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From risk to opportunity Lecture 11 John Hey and Carmen Pasca

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Lecture 11 Implications of EUT Finding your (EU) utility function… … two different ways. Defining risk aversion and risk loving. Defining two indices of risk aversion. Some special utility functions with nice properties. Examples of its use in economics: 1.The theory of the competitive firm facing price uncertainty. 2.The life-cycle savings problem under income risk.

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Lecture 11 Implications of EUT: Finding your (EU) utility function 1 Finding your utility function over [x, X]. Here we are using x to denote the lower bound and X the upper bound of the interval over which we are going to find your (EU) utility function. There are lots of ways to find it. Here is just one. Put u(x)=0 and u(X)=1. To find the utility value for some intermediate amount x i answer the question: “what probability u i in the gamble [X,u i ;x,(1-u i )] makes you indifferent between that gamble and x i ?”. It immediately follows that u(x i ) = u i. Repeat for lots of different values of x i. Example, put x= €0 and X= €100. Suppose you are risk-averse and you are indifferent between €50 and the gamble [€100,0.75; €0,0.25] then for you u(€50) = 0.75. (Note EX=€75>€50.)

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Lecture 11 Implications of EUT: Finding your (EU) utility function 2 Here is another way: interpolation. Suppose that you have already found x a and x b such that, for you, u(x a )=a and u(x b )=b. To find your utility value half-way in-between answer the question: “what amount of money x (a+b)/2 makes you indifferent between that amount and the 50-50 gamble between x a and x b ; that is the gamble [x a,½;x b,½]?”. It immediately follows that u(x (a+b)/2 ) = (a+b)/2. Example, suppose a=0.5, x a =25; b=0.7, x b =49 and you are indifferent between €36 and a 50-50 gamble between €25 and €49 then for you u(36) = 0.6. Note this latter gamble has expected value €37 – you are risk-averse (and your function is concave between €25 and €49).

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Lecture 11 Implications of EUT: Certainty Equivalent For a given individual we define his or her certainty equivalent, CE, of some lottery/gamble as the amount of money, which, if received with certainty, the individual regards as the same as the lottery. So u(CE) = Eu(X) where X is the amount received in the lottery, CE denotes the Certainty Equivalent and where u(.) is the individual’s utility function. Example: lottery is 50:50 chance of €16 or €4. (Note that EX = 10.) Suppose u(x) = x 0.5. Then Eu(X) = 0.5u(16) + 0.5u(4) = 0.5(4)+0.5(2) = 3. And hence the CE is given by u(CE)=3. Hence CE = 9.

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Lecture 11 Implications of EUT: Risk Premium For a given individual we define his or her risk premium, RP, for some lottery/gamble as the amount of money he or she would pay to convert the lottery into its expected value. So RP = EX – CE, where CE is the individual’s certainty equivalent for the gamble. Example: lottery is 50:50 chance of €16 or €4. (Note that EX = €10.) Suppose u(x) = x 0.5. Then Eu(X) = 0.5u(16) + 0.5u(4) = 0.5(4)+0.5(2) = 3. And hence the CE is given by u(CE)=3. That is CE = €9. And so the RP = 10 – 9 = 1; the individual would pay up to €1 to exchange the lottery for the certainty of €10.

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Lecture 11 Implications of EUT: Risk aversion We define a risk-averse person as one who (always) prefers a certainty to a risk with the same expected value. So his or her certainty equivalent for some lottery is (always) less than the Expected Value of the lottery; the risk premium is always positive. This implies that his or her utility function is (everywhere) concave. Let us continue with the example where u(x) = √x = x 0.5 (concave) and where the lottery is a 50:50 chance of 16 or 4. What is the expected value of this lottery? 0.5(16) + 0.5(4) = 10. And the CE? 9. See the next slide.

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Lecture 11 Implications of EUT: Concave utility Gamble pays €4 with probability ½ and €16 with probability ½. Expected Value is €10 Certainty equivalent is €9 because u(9) = 3 = ½ u(4) + ½ u(16) = EU(X) Risk Premium = €1 = €10 - €9 = EX- CE

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Lecture 11 Implications of EUT: risk attitudes An individual is everywhere risk-averse (-neutral, -loving)… …if his or utility function is always concave (linear, convex) … if his or her certainty equivalent for some risk is always less than (equal to, more than) the expected value of the risk. … if he or she is always willing to pay a positive (a zero, a negative) amount to turn a risk into a certainty with the same expected value. The degree of concavity (convexity) indicates the degree of risk-aversion (loving).

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Lecture 11 Implications of EUT: measuring risk attitude The degree of concavity indicates the degree of risk aversion. Concavity of a function is to do with its second derivative. But as the function is unique only up to a linear transformation, it has to be divided by the first derivative. Absolute risk aversion index = -u”(x)/u’(x) Relative risk aversion index = -xu”(x)/u’(x)

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Lecture 11 Implications of EUT: CARA and CRRA For one who has Constant Absolute Risk Aversion: If we add some constant to all the outcomes of a gamble, the CE of that gamble rises by the same constant and hence the Risk Premium stays the same. From -u”(x)/u’(x) = r we get u(x) is proportional to –e -rx [unless r=0 in which case is proportional to x] If X is N(μ,σ 2 ) then Eu(X) proportional to –exp(-rμ+r 2 σ 2 /2). For one who has Constant Relative Risk Aversion: If we multiply by some constant to all the outcomes of a gamble, the CE of that gamble is multiplied by the same constant and hence the Risk Premium is multiplied by the same constant. From -xu”(x)/u’(x) = r we get u(x) is proportional to x 1-r [unless r=1 in which case is proportional to ln(x)] Note that the proportionality results from the fact that the utility function is unique only up to a linear transformation.

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Lecture 11 Implications of EUT: the perfectly competitive firm Consider the perfectly competitive firm under output price uncertainty. p, the price, is risky with known density function. The cost function c(.) is known. The firm wants to maximise the Expected Utility of profits = π = px – c(x) by its choice of x, the output. Choose x to maximise Eu(π)=Eu[px – c(x)]. FOC is that E{u’(π)[p-c’(x)]} = 0. From this we can show c’(x) < Ep Firm produces less under risk. See Hey JD Uncertainty in Economics, Martin Robertson 1979 (now way out of print).

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Lecture 11 Implications of EUT: life-cycle savings Life-cycle consumption/savings problem under income risk. Objective to maximise u(C 1 ) + ρu(C 2 ) + ρ 2 u(C 3 ) + … subject to W t+1 = R(Y t – C t + W t ) for all t. C, Y and W are consumption, income and wealth; ρ and R are the discount rate and the rate of return (1 plus the rate of interest). In general it can be shown that the optimal strategy is C* = a + b W, and that b=(R-1)/R So the marginal propensity to consume (out of wealth) depends only on the rate of interest/return. When the utility function is CARA, with r the index of absolute risk aversion and the distribution of income is N(μ,σ 2 ) it can also be shown (assuming r > 0) that a = μ – ½r(R-1)σ 2 –ln(Rρ)/[r(R-1] So the intercept of the consumption function depends positively on the mean of the income distribution and negatively on the variance; also if Rρ < 1 then increases in r and in R both lead to decreases in the intercept. Hey J D, “Optimal Consumption under Income Uncertainty”, Economic Letters, 5, 1980, 129-133.

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Lecture 11 Implications of EUT: Conclusions The great joy of EUT is its elegance and tractability. It is easy to find your (EU) utility function. It is concave (linear, convex) where you are risk- averse (-neutral, -loving). The degree of risk-aversion can be measured by the degree of concavity of the utility function (using either an absolute or a relative measure). CARA and CRRA are to useful special cases… … which lead to insightful results.

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Lecture 11 Goodbye!

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Lecture 11 Implications of EUT

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