Transcript π 2 u`(c 2 ) - User Web Areas at the

Microeconomics 2
John Hey
Chapters 23, 24 and 25
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CHOICE UNDER RISK
Chapter 23: The Budget Constraint.
Chapter 24: The Expected Utility Model.
Chapter 25: Exchange in Markets for Risk
Remember the Health Warning: this is one of my research
areas...
I have changed the PowerPoints for chapters 23 and 24...
...I was not happy with them.
Note that the lecture (Maple) file contains a lot of material
which you will NOT be examined on.
You will be examined on this PowerPoint presentation* and
not the lecture (Maple) file. The same with lecture 23.
*Except for some technically difficult bits which I note.
A bet here and now
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I intend to sell this bet to the highest bidder.
We toss a fair coin...
... if it lands heads I give you £20.
... If it lands tails I give you nothing.
We will do an “English Auction” – the student
who is willing to pay the most wins the auction,
pays me the price at which the penultimate
person dropped out of the auction, and I will play
out the bet with him or her.
Revision: Expected Values
• Suppose some risky/random variable, call it C,
takes the values c1 and c2 with respective
probabilities π1 and π2, then the Expected
Value of C is given by
• EC = π1 c1 + π2 c2
• Intuitively it is the value of C we can expect ...
• ...on average, after a large number of
repetitions.
• It is also the weighted average of the possible
values of C weighted by the probabilities.
Expected Utility Model (ch 24)
• This is a model of preferences.
• Suppose a lottery yields a random variable C
which takes the value c1 with probability π1 and
the value c2 with probability π2 (where π1 + π2 = 1).
• Expected Utility theory says this lottery is valued
by its Expected Utility:
... Eu(C) = π1u(c1)+ π2u(c2)
• where u(.) is the individual’s utility function.
• In intuitive terms the value of a lottery to an
individual is the utility that the individual expects
to get from it.
The Utility Function
• This is crucial. Tutorial 8 shows you one way of finding yours. Find
your function before the tutorial.
• Here is another way (there are lots).
• First calibrate the function on the best and worst...
• ...suppose £1000 is the best and £0 the worst. Put u(£1000)=1 and
u(£0)=0.
• Now to find your utility of some intermediate outcome, say £500, ask
yourself the following question:
• “For what probability p am I indifferent between £500 and the
gamble which gives me £1000 with probability p and £0 with
probability (1-p)?”
• This p is your utility of £500. u(£500) = p.
• Why? Because the expected utility of that gamble is p*1+(1-p)*0 = p.
Extensions and Implications
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You can repeat the above for different values of the intermediate outcome
(£500 above), and you can draw a graph of the function. What shape does it
have and what does the shape tell us?
Just consider the u(£500) and the graph composed of the 3 points.
If u(£500) = 0.5 then you are indifferent between the certainty of £500 and
the 50-50 gamble between £1000 and £0. This gamble has expected value
= £500. You are ignoring the risk: you are risk-neutral; the graph is linear.
If u(£500) > 0.5 then you are indifferent between the certainty of £500 and a
gamble between £1000 and £0 where the probability of winning £1000 is
more than 0.5. This gamble has expected value > £500. You want
compensation for the risk; you are risk-averse; the graph is concave.
If u(£500) < 0.5 then you are indifferent between the certainty of £500 and a
gamble between £1000 and £0 where the probability of winning £1000 is
less than 0.5. This gamble has expected value < £500. You like the risk;
you are risk-loving; the graph is convex.
Normalisation
• Note that we normalised (like temperature).
• So our function is unique only up to a linear
transformation.
• What does this mean?
• That if u(.) represents preferences then so does
v(.)=a+bu(.).
• Why? Because if X is preferred to Y then Eu(X)
> Eu(Y) and hence Ev(X) = a+bEu(X) >
a+bEu(Y) = Ev(Y).
Measuring risk attitudes
• Certainty Equivalent, CE, of a gamble G
for an individual is given by u(CE) = U(G).
• CE < (=,>) EG if risk averse (neutral,
loving).
• The Risk Premium, RP, is given by
• RP = EG-CE, the amount the individual is
willing to pay to get rid of the risk.
• RP > (=,<) 0 if risk averse (neutral, loving).
Measuring risk aversion
• How risk-averse an individual is is given by the degree of
concavity of the utility function.
• Concavity is measured by the second derivative of the
utility function –u”(c)
• Because the utility function is unique only up to a linear
transformation, we need to correct for the first derivative
u’(c).
• Our measure of the degree of (absolute) risk aversion is
thus
• -u”(c)/u’(c)
Constant (absolute) risk aversion
• Suppose our measure is constant
• -u”(c)/u’(c) = r, where r is constant.
• Integrating twice we get
• u(c) is proportional to –e-rc.
• This is the constant absolute risk averse
utility function.
• (For reference/interest the constant relative risk averse
utility function is proportional to cr)
A nice result for the keenies (not to be examined)
• Suppose an individual with a constant absolute
risk aversion utility function –e-rc faces a c which
is normally distributed with mean μ and variance
σ2 then (see next slide) his/her expected utility is
– exp(-rμ+r2σ2/2) and so his/her CE is μ-rσ2/2
and his/her Risk Premium is rσ2/2, which
increases with risk aversion and with variance,
but does not depend on the mean.
• Nice! But this does not depend on normality...
(see Maple after next slide)
A proof* for the keenies (for the case when u(c) = –e-rc)
• EU for discrete: Eu(C) = π1u(c1)+ π2u(c2)
• EU for continuous: Eu(C) = ∫u(c)f(c)dc where f(.)
is the probability density function of C.
• If c is normal with mean μ and variance σ2 then
f(c)= exp[-(x-μ)2/2σ2]/(2πσ2)1/2
• Thus Eu(C)=-∫exp(-rc)exp[-(x-μ)2/2σ2]/(2πσ2)1/2dc
• = – exp(-rμ+r2σ2/2) ∫exp[-[x-(μ-rσ2]2]/2σ2]/(2πσ2)1/2dc
• = – exp(-rμ+r2σ2/2)
• because the integral is that of a normal pdf.
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*This will not be examined.
Remember the conclusion from lecture 23?
• In a situation of decision-making under risk we have
shown that the constraint with fair markets is
• π1c1 + π2c2 = π1m1 + π2m2
• (starts with m1 and m2 and trades to/chooses to
consume c1 and c2).
• Note that the ‘prices’ are the probabilities (State 1
happens with probability π1 and State 2 with
probability π2 = 1-π1)
• So the slope of the fair budget line is -π1/π2.
• We now consider what an Expected Utility
maximiser will do in such a situation.
Indifference curves in (c1,c2) space
• Eu(C) = π1u(c1)+ π2u(c2)
• An indifference curve in (c1,c2) space is
given by π1 u(c1)+ π2 u(c2) = constant
• If the function u(.) is concave
(linear,convex) the indifference curves in
the space (c1,c2) are convex (linear,
concave).
• The slope of every indifference curve on
the certainty line = -π1/π2 (see next slide).
The slope of the indifference curves along the certainty line (c1=c2)
• An indifference curve in (c1,c2) space is
given by π1 u(c1)+ π2 u(c2) = constant
• Totally differentiating this we get
• π1 u’(c1)dc1+ π2 u’(c2)dc2 = 0 and hence
• dc2/dc1 = -π1 u’(c1)/π2 u’(c2)
• and so, putting c1= c2 we get
• dc2/dc1 (if c1= c2) = -π1 /π2
• Does this remind you of something?
Risk-averse
• Eu(C) = π1 u(c1)+ π2 u(c2)
• u(.) is concave
• An indifference curve is given by
π1 u(c1)+ π2 u(c2) = constant
• Hence the indifference curves in the space
(c1,c2) are convex. (Prove it yourself or see book or tutorial 8.)
• The slope of every indifference curve on
the certainty line = -π1/π2
Optimal choice π1= π2= 0.5 with fair insurance/betting
Optimal choice π1= 0.4,π2=0.6 with fair insurance/betting
More generally
• It follows immediately from the fact that the
slope of the fair budget line is -π1/π2 and
that the slopes of the indifference curves
along the certainty line are also -π1/π2
that...
• ...a risk-averter will always chose to be
fully insured in a fair market.
• Is this surprising/interesting?
Risk neutral
• Eu(C) = π1 u(c1)+ π2 u(c2)
• u(c)= c : the utility function is linear
• An indifference curve is given by
π1 c1+ π2 c2 = constant
• Hence the indifference curves in the space
(c1,c2) are linear. (Prove it yourself or see book or tutorial 8.)
• The slope of every indifference curve =
-π1/π2
Optimal choice π1= π2= 0.5
Risk-loving
• Eu(C) = π1 u(c1)+ π2 u(c2)
• u(.) is convex
• An indifference curve is given by
π1 u(c1)+ π2 u(c2) = constant
• Hence the indifference curves in the space
(c1,c2) are concave. (Prove it yourself or see book or tutorial 8.)
• The slope of every indifference curve on
the certainty line = -π1/π2
Optimal choice π1= 0.4,π2=0.6
Chapter 24
• Phew!
• Goodbye!