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Microeconomics 2

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...

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Microeconomics 2

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  1. Microeconomics 2 John Hey

  2. 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.

  3. 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.

  4. Revision: Expected Values • Suppose some risky/random variable, call it C, takes the values c1and c2 with respective probabilities π1andπ2, 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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).

  9. 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).

  10. 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)

  11. 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)

  12. 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 μand variance σ2then (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)

  13. 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.

  14. 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 m1and m2and trades to/chooses to consume c1and 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.

  15. 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).

  16. 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= c2we get • dc2/dc1 (if c1= c2)= -π1 /π2 • Does this remind you of something?

  17. Risk-averse • Eu(C) = π1 u(c1)+ π2 u(c2) • u(.) isconcave • 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

  18. Optimal choice π1= π2=0.5 with fair insurance/betting

  19. Optimal choice π1=0.4,π2=0.6 with fair insurance/betting

  20. 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?

  21. 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

  22. Optimal choice π1= π2=0.5

  23. Risk-loving • Eu(C) = π1 u(c1)+ π2 u(c2) • u(.) isconvex • 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

  24. Optimal choice π1=0.4,π2=0.6

  25. Chapter 24 • Phew! • Goodbye!

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