model views decision making under risk as a choice between
alternatives. Decision makers are assumed to have a preference ordering defined
over the probability distributions for which the axioms of the EU model hold
(Mas-Colell et al., 1995). Risky
alternatives can be evaluated under these assumptions using the expected
utility function
.
In
maximizing the decision maker’s utility, consider a risk prospect in which the
decision maker does not know ex-ante which state of the world will occur.
However he can list the various alternatives and can attach probabilities to
them. For simplicity, assume two possible states of the world, state 1 and
state 2, with respective probabilities p1 and p2 and denote 
the individual’s
monetary gain if state 1 occurs and 
if state 2 occurs. The individual must choose ex-ante between
the risky bundles
. Ex-post, the individual gets or depending upon which
state of the world has occurred. If the decision maker’s preference ordering
over risky alternatives satisfies all the axioms of expected utility, including
the independence and continuity axioms (see next section), then there exists a
vNM expected utility function. This vNM expected utility function reflects the
decision maker’s choice as if he maximizes utility of the different states weighted
by the probabilities for each state to occur.
vNM
began by stating that utility maximization is a rational goal when a decision
maker is faced with risky choices. In this framework, an individual will
evaluate the expected value and objectively given probability of occurrence of
each alternative. This evaluation is carried out by first entering the
probabilities and expected outcomes into an individual’s utility function. It
is then a matter of selecting the combination of available alternatives that maximizes
the function. The manner in which individuals choose among available
alternatives is then dependent upon their utility function. For this setting
the vNM expected utility function can be specified as:
(1)
where 
is the vNM expected utility function,
is the utility of the 
element of a vector of possible outcomes, and 
is the probability of
outcome 
, 
. The vNM expected utility function
, defined up to a positive linear transformation,
characterizes both the utility of the outcome and the individual’s attitude toward risk. The curvature of this
utility function contains information about the degree of individual’s risk
aversion (Mas-Colell et al., 1995:
173).
Axioms of the expected utility theory
There are
three main axioms in the expected utility framework. They are defined over a
binary relation where:
denotes weak preference,
denotes strict preference, and
~ denotes
indifference.
For
preferences over probability distributions 
that are defined over
a common (discrete or continuous) outcome vector
. The three axioms that are necessary and sufficient for the
expected utility representation 
over preferences are:
Axiom O (Order):
The binary
relation on
is asymmetric and
transitive. The asymmetric part of axiom 
says that the decision maker will not both prefer 
to 
and prefer to. According to expected utility theory, it is irrational to
hold a definite preference for over and a definite preference for over at a time. However, there is a possibility that neither nor is preferred (i.e. 
, the decision maker is indifferent between and ).
The
transitivity part of axiom holds if and only if both and ~are transitive,
i.e., for all , (
, and 
) 
; (and 
)
. Transitivity implies that it is impossible to face the
decision maker with a sequence of pair wise choices in which preferences appear
to cycle. For example, a decision maker feels that an apple is at least as good
as a banana and that a banana is at least as good as an orange but then also
preferring an orange over an apple.
Axiom C (Continuity):
For all with and there exists 
such that: 
and 
. This axiom gives continuity to the preferences. Continuity
means that small changes in probabilities do not change the nature of the
ordering between two lotteries (see Mas-Colell et al., 1995: 171). Continuity rules out lexicographic preferences.
Axiom I (
For all and for all
, if, then
. This axiom states that preferences over probability distributions
should only depend on the portions of the distributions that differ ( and ), not on their common elements (
) and of the level of 
that defines the linear
combination. In other words, if we mix each of two lotteries with a third one,
then the preference ordering of the two resulting mixtures does not depend on
the particular third lottery used.
Axioms
O, C, and I can be shown to be necessary and sufficient for the existence of a
function on the outcomes 
that represents
preferences through. The role of the order, completeness and continuity axioms
are essential to establish the existence of a continuous preference function
over probability distributions. It is the independence axiom which gives the
theory its empirical content and power in determining rational behavior. That
is, the preference function is constrained to be a linear function over the set
of probability distribution functions, i.e. linear in probabilities (Machina,
1982: 278).
If
an individual obeys the expected utility axioms, then a utility function can be
formulated that reflects the individual preferences (Mas-Colell et al., 1995: 175; Robison and et al., 1984: 13). Further individual’s
risk attitude can be inferred from the shape of his/her utility function. Since
vNM (1947), the expected utility model has been the dominant model in
predicting choice behavior under risk. Starting with the well-known paradox of
Allais (1953), however, a large body of experimental evidence has been
documented which indicates that individuals tend to violate the axioms
underlying the expected utility model systematically. This empirical evidence
has motivated researchers to develop alternative theories of choice under risk
able to accommodate the observed patterns of behavior. A wave of theories
designed to explain the violation of expected utility theory began to emerge at
the end of the 1970. Examples are prospect theory (Kahneman and Tversky, 1979),
regret theory (Loomes and Sugden, 1982), dual theory (Yaari, 1987), cumulative
prospect theory (Tversky and Kahneman, 1992), and rank-dependent utility
(Quiggin, 1993). For a thorough review see Starmer (2000). In the empirical
literature prospect theory is the dominant theory. Therefore, it will be
discussed in section 2.2.
2.1.1 Violation of the independence axiom
The
common consequence effect. The well-known risky choice
provided by Allais is given in a paper by Kahneman and Tversky (1979). They
synthesize the work by Allais and by others who have shown experimental
violations of expected utility. The Allais paradox depicted in Table1 is the
leading example of this class of anomalies. There are two different choice
sets, for each choice set there are two lotteries from which you can choose.
For example, in lottery A1 there is a guaranteed payoff of $1M and there is
zero probability of winning nothing. In lottery A2 there is a 0.10 probability
of winning $5M, a 0.89 probability of winning $1M, and a 0.01 probability of
winning nothing. Then one has to choose between A1 and A2, and between A3 and
A4. Where 
are lotteries.
Table 1 The Allais paradox: the common consequence effect
|
Choice 1 |
A1 |
{1 M, 1; 0
M, 0} |
A2 |
{5 M, 0.1;
1 M, 0.89; 0 M, 0.01} |
|
Choice 2 |
A3 |
{5 M, 0.1;
0 M, 0.9} |
A4 |
{1 M, 0.11;
0 M, 0.89} |
Note outcomes are in Dollars and 1M = $1,000,000.
Many agents
prefer lottery A1 to A2 and prefer lottery A3 to A4. This empirical tendency
directly contradicts expected utility theory. According to expected utility theory 
if and only if 
. Subtracting 
from each side, it follows that
. Adding 
to both sides, we have 
which holds if and only if
. Thus, from expected utility theory, one can deduce that