EXPERIMENTAL ECONOMICS An Elementary Introduction

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EXPERIMENTAL ECONOMICSAn Elementary Introduction

• By
• Christian Seidl,
• University of Kiel, Germany

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• Contents
• Expected Utility
• 1.1 Theory
• 1.2 Experimental Design
• 1.3 Pitfalls
• 1.3.1 Allais Paradox
• 1.3.2 Ellsbergs Paradox
• 1.3.2 Preference Reversal
• 1.3.4 Response-Mode Bias
• 2. Main Fields of Experimental Economics
• 2.1 General
• 2.2 Own Research
• 2.2.1 Equitable Income Taxation
• 2.2.2 Acceptance of Distributional Axioms
• 2.2.3 Background Context Effects
• 2.2.4 Own Research Other than Mentioned
  Above
• 3. How Can We Evaluate Income Distributions?
• 4. Aim of the Experiment

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• EXPECTED UTILITY
• 1.1 Theory
• In their famous book on Game Theory and Economic
  Behavior, von Neuman and Morgenstern devised a
  method to represent preferences among possible
  actions, ai, i1,,n, by the expected value of
  the utilities of the results, pij, i1,,n,
  j1,,k, of a decision problem under risk, i.e.,
  with given probabilitiers, pj, j1,,k, of the
  possible states of the world which, together with
  the chosen action, engender the results of the
  decision problem.
• More precisely, consider a decision matrix

p1 p2 pk
a1 p11 p12 p1k
a2 p21 p22 p2k

an pn1 pn2 pnk
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Provided that some common-sense axioms, which
every rational decision-maker would approve,
hold, von Neumann and Morgenstern showed the
existence of a real-valued utility function u()
defined on the space of results such that
(1)
They also showed that the utility function u()
is cardinal, i.e., unique up to a positive linear
transformation. Whereas von Neumann and
Morgenstern assumed the probabilities of the
states of the world as given, later on, Savage
devised a more general theory based on a
qualitative probability relation and derived
subjective probabilities jointly with the
respective utility function. In this
introduction, I will not go into details.
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To exemplify expected utility in a simple
framework, let us restrict the decision problem
to three outcomes such that
Obviously, expected utility is
Substituting p21-p1-p3 gives us
(2)
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This formula allows us to arrange this simple
lottery in terms of a Marschak-Jensen-Machina
triangle
p1
1.0
(2)
p2
p3
p1
p3
1.0
As the utilities of the three results are given,
(2) represents an indifference curve for u as a
linear function of p1 in terms of p3 and u (red
line). Let us replace u by û, ûgtu, then the
indifference curve moves in a parallel way to the
North-West of this triangle (green line). Hence,
the family of indifference curves are parallel
lines with the North-West preference direction.
Note that this applies to expected utility
theory.
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1.2 Experimental Design Experimental elicitation
of utility usually works by way of a series of
binary lotteries. Suppose we have four results
e.g., pay-offs
and two probabilities p and q. then we consider
two binary lotteries
(3)
with expected utility values
(4)
(5)
Notice that for ?2?3 the second lottery
reduces to a single result, ?2.
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Parameter elicitation follows the pattern that
one component of the lotteries in (3) is taken as
variable, and the values of the other components
are fixed. The subject is asked to choose the
value of the variable component such that the
subject is indifferent between the two lotteries.
Then we insert the chosen value of the variable
in the expression (4) or (5), equate both
expressions, and derive points of the von
Neumann-Morgenstern utility function by simple
rearrangements of (4) and (5).
(4)
(3)
(5)

• Consider first ?2?3 this means that we look
  for component values which render the first
  lottery in (3) indifferent to p2.
• This constitutes the group of standard-gamble
  methods
• Certainty equivalence ?2 is the variable
  component.
• Value equivalence ?1 or ?4 is the variable
  component.
• Probability equivalence p is the variable
  component.

u(p2)
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Consider second then
again we look for component values which render
the lotteries in (3) indifferent. This
constitutes the group of paired-gamble
methods (iv) Value equivalence exactly one
component out of ??1,?2,?3,?4? is the variable
component. (v) Probability equivalence one out
of the two probabilities, p or q, is the variable
component lottery equivalence if ?3?40 for
real-valued results.

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We illustrate utility elicitation using the
standard-gamble method which was used originally.
Notice that, as the values ?j are given, this
method provides both the values of the domain ?j
and the range u(?j) of the utility function.
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In the Mosteller and Nogee experiment, ?k denoted
the loss of 5 Cents, i.e., u(?k)-1, and ?1 meant
refusal to participate in the lottery, i.e.,
u(?1)0. Keeping the loss of 5 Cents constant,
Mosteller and Nogee repeated this experiment for
seven different values of pj, which provided them
seven payoffs ?j and seven utility values u(?j)
on the von Neumann-Morgenstern utility schedule
in addition to u(?1)0 and u(?k)-1.
Proceeding this way further by sequentially using
a subjects certainty equivalents as prizes for
lotteries which share the probability p for the
better reward of the respective lottery, one can
derive arbitrarily many points on the subjects
von Neumann-Morgenstern utility function.
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In addition to that, Becker, deGroot, and
Marschak (1964) devised an ingenious procedure to
elicit subjects true certainty equivalents
although it is similar, it was developed
independently of Vickreys (1961) famous
sealed-bid second price option which induces
subjects to truthfully reveal their valuation for
an object at auction Subjects are asked for
their selling price of a binary lottery. Then a
buying price is randomly chosen by the
experimenter. If it exceeds the stated selling
price, then the subject gets the buying price if
not, then the respective lottery is played out.
Under the precepts of expected utility theory, it
never pays for the subject to cheat with respect
to the revelation of his or her true certainty
equivalent if the stated certainty equivalent
exceeds the true one, then the subject runs the
risk of playing out the lottery, whereas the
subject would have preferred the buying price
(provided that it is between the stated and the
true certainty equivalent). If the certainty
equivalent understates the true one, then the
subject runs the risk of getting the buying price
(provided that it is between the true and the
stated certainty equivalent) , whereas the
subject would have preferred playing out the
lottery.
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1.3 Pitfalls Recall that expected utility theory
rests on plausible common-sense axioms which
engender the shape of a von Neumann-Morgenstern
utility function by mathematical reasoning. Yet
experimental research has shown that subjects
actual behavior deviates from expected utility
theory. Subsequently this led to the proposal of
other models to capture subjects behavior, such
as prospect theory, state-dependent utility
theory, causal utility theory, regret theory,
range-frequency theory, etc. Note, however, that
it were the pitfalls of expected utility theory
as pointed out by experimental results, which
triggered the development of later theories of
decisions under risk and uncertainty. In the
following, pitfalls of expected utility theory
are illustrated by four famous examples. 1.3.1
Allais Paradox Consider four lotteries S, S,
S, and S with at most 3 results in a Marschak
triangle such that the slopes of the lines
connecting S and S on the one hand and S and
S on the other are parallel. This means
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which gives us the Marschak triangle
p1
1.0
S
S
S
S
p3
1.0
Suppose a subjects preferences are characterized
by the family of red indifference curves. Then
this subject must prefer S to S whenever it
prefers S to S, and vice versa for the green
indifference curves.
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Lets now come to the gist of Allais Paradox.
Nobel laureate Maurice Allais presented the
following choice problem to subjects (among them
Leonard Savage) and asked them for their
preferences for the following pairs of lotteries
all payoffs in (old, i.e., 1952) French
Francs S Certainty of receiving 100
million Francs 10 chance of winning 500
million Francs S 89 chance of winning 100
million Francs 1 of winning nothing
S 11 chance of winning 100 million
Francs 89 chance of winning
nothing S 10 chance of winning 500 million
Francs 90 chance of winning nothing
Allais observed that many subjects preferred S
to S but also S to S, which contradicts
expected utility because we have
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Illustrating this in terms of a Marschak triangle
shows
p1
1.0
S
S
p3
S
1.0
S
Preferring S to S and S to S violates the
condition that the family of utility indifference
curves have to be parallel lines under expected
utility. Hence, this example shows violation of
expected utility.
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1.3.2 Ellsbergs Paradox Ellsbergs Paradox
concerns ambiguity, i.e. it focuses on the
probability structure. Ellsberg (1961) posed the
following problem to subjects Consider an urn
containing 100 balls. You know there are 33 red
balls but you do not know the composition of the
remaining 67 black and yellow balls. One ball is
to be drawn at random from an urn. Which do you
prefer S or S? S Receive 1000 if a red
ball is drawn S Receive 1000 if a black ball
is drawn Which do you prefer S or S? S
Receive 1000 if a red or a yellow ball is
drawn S Receive 1000 if a black or a yellow
ball is drawn As drawing a yellow ball in the
second pair of lotteries implies that the subject
gets 1000 anyway, the preferences should be the
same across both pairs of lotteries, i.e.
However, many subjects preferred S to S and S
to S.
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Let us express Ellsbergs Paradox in terms of a
Marschak triangle. As we have only two payoffs,
?11000 and ?30, p20, all lotteries lie on the
diagonal boundary of the Marschak triangle
S 1000 if red ball S 1000 if black ball
p1
1.0
S 1000 if red or yellow ball S 1000 if
black or yellow ball
S if more black than red balls
S
S if less black than red balls
S if more black than red balls
0.33
S
S if less black than red balls
p3
1.0
McCrimmon and Larsen (1979) found for a
probability of 0.33 for red balls a maximum of
70 of Ellsberg-type violations of expected
utility, but falling to 20 for a probability of
0.2 for red balls and to 0 for a probability of
0.5 for red balls.
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1.3.3 Preference Reversal
That is, eliciting subjects preferences for
lotteries contradicts their preferences when
eliciting them in terms of certainty equivalents
and vice versa. The preference reversal
phenomenon was discovered by Lindman (1965 1971)
and Slovic and Lichtenstein (1968 1971). Their
experiments showed that this phenomenon can
mainly be observed for lottery comparisons
between a P-bet, i.e., a binary lottery which
accords a high probability of winning a modest
amount and a low probability of losing (or
winning) an even more modest amount, and a -bet,
i.e., a binary lottery which accords a low
probability of winning a high amount and a large
probability of losing (or winning) a modest
amount. These authors observed that many subjects
preferred the P-bet to the -bet, but indicated a
higher certainty equivalent for the -bet than
for the P-bet.
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The next table shows the results from three
Lichtenstein and Slovic experiments
Meanwhile, there are hundreds of papers which
investigated the preference-reversal phenomenon
for a survey see Seidl (2002). The
preference-reversal phenomenon was extended to
more general lotteries and compared with equally
structured income distributions by Camacho,
Seidl, and Morone (2005) . They modeled
lotteries and income distributions by appropriate
distributions of 100 tally marks, where P-bets
were represented by negatively skewed and -bets
by positively skewed distributions. In addition
to that, bimodal, symmetric, and uniform
distributions were tested. Camacho, Seidl, and
Morone found four patterns of preference
reversals, where two of them were only observed
for income distributions. They also observed more
preference reversals for income distributions
than for lotteries. For income distributions the
equally distributed equivalent income replaced
the certainty equivalent. The transfer principle
was violated in more than 50 of the cases.
Negatively skewed distributions engendered
greater happiness.
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• 1.3.4 Response-Mode Bias
• Seidl and Traub (1999) compared von
  Neumann-Morgenstern utility functions derived
  according to the certainty-equivalence method
  with utility functions derived according to the
  lottery-equivalence method. The observed
• Dependence of the utility functions on the
  probability used for their generation.
• Even when the same generating probability was
  used, the two types of utility functions are
  different.
• The correlation of risk attitudes are higher for
  the same response mode than for different
  response modes.
• For risk averse risk loving subjects, a
  utility function derived with a higher generating
  probability dominates is dominated by a utility
  function derived with a lower generating
  probability.

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2. MAIN FIELDS OF EXPERIMENTAL ECONOMICS 2.1
General Experimental economics has been applied
to behavioral checks of economic theory in many
fields. The first comprehensive books on
experimental economics were published in the
90s John D. Hey, Experiments in Economics,
Blackwell, Oxford and Cambridge MA 1991. Vernon
L. Smith, Papers in Experimental Economics,
Cambridge University Press, Cambridge
1991. Douglas D. Davis and Charles A. Holt,
Experimental Economics, Princeton University
Press, Princeton NJ 1993. John H. Kagel and Alvin
E. Roth (eds.), Handbook of Experimental
Economics, Princeton University Press, Princeton
NJ 1995.
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• The main fields to which experiments were
  applied, are
• decision making
• bargaining
• game theory
• market organization
• public goods
• coordination problems
• industrial organization
• auctions
• fairness
• asymmetric information
• voting and social choice

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In this short introduction I will single out the
voluntary provision of public goods and present
the most basic experiment in this field Each of
four subjects in an experiment is given 5.00 to
be used for two purposes. They may put part of it
in an envelope to be used for a group project
i.e., a public good or kept to be used as a
private pay-off. The experimenter will collect
the contributions for the group project, total
them up, double the amount, and then divide this
money equally among the subjects of the group.
Note that the private benefits resulting from
this operation is only half of the total
contribution. What does theory tell us about
this situation? Game theory tells us that no one
will ever contribute anything. Each subject will
free ride instead. The dominant strategy is to
contribute nothing because each 1.00
contributed yields only 0.50 to its
contributor, no matter what others do. In
contrast to that, the group as a whole would be
best off if all participants contributed 5.00
because this would give an additional 5.00 to
each participant. However, the best result for an
individual subject would be if he or she
contributes nothing (i.e., keep all 5.00) and
all the others contribute 5.00 each. This
results in 12.50 for the first subject and
7.50 for all others. However, if all subjects
follow this reasoning, nobody will contribute and
all end up with 5.00 each. If just one subject
contributes all 5.00, this subject would end up
with 2.50 and the three other subjects would
end up with 7.50 each. So it is better for him
or her to contribute nothing. This holds, vice
versa, for all subjects.
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The set of Pareto-optimal states is
(10,10,10,10), (12.5,7.5,7.5,7.5),
(7.5,12.5,7.5,7.5), (7.5,7.5,12.5,7.5),
(7.5,7.5,7.5,12.5), the equilibrium resulting
from uncoordinated individual actions is
(5,5,5,5). This theoretical reasoning seems to
invalidate the model of voluntary contributions
for the provision of public goods which was
proposed by Mazzola, de Viti de Marco, Sax,
Lindahl, Samuelson and others, as Wicksell
(1896) had surmised. Hence, is a system of
mandatory taxation the only solution for
providing public goods? Experimental results
show that neither theory is right. Some subjects
contribute nothing, some contribute their whole
budget. Generally, total contributions can be
expected to lie between 8.00 and 12.00,
or 40 and 60 of the group optimum. Moreover, as
this experiment is repeated with the same
subjects, it was observed that contributions
decrease with the number of rounds, which
reflects disappointment of the original
contributors with the free riders. Gradually free
riding gains ground. Why should we care about
public goods experiments? Now, the desired
outcome is that anybody contributes 5.00.
Experimental evidence showed that voluntary
contributions will not produce the desired
outcome. Hence, we have to look for institutions
which causes the outcome to be closer to the
group optimum. To achieve that we have to
anticipate how individual choice will change as
the institutions change. Experiments are the only
way to perform this task. Recently, an
anonymous private person in Germany donated 5
million to buy a ship for an organization of
rescuing ship-wrecked people.
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2.2 Own Research Let me now give you a short
account of my own research in experimental
economics. 2.2.1 Equitable Income
Taxation Stefan Traub and I investigated the
perception of equitable income taxation by means
of field data. We polled some 200 employees in
firms twice. First, they got information about
the current tax burden for singles in Germany.
Then we asked them for the equitable taxation for
four household types (single, couple without
children, couple with one child, couple with two
children) and five income levels. Subjects
started with the tax burden of singles and were
asked for the equitable taxation of other
household types. We collected the questionnaires
without leaving them a copy. Some ten days later,
we told them that a tax reform was envisaged and
indicated the taxes for the households with two
children. These were precisely the same taxes as
proposed by the respective subjects. Then
subjects were asked for the equitable taxes of
the other household types. Interestingly enough,
subjects indicated lower taxes for the other
household types than in the first round. They
also disapproved of the German tax-splitting
boon. This applied to married couples (the
beneficiaries of the splitting boon) as well.
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2.2.2 Acceptance of Distributional
Axioms Elizabeth Harrison and I polled hundreds
of students by questionnaire whether they obey
the main axioms of income inequality measurement.
We observed violations amounting to 20 to 40 of
the responses. In particular, the transfer axiom
was very often violated, but also the population
principle. 2.2.3 Background Context
Effects Subjects perceive phenomena in relation
to their background. Psychologists observed that
the darkness of grey squares is upgraded if they
are presented in the context of many lighter grey
squares, and downgraded if they are presented in
the context of many darker grey squares. In joint
work with Stefan Traub and Andrea Morone, we
embedded certain income levels in a context of
several distributions, viz. bimodal, normal,
negatively skewed, and positively skewed. We
observed background context effects an income
level received better grades when it was
presented in the frame of a positively skewed
distribution than when it was presented in the
frame of a negatively skewed distribution. In
spite of that, mean income was lowest for the
positively skewed distribution and highest for
the negatively skewed distribution.
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Moreover, we struck another paradox whereas
individual income satisfaction with a certain
income level is highest for a background of
positively skewed distributions, a Harsanyi-type
social welfare function demonstrates higher
average social welfare for a background of
negatively skewed distributions. This paradox
results from the weighting of income satisfaction
with the stratified frequency of the involved
subjects although an individual has a bit less
satisfaction with his or her income under a
background of a negatively skewed distribution,
there are more individuals with higher incomes in
a negatively skewed distribution. This frequency
effect overcompensates the higher income
satisfaction under the background of a positively
skewed income distribution. Compare Derek
Parfits (1984) repugnant conclusion! 2.2.4 Own
Research Other than Mentioned Above Testing
Decision Rules for Multiattribute Decision Making
Knock-Out for Descriptive Utility or
Experimental Design Error? A New Test of Image
Theory Stochastic Independence of
Distributional Attitudes and Social Status. A
Comparison of German and Polish Data
Friedman, Harsanyi, Rawls, Boulding Or Somebody
Else? An Experimental Investigation of
Distributive Justice The Performance of Peer
Review and a Beauty Contest of Referee Processes
of Economics Journals Lorenz Meets Rating but
Misses Valuation An Experimental Study on
Individual Choice, Social Welfare and Social
Preferences
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3. HOW CAN WE EVALUATE INCOME DISTRIBUTIONS?
Problem 1 Consider two income distributions
A 10, 20, 30, 50, 70,
80, 90 B 50, 50,
50, 50, 50, 50, 50 Question Which of the two
income distributions is more equally distributed?
Obviously B is more equally distributed than A,
although some sub-jects might prefer to live in a
society with income distribution A.
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Problem 2 Consider the income distributions

C 100, 200, 300, 500, 700, 800, 900
B 50, 50,
50, 50, 50, 50, 50
Obviously B
Questions (i) Which of the two income
distributions is more equally distributed? (ii)
Which of the two income distributions generates
more income for the economy? (iii) Which of the
two income distributions generates more welfare
for the economy? (iv) Would you rather live
in a society with income distributions B or C if
your own income position will be later on
determined by chance?
Obviously C
The answer depends on the subjects social
welfare function. For welfarist social welfare
functions, C generates more welfare. If equality
preferences enter the social welfare function, B
might also emerge as generating higher welfare.
The answer depends on your distributional
preferences and your risk attitude.
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Problem 3 Consider three income distributions
A 10, 20,
30, 50, 70, 80, 90
D 10, 20, 30, 50, 70, 70, 90
E 10, 10,
30, 50, 70, 80, 90
Questions (i) Is D more equally or more
unequally distributed than A? (ii) Is E more
equally or more unequally distributed than A?
In D, as compared with A, the second richest
person loses 10 monetary units. In E, as compared
with A, the second poorest person loses 10
monetary units. There is no right or wrong
answer it is up to the view of the beholder
whether D or E is considered more equally or more
unequally distributed than A.
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Problem 4 Consider three income distributions
A 10, 20, 30,
50, 70, 80, 90
F 10, 30, 30, 50, 70, 70, 90
G 20, 20, 30, 50, 70, 80,
80
F comes about from A if 10 monetary units are
transferred from the second richest to the second
poorest income recipient.
Questions (i) Is F more equally or more
unequally distributed than A? (ii) Is G more
equally or more unequally distributed than
A? (iii) Is D more equally or more unequally
distributed than F?
G comes about from A if 10 monetary units are
transferred from the richest to the poorest
income recipient.
G can come about from F in two ways (i) the
second poorest transfers 10 monetary units to the
poorest income recipient, and the richest
transfers 10 monetary units to the second richest
income recipient thus, we have two progressive
transfers. (ii) The richest transfers 10 monetary
units to the poorest income recipient, and the
second poorest transfers 10 monetary units to the
second richest income recipient thus, we have a
progressive and a regressive transfer.
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Problem 5 Consider three income distributions
A 10,
20, 30, 50, 70, 80, 90
H 5, 20, 30, 50, 70, 70, 90

J 15, 20, 30, 50, 70, 80, 80
H comes about from A by an income loss of 5
monetary units of the poorest income recipient
and by an income loss of 10 monetary units of the
second richest income recipient. This depicts
income changes which point in the same direction.
J comes about from A by a gain of 5 monetary
units of the poorest income recipient and a loss
of 10 monetary units of the richest income
recipient.
Questions (i) Is H more equally or more
unequally distributed than A? (ii) Is J more
equally or more unequally distributed than
A? (iii) Is J more equally or more unequally
distributed than H?
Again there is no right or wrong answer to these
questions.
J can come about from H in two ways (i) the
poorest income recipient receives 10 monetary
units and the richest income recipient transfers
10 monetary units to the second richest income
recipient. (ii) The richest income recipient
transfers 10 monetary units to the poorest income
recipient and the second richest income recipient
receives 10 monetary units.
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4. AIM OF THE EXPERIMENT You will be shown a
model income distribution in terms of EUROs. The
computer will add or subtract EURO 100 to or from
one persons income in this income distribution.
Then the computer will determine some other
person in this income distribution and will ask
you to change this persons income such that the
former degree of income inequality in this income
distribution is restored, according to your
perception. Please, note that income changes are
caused by external events, such as the correction
of errors of monetary transactions which happened
in the past, tax refunds or tax payments from
past incomes, bonus premiums for last year or
re-payment of erroneous too high salary in the
last year, etc. Moreover, your suggestion of
changing another persons income is a purely
hypothetical correction which would, according to
your perception, restore the former degree of
income inequality in this society. It does not
mean actually giving income to or taking off
income from this person. Just imagine which
external income change which occurred to this
specified person would restore the former degree
of income inequality in this society.