Microeconomics 2

John Hey

Chapters 23, 24 and 25

• 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

• 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

• 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. • *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!