Lecture 5 Consumer Choice under uncertainty

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Lecture 5Consumer Choice under uncertainty

• Simple lottery
• A simple lottery L is a list L(p1pn) with pn?0
  for all n and , where pn is the probability of
  the outcome n occurring
• We can define more complex lotteries (lotteries
  over lotteries)
• Compound lottery
• Given k simple lottery , k1K and some
  probability that that a lottery Lk occurs, then
  we can define a compound lottery , which is the
  risky alternative that give Lk with probability

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Lotteries

• Example
• Prize100
• Compound lottery A
• L1(100,0,0) occurs with probability 1/3
• L2(1001/4, 100 3/8, 1003/8) with probability
  1/3
• L3((1001/4, 1003/8, 1003/8) with probability
  1/3
• Compound lottery B
• L1(1001/2, 1001/2, 0) with probability ½
• L2(1001/2, 0, 1001/2) with probability ½
• Do you prefer A or B?

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Reduced Lotteries

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Reduced lotteries

• Given all the possible outcomes, we can apply the
  same logic to compute the probability of each
  outcome and hence find the Reduced Lottery

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Reduced lotteries
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Preference over compound lotteries

• Since the consumer only cares about the
  distribution of final outcomes, he will be
  indifferent between to Compound Lotteries that
  deliver the same Reduced Lottery
• Do you agree?

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Preferences over lotteries

• Remember when we defined preferences over a set
  of goods.
• Again now we have to define preferences but over
  a space of lotteries - AXIOMS
• We require
• Continuity - given two lotteries, small changes
  in probabilities do not change the ordering
  between two lotteries
• Indipendence if we mix each of the two lotteries
  with a third one, the preference ordering of the
  two resulting mixtures does not depend on the
  particular third lottery that is used.
• Note the difference with the consumer choice
  under certainty here when we consider three
  alternative lotteries L1, L2 and L3, the consumer
  cannot consumer L1 and L2 together or L1 and L3
  together etc. but he has to choose between
  mutually exclusive alternatives!

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Expected Utility
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Expected Utility

• Expected Utility theorem
• If preferences over lotteries satisfy the
  continuity and the independence axioms, then
  preferences can be represented by a utility
  function with expected utility form
• Indifference curves
• Since the Von-Neuman-Morgestern utility is linear
  in probabilities, if we represent the
  indifference curves on the lottery space, we will
  obtain parallel straight lines
• If not straight lines indipendence axiom not
  satisfied
• If not parallel indipendence axiom not satisfied
  

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Consumer choice under uncertainty

• If preferences admit expected utility form
  representation, then the consumer choice under
  uncertainty reduces to the maximisation of his
  expected utility, given his budget constraint

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Money Lottery and risk adversion

• Solving problems under uncertainty we can see
  that individuals show some degree of risk
  aversion
• Where does risk aversion come from?
• In other words, which property of the utility
  function in the expected utility framework
  implies risk aversion?
• Suppose that we define utility over monetary
  lotteries. Let x be certain amount of money an
  individual receives
• And u(x) the utility associated x. We can show
  that risk aversion is implies by CONCAVITY of
  u(x).UNIQUENESS
• Is the utility function unique?
• NO - - ANY MONOTONIC TRANSFORMATION REPRESENTS
  THE SAME PREFERENCES
• UTILITY IS AN ORDINAL CONCEPT NOT A CARDINAL
  CONCEPT!

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Utility function
Risk aversion

• Concave utility function

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Utility function
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Risk Aversion

• Risk aversion can also be seen using two other
  concepts
• Certainty equivalent the sure amount of money
  that you are willing to accept instead of the
  lottery
• Probability premium the excess in probability
  over fair odds that makes an individual
  indifferent between a certain outcome and a
  gamble
• If an individual is risk averse you expect that
• The sure amount of money he is willing to accept
  instead of the gamble should be less then the
  expected value of the money he would get from the
  gamble (he is willing to take less and avoid the
  risk)
• An individual accepts the risk only if he is
  offered better than fair odds

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Measures of Risk Aversion
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Measures of Risk Aversion
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Risky choices
Risky Choices

• Examples of risky choices
• Insurance
• Investment in risky asset
• Comparisons of different payoffs
• Suppose that you are faced with the choice of
  comparing different risky investments
• Which type of statistics about the lottery would
  you use to choose among investments?

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Risky Choices

• level of return (average)
• dispersion of returns
• If you know the distribution function, you can
  use this information to compare lotteries

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Stochastic dominance

• First order stochastic dominance F(.)
  first-order stochastically dominates G(.) if

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Stochastic dominance

• Now, suppose that you can change F(.) in a way
  that preserves the mean but changes the variance.
  Suppose that G(.) is the function that you obtain
  from this transformation, that is G(.) is a
  mean-preserving spread of F(.)
• Ex F(.) such that with probability p1/2 you get
  2 and with prob p1/2 you get 3, hence the
  average payoff is 5/2
• Let be G(.) such that with probability p1/4 you
  can get (1,2,3,4) hence the average payoff is
  again 5/2
• Would you prefer F(.) or G(.)?

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Stochastic dominance

• If you are risk averse, you will prefer F(.) to
  G(.).
• Indeed, we can prove that F(.) second-order
  stochastically dominates G(.)
• Definition
• Second order stochastic dominance given F(.) and
  G(.) with the same mean, F(.) second-order
  stochastically dominates (or is less risky than)
  G(.) if

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Stochastic dominance

• Result if G(.) is a mean preserving spread of
  F(.), then F(.) second-order stochastically
  dominates G(.)
• Proof
• Let x be a lottery distributed according to F(.).
  Suppose that we further randomize x so that the
  final payoff is xz where z is distributed
  according the the function H(z) with zero mean.
  Therefore, xz has the same mean as x but
  different variance. We define G(.) final reduced
  lottery, i.e. the function that is assigning a
  probability to each x using the transformation of
  F(.) we have just described. Hence G(.) is a mean
  preserving spread of F(.).

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Stochastic dominance
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Expected Utility theory a calibration exercise

• Consider the following gamble
• You can loose 10 with probability 50 and gain
  11 with probability 50
• Do you accept this gamble?
• Consider now these alternative gambles
• Loose 100 with 50 prob. and win 110 with 50
  prob.
• Loose 100 with probability 50 and win 221 with
  50 prob.
• Loose 100 with probability 50 and win 2000
  with 50 prob.
• Loose 100 with probability 50 and win 20,000
  with 50 prob.
• Loose 100 with probability 50 and win 1
  million with 50 prob.
• Loose 100 with probability 50 and win 2
  millions with 50 prob.
• Which bet will you be willing to accept?

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Source M. Rabin and R.H.Thaler (2001),
Anomalies- Risk Aversion, Journal of Economic
Perspectives, pages 219-232
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Expected Utility theory a calibration exercise

• Problem the expected utility theory delivers
  implausible excessive degree of risk aversion!