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 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 £ 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 c 1 and c 2 with respective probabilities π 1 and π 2, then the Expected Value of C is given by EC = π 1 c 1 + π 2 c 2 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 c 1 with probability π 1 and the value c 2 with probability π 2 (where π 1 + π 2 = 1). Expected Utility theory says this lottery is valued by its Expected Utility:... Eu(C) = π 1 u(c 1 )+ π 2 u(c 2 ) 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 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 c r )

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 σ 2 then (see next slide) his/her expected utility is – exp(-rμ+r 2 σ 2 /2) and so his/her CE is μ-rσ 2 /2 and his/her Risk Premium is rσ 2 /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) = π 1 u(c 1 )+ π 2 u(c 2 ) 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 σ 2 then f(c)= exp[-(x-μ) 2 /2σ 2 ]/(2πσ 2 ) 1/2 Thus Eu(C)=-∫exp(-rc)exp[-(x-μ) 2 /2σ 2 ]/(2πσ 2 ) 1/2 dc = – exp(-rμ+r 2 σ 2 /2) ∫exp[-[x-(μ-rσ 2 ]2]/ 2 σ 2 ]/(2πσ 2 ) 1/2 dc = – exp(-rμ+r 2 σ 2 /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 π 1 c 1 + π 2 c 2 = π 1 m 1 + π 2 m 2 (starts with m 1 and m 2 and trades to/chooses to consume c 1 and c 2 ). Note that the ‘prices’ are the probabilities (State 1 happens with probability π 1 and State 2 with probability π 2 = 1-π 1 ) So the slope of the fair budget line is -π 1 /π 2. We now consider what an Expected Utility maximiser will do in such a situation.

Indifference curves in (c 1,c 2 ) space Eu(C) = π 1 u(c 1 )+ π 2 u(c 2 ) An indifference curve in (c 1,c 2 ) space is given by π 1 u(c 1 )+ π 2 u(c 2 ) = constant If the function u(.) is concave (linear,convex) the indifference curves in the space (c 1,c 2 ) are convex (linear, concave). The slope of every indifference curve on the certainty line = -π 1 /π 2 (see next slide).

The slope of the indifference curves along the certainty line (c 1 =c 2 ) An indifference curve in (c 1,c 2 ) space is given by π 1 u(c 1 )+ π 2 u(c 2 ) = constant Totally differentiating this we get π 1 u’(c 1 )dc 1 + π 2 u’(c 2 )dc 2 = 0 and hence dc 2 /dc 1 = -π 1 u’(c 1 )/π 2 u’(c 2 ) and so, putting c 1 = c 2 we get dc 2 /dc 1 (if c 1 = c 2 ) = -π 1 /π 2 Does this remind you of something?

Risk-averse Eu(C) = π 1 u(c 1 )+ π 2 u(c 2 ) u(.) is concave An indifference curve is given by π 1 u(c 1 )+ π 2 u(c 2 ) = constant Hence the indifference curves in the space (c 1,c 2 ) are convex. (Prove it yourself or see book or tutorial 8.) The slope of every indifference curve on the certainty line = -π 1 /π 2

Optimal choice π 1 = π 2 = 0.5 with fair insurance/betting

Optimal choice π 1 = 0.4,π 2 =0.6 with fair insurance/betting

More generally It follows immediately from the fact that the slope of the fair budget line is -π 1 /π 2 and that the slopes of the indifference curves along the certainty line are also -π 1 /π 2 that......a risk-averter will always chose to be fully insured in a fair market. Is this surprising/interesting?

Risk neutral Eu(C) = π 1 u(c 1 )+ π 2 u(c 2 ) u(c)= c : the utility function is linear An indifference curve is given by π 1 c 1 + π 2 c 2 = constant Hence the indifference curves in the space (c 1,c 2 ) are linear. (Prove it yourself or see book or tutorial 8.) The slope of every indifference curve = -π 1 /π 2

Optimal choice π 1 = π 2 = 0.5

Risk-loving Eu(C) = π 1 u(c 1 )+ π 2 u(c 2 ) u(.) is convex An indifference curve is given by π 1 u(c 1 )+ π 2 u(c 2 ) = constant Hence the indifference curves in the space (c 1,c 2 ) are concave. (Prove it yourself or see book or tutorial 8.) The slope of every indifference curve on the certainty line = -π 1 /π 2

Optimal choice π 1 = 0.4,π 2 =0.6

Chapter 24 Phew! Goodbye!