Applied Microeconomics

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Applied Microeconomics

• Decision-Making Under Uncertainty

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Outline

• Uncertainty
• Expected utility theorem
• Risk aversion
• Arrow Pratt measure of risk aversion
• CARA/DARA utility

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Readings

• Kreps chapters 15 and 16
• Perloff chapter 17

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St. Petersburg Paradox

• How much would you pay to participate in the
  following gamble? (Nicholas Bernoulli, 1713)
• A fair coin will be tossed until a head appears
• If the first head appears on the nth toss, then
  the payoff is 2n ducats

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St. Petersburg Paradox

• The expected value of the gamble p is Exi___
• Daniel Bernoullis solution (1738)
• people's utility from wealth, u(W), is not
  linearly related to wealth (W) but rather
  increases at a decreasing rate - the famous idea
  of diminishing marginal utility, u(W) gt 0 and
  u(W) lt 0
• a person's valuation of a risky venture is not
  the expected return of that venture, but rather
  the expected utility from that venture
• Ep(u(xi))___

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Uncertainty

• Let X(x1,x2,,xn) be a set of outcomes
• Let p(p1,,pn) be a simple lottery on the set of
  outcomes
• pi is the probability of outcome xiÎX occurring
• pi³0 for all xi in X and å i1npi1
• Define D(X) as the set of lotteries on X

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Example

• Suppose a coin is tossed and we win 100 Euros if
  heads comes up and nothing if tales comes up
• The set of outcomes in this case is X(100,0)
• A fair coin represents the simple lottery
  p(0.5,0.5) over X
• A manipulated coin could for instance give us the
  simple lottery q(0.3,0.7) over X

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Compound Lotteries

• Suppose we have two simple lotteries p and q
• Then we can define a compound lottery over the
  two lotteries with probability a you win lottery
  p and with probability 1-a you win q
• But this is the same as the simple lottery
  r(ap1(1-a)q1,,apn(1-a)qn)
• Example if with probability 0.5 we win the fair
  coin lottery and with probability 0.5 the
  manipulated coin lottery, this is equivalent to
  the simple lottery r(0.4,0,6)

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Preferences over Lotteries

• Suppose now that decision makers (DMs) have
  preferences over the simple lotteries on X
• We will write p³hq to denote that the DM weakly
  prefers p to q
• We will write pgthq to denote that the DM strictly
  prefers p to q

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Axioms

• A.1 ³ h is complete, i.e. either p ³ h q or q ³
  h q for all p,qÎD(X).
• A.2 ³ h is transitive, i.e. if p ³ h q and q ³ h
  r, then p ³ h r for all p, q, r Î D(X)
• A.3 ³ h is continuous if p,q,rÎD(X),
  amÎ0,1, limm?8 ama , and amp(1-am)q³ hr for
  all m, then ap(1-a)q³ hr
• A.4 Independence Axiom if p,q,rÎD(X) and a
  Î0,1, then p ³ hq if and only if ap(1-a )r³
  haq(1-a )r

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The Expected Utility Theorem(Neumann
Morgenstern 44)

• ³ h satisfies (A.1), (A.2), (A.3) and (A.4) if
  and only if there is a Bernoulli function u(xi)
  such that for all p,qÎD(X), p ³ hq Û
  U(p)Epu(xi) ³ Eq u(xi)U(q)
• If u(xi) is such a function, then any other
  function v(xi) representing the preferences must
  be of the form v(xi)bu(xi)c, where bgt0
• A DM whose preferences satisfy the four axioms is
  said to have a von Neumann-Morgenstern (vNM)
  utility function

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Example

• A DM with vNM utility has the Bernoulli function
  u(x)ln(x)
• Which of the following two lotteries does he
  prefer
• p(0.5,100,0.5,1)
• q(1,50)
• Answer compute the expected utilities
• U(p)0.5ln(100)0.5ln(1)2.3
• U(q)ln(50)3.9

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Remarks

• Lotteries are over final outcomes
• Suppose a DM with vNM utility function and wealth
  1 million Euros is choosing between a lottery
  that gives him 1 extra Euro with probability 0.5
  and 0 with prob. 0.5 and a lottery that pays off
  90 cent with certainty
• To decide he needs to compare the lotteries
  p(1,000,0010.5,1,000,0000.5) and
  q(1,000,000.901)

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Remarks

• If U(p) represents the preferences of the
  decision maker, than so does V(p)aU(p)b for agt0
• Outcomes do not have to be monetary payoffs, but
  can for instance be bundles of goods

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Certainty Equivalent

• Suppose a DM is indifferent between 40 Euros for
  sure, and a lottery p that gives 100 Euros with
  prob. 0.5 and 1 Euro with prob. 0.5,
• Then 40 Euros is his certainty equivalent, CE(p),
  of the lottery p
• Mathematically, this can be expressed as
  u(CE(p))Epu(xi)U(p)

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Example

• Suppose a vNM decision maker with zero wealth has
  the Bernoulli function u(x)-e-0.01x
• What is his certainty equivalent of the lottery
  p(0.5,100,0.5,1)?
• Answer solve the equation 0.5(-e-1)0.5(-e-0.01)
  -e-0.01CE(p)
• Solution CE(p)-100ln(0.5e-10.5 e-0.01)38.7

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Risk Aversion

• Note that CE(p)38.7ltEp(xi)50.5
• Hence, the DMs money value of the lottery is
  less than its expected value
• A decision maker who has a lower certainty
  equivalent than expected value of any risky
  lottery, is said to be risk avert
• The difference between the two is known as the
  risk premium RP(p) Ep(xi)-CE(p)
• In the above example RP(p)50.5-38.711.8

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Risk Neutrality

• On the other hand, a decision maker whose
  certainty equivalent and expected value are equal
  for any lottery on X is said to be risk neutral
• A decision maker who has a weakly higher
  certainty equivalent than expected value for any
  lottery, is said to be risk loving
• What are your risk preferences?

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Example

• Suppose a decision maker has the Bernoulli
  function u(x)x. Is he risk averse/loving/neutral?
  
• Answer since u(Exi)ExiEpu(xi), he is
  risk neutral
• Moreover, any risk neutral decision maker must
  have a Bernoulli function of the form u(x)axb,
  where agt0
• What about more general Bernoulli functions?

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Risk Aversion and Concavity

• A decision maker is risk-averse ? he has concave
  utility function
• With a differentiable Bernoulli function this is
  equivalent to u(x)0 for all x in X

CEp
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Risk Aversion and Concavity

• A decision maker is risk loving ? he has convex
  utility function
• A decision maker is risk neutral ? he has linear
  utility function

CEp
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Comparing Risk Aversion

• How do we compare the risk aversion of different
  decision makers?
• Arrow-Pratt measure of absolute risk aversion
  RA(W)-u(W)/u(W)
• Examples
• u(W)ln(W) gives RA(W)1/W Absolute risk
  aversion decreasing in wealth, DARA utility
• u(W)a-be-cW gives RA(W)c Absolute risk
  aversion constant at all wealth level, CARA
  utility

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Conclusions

• If the DMs preferences over lotteries satisfies
  four axioms, then they can be represented by an
  expected utility function
• A DMs certainty equivalent of a lottery is the
  amount of money that makes him indifferent
  between participating in the lottery and taking
  the money
• The risk premium is the difference between the
  expected value and the certainty equivalent
• A risk averse vNM DM has concave Bernoulli
  function and lower certainty equivalent than
  expected value for all lotteries over the set of
  outcomes
• The Arrow-Pratt measure of absolute risk aversion
  makes it possible to compare the risk aversion of
  different DMs