Utility and Choice Under Uncertainty

Transcription

1 Introduction to Microeconomics Utility and Choice Under Uncertainty The Five Axioms of Choice Under Uncertainty We can use the axioms of preference to show how preferences can be mapped into measurable utility. 1. Completeness: For the entire set S of uncertain alternatives an individual can say either that outcome x is preferred to y or y is preferred to x or the individual is indifferent between the two x y or y x or x = y 2. Transitivity If x > y and y > z then x > z this ensures that utility curves cannot cross one another 3. Strong Independence says that if the individual is indifferent as to x and y, then he or she will also be indifferent as to a first gamble, set up between x with probability α and a outcome z, and a second gamble set up between y with probability α and the same outcome z. If x~y then G(x, z: α)~g(y, z: α) If Volvo~Saab therefore G(Volvo, Suburu: )~G(Saab, Suburu: ) 4. Measurability: If outcome y is preferred less than x but more than z, then there is a unique (probability) that the individual will be indifferent between y and a gamble between x with probability and z with probability (1 ). If Ferrari Alpha Romeo Fiat then! such that G(Ferrari, Fiat: )~Alpha Romeo If = 1 Ferrari Alpha Romeo, and = 0 Fiat Alpha Romeo If then gamble is less attractive and Alpha Romeo is more attractive If then the gambe is more attractive because Ferrari Alpha Romeo Essentially measures Alpha Romeo with respect to the gamble. 5. Ranking: If alternatives y an u both lie somewhere between x and z and we can establish gambles such that an individual is indifferent between y and a gamble between x (with probability 1 ) and z, while also indifferent between u and a second gamble, this time between x (with probability 2 ) and z, the if 1 > 2 the y is preferred to u. Two additional assumptions to the standard preference assumptions are that of non-satiation and rational behaviour. 1

2 Utility Functions The two important properties of utility functions are: 1. Order preserving 2. Expected Utility can be used to rank combinations of risky alternatives Proving utility functions are order preserving Consider the set of risky outcomes Q which is assumed to be bounded above by outcome k and below by outcome l. Now consider to outcomes between k and l called x and y. And k x l or k x l k y l or k y l Axiom 4 allows us to choose unique probabilities for x and y in order to construct the gambles for x and y. x~g k, l: (x) and y~g k, l: (y) Now by using axiom 5 we can rank the utilities. We do this by using the probabilities so that α(x) and α(y) can be interpreted as numerical utilities that uniquely rank x and y. If (x) > (y), then x y If (x) = (y), then x = y If (x) < (y), then x y The max and min outcomes k and l may be assigned any number eg. k = 1000 and l = 0. Then by forming simple gambles we can assign cardinal utility numbers to the intermediate outcomes x and y. 2. Proving Expected Utility can be used to rank combinations of risky alternatives Lets introduce as third alternative z which will not affect the relationship between x and y (axiom 3). By axiom 4 there must be a unique probability β(z), that would make us indifferent as to outcome z and a gamble between x and y. The Gambles α(x) k α(y) k x y 1-α(x) l 1-α(y) l 2

3 In the figure below we can trace the branches of the decision tree and show that the individual will be indifferent between z and a gamble between x and y. The Outcome z compared with a gamble between x and y α(x) k β(z) x z 1-β(z) 1-α(x) α(y) l k y 1-α(y) l Now we can relate z to the elemental prospects a and b and if we trace the branches in the above diagram we will be indifferent between z and outcome k with probability γ = β(z)α(x) + 1 β(z) α(y) and outcome l with probability (1 γ). This can be written as the following: The outcome z related to the prospects of k z~g[k, l: β(z)α(x) + (1 β)(z)α(y)] γ = β(z) α(x) + (1 β(z)α(y) k z l 1 γ = β(z) 1 α(x) + (1 β(z)(1 α(y)) Now we have already shown that Axioms 4 and 5 let us represent the utility of x and y by their probabilities. This means that (x) = U(x)and (y) = U(y) so now we can write the above gamble as the following: U(z) = β(z)u(x) + 1 β(z) U(y) This shows that the ranking function for risky alternatives is expected utility. 3

4 U(z) = β(z)u(x) + 1 β(z) U(y) This says that the utility of z is equal to the probability of x times its utility plus the probability of y times its utility. This is an expected utility that represents a linear combination of the utilities outcomes. This can be written as the following where the expected utility of wealth is represented by W: E[U(W)] = p i U(W i ) Investors will calculate expected utility of wealth for all possible alternative choices and then choose the outcome that maximises their expected utility of wealth. i Example 1 How to Create a Utility Function Suppose an there will be a loss of -100 utility units when we lose $2000. Also suppose that there is a gamble with a probability of α=0.7 winning $2000 and a probability of (1-α)=.3 of losing $2000. The probability of winning must be this amount in order for the individual to be indifferent between $0 and the gamble. What is the probability that would make us indifferent between making the gamble and having $0 with absolute certainty? Or 0~G(2000, 2000: α) U(0) = αu(2,000) + (1 α)u( 2000) Using the probabilities and the above formula we will solve for the probability (1 α)u( 2000) U(2000) = α = (0.3)U( 100) 0.6 = 50 utility units We repeat this procedure for different payoffs in order to create a whole utility function. But Remember all utility functions are specific to the individual and we cannot compare interpersonal utility functions. Monotonic Transformation: A monotone transformation is a transformation that preserves inequalities of its arguments. That is, if T is a monotone transformation, then: (i) iff x>y, then Tx>Ty, and iff x<y, then Tx<Ty. Changes in utility between any two wealth levels have exactly the same meaning on the two utility functions; that is, one utility function is just a transformation of the other. Utility curves are ranking devices and if we scale two bundles that have a set utility level by the same amount the ratio between them will remain. U 1 = 100 and U 2 = 50 Now we multiply both utilities by a scalar which could be 100 and the ratio between the two will remain constant at 2. Therefore, 100(U 1 ) = 100(100) = 10,000 and (100)U 2 = 100(50) =

5 = = 2 Expected Utility Theory Expected Utility Theory Intuition Expected values do not work, and the average outcomes are not so good. The second best way works, where the preformed object is transferred into the utility space then we use expected values. Basically we just transfer into the utility space then solve and transfer back to the real space. An Example is Below with Logarithmic Utility Function Risk Aversion Firstly we should assume there is non-satiation of wealth and therefore the marginal utility of wealth is increasing function. Also suppose that there is a gamble between two prospects a and b. Would a person rather receive $100 for sure or would this person rather enter a gamble which pays $1000 with a 10% probability or $0 with a 90% probability? The actuarially fair price is $100 because this is the expected outcome of the gamble. Risk Lover: Will prefer to take the risk Risk Neutral: will be indifferent between the $100 with certainty and the gamble. Risk Averse: Will prefer the of $100 with certainty Now suppose we have a logarithmic function U(W) = ln(w). The gamble is 80% chance of $5 and a 20% chance of $30. The actuarially fair price is the expected value of the gamble which is $10. E(W) = 0.8(5) + 0.2(30) = $10 Lets suppose if an individual receives $10 with certainty they will get 2.3 utility units. The other possibility is the utility from the gamble which is equal to the expected utility provided by the wealth of the gamble. Logarithmic Utility Function Example U(W)=ln U = 3.4 U = 2.3 U = 1.97 U = W E[U(W)] = 0.8(1.61) + 0.2(3.4) =

6 2.3 > 1.97 Utility (Expected Wealth acturial value of gamble) > Utility (Expected wealth from gamble) Because this amount is less than the amount of utility received from declining the gamble and having the $10 actuarially fair price the person will be risk averse. Risk Premium: The max amount of wealth a person is willing to give up in order to avoid the gamble. In the last example and given a logarithmic function the individual will be willing to pay $2.83 in order to avoid the gamble. This is because the level of wealth that provides 1.97 utility units if $7.17 but the individual has an expected wealth of $10. This is also known as the Markowitz risk premium. If the insurance cost is less than $2.83 the individual will buy it. Risk Premium: The difference between an individual s expected wealth, given the gamble and the level of wealth that individual would accept with certainty if the gamble were removed. Example 2 Risk Premiums A risk averse person has the same logarithmic utility function already discussed and has current wealth $10. The gamble is 10% chance of $10 or 90% chance of $100. Current wealth = $10 Expected Wealth = Current wealth + expected outcome of the gamble = $101 Risk Premium = $ =$8.26 CEW = Acturial value of gamble = $92.76 But does this make sense. This measures in dollars the risk premium associated with the gamble. For a risk averter the risk premium is always positive but the cost of gamble can be either positive or negative. The gamble is favourable we will always win if we accept it so even a risk averse investor would be willing to pay for this gamble. Cost of gamble = current wealth CEW = $10 $92.76 = $82.76 This means that a person would be willing to pay $82.76 in order to take the gamble. U(W)=ln Logarithmic Utility Function Example UE(W) = 4.7 E(U(W)) = 2.3 U = 1.97 U = 1.61 Cost of RP W 0 W 1 CEW EW W 2 6 W

7 Some further intuition α = Co=10 1-α= EW = 0.1(20) + 0.9(110) = 101 ln(w) = U(w) E[U(W)] = 0.1ln ln110 = 4.53 Utils U(W) = ln(w) = E[U(W)] = 4.53 utils W = e 4.53 = Although your expected wealth is 101 you feel like you only have $ This represents Risk aversion because the gamble gives higher payout but you are only willing to pay $92.76 to get $101. willing to give up wealth for CEW Inverse utility of expected utility = U 1 (E[U(W)]) Risk Premium = EW CEW = How much you are willing to pay for the Gamble or Cost of Gamble = CEW W 0 = = $82.76 Example 3 Risk Premiums What would happen if we scale up the original wealth so instead of beginning at $10,000 they have initial wealth of $20,000? (Think Monotonic Transformation) α = Co=20 1-α=

8 Logarithmic Utility Function U(W)=ln UE(W) = 4.7 E(U(W)) = 4.63 U = U = 2.99 Cost of Gamble RP W W 0 W 1 CEW EW W 2 W 0 = 20 EW = 0.1(30) + 0.9(120) = 111 ln(w) = U(w) E[U(W)] = 0.1ln ln120 = 4.64 Utils U(W) = ln(w) = E[U(W)] = 4.64 utils W = e 4.64 = = CEW Although your expected wealth is 111 you feel like you only have $ This represents Risk aversion because the gamble gives higher payout but you are only willing to pay $ to get $111. willing to give up wealth for CEW Inverse utility of expected utility = U 1 (E[U(W)]) Risk Premium = EW CEW = = $7.46 How much are you willing to pay for the gamble or in other words the cost of the gamble Cost of Gamble = CEW W 0 = The Expected value and expected of utility of wealth both increases by 10. The CEW increases by more than $10, so this shows that as the person increases wealth they become less risk averse. For instance if you had $1 bet and $1 income you would be very risk averse compared to a $1 bet for someone with a $1Bn. This is also reflected in the risk premium which decreases because they do not have to be compensated as much for the risk. Also the amount the person is willing to pay for the gamble increases as they become less risk averse when income increases. 8

9 Risk Aversion Risk aversion means that their utility functions are strictly concave and increasing. 1. This means that they always prefer more wealth to less wealth (Marginal utility of wealth is positive) 2. The marginal utility of wealth is a decreasing function as their wealth increases (Negative second derivative) Risk Aversion, Neutrality and Loving If U[E(W)] > E[U(W)] = Risk Aversion If U[E(W)] = E[U(W)] = Risk Neutrality If U[E(W)] < E[U(W)] = Risk Loving Strict Concavity = Risk Averse Linearity = Risk Neutral Convexity = Risk Loving PRATT (1964) & ARROW (1971) Both Pratt and Arrow provide a specific definition of risk aversion: The risk premium is the amount that must be added to make the person indifferent between it and the actuarial value of the gamble. An individual has wealth of W and then the individual is presented with a actuarial neutral gamble of Z where Z = 0. What risk premium π(w, Z ) must be added to the gamble to make her indifferent between the gamble and the actuarial value of the gamble? This can be represented in the following equation by: E[U(W + Z )] = U W + E Z π(w, Z ) The left hand side is the expected utility of the current level of wealth, given the gamble. Its utility must equal the utility of the right hands side, that is the current level of wealth W plus the utility of the actuarial value of the gamble, E Z minus he risk premium, π(w, Z ). We use Taylor s series expansion to expand the utility function of wealth around both sides of the above equation. Eventually after manipulation we derive an expression for the risk premium π = 1 2 σ Z 2 U (W) U (W) Then from this we can define the ARA Absolute Risk Aversion. ARA: Measures the level of risk aversion for a given level of wealth. The Pratt Arrow definition provides good intuition of people s behaviour when faced with risk. ARA will probably decrease as wealth decreases. For example a $1000 gamble may seem trivial for a billionaire but for a homeless person they would probably be quite risk averse when facing this gamble. ARA = U (W) U (W) 9

10 RRA: Relative Risk Aversion: is the absolute risk aversion measure multiplied by the level of wealth to obtain what is known as relative risk aversion. Remember the risk premium will stay the same as long as the value of the gamble increases at the same pace as wealth. However it can increase or decrease depending on the values. ARA = W U (W) U (W) These measures of risk play an important role when examining different types of utility functions to see whether or not they have decreasing ARA and constant RRA. For example the quadratic utility function exhibits increasing ARA and increasing RRA. This does not make sense intuitively. Would a billionaire who loses $500m lose more utility than the same person started with $20,000 and ends up with $10,000. Comparisons of Risk Aversion in the Small and in the Large Arrow Pratt definition involves ARA and RRA which provide good intuition, but it assumes the risks are small and actuarially neutral. The Markowitz concept is not limited by this. Example 4 Risk Measures Logarithmic utility function and a level of wealth of $20,000 is exposed to two different risk: (1) 50/50 chance of gaining or losing $10 and (2) an 80% chance of losing $1000 and a 20% chance of losing $10,000. What is the risk premium required by the individual with each of these risks? If the risk is small Arrow Pratt measure should do a good job. The variance of (1) is π = 1 2 σ Z 2 U (W) U (W) σ 2 2 Z = p i (X i E(X)) = 1 2 (20,000 20,000) (19,990 20,000)2 = 100 The ratio of the second and first derivates of logarithmic utility function evaluated at ta level of wealth of $20,000 is, U (W) = 1 W U (W) = 1 W 2 U (W) U (W) = 1 w = 1 20,000 Therefore using definition of Arrow Pratt risk premium we will have π = 1 2 σ Z 2 U (W) 100 U = (W) 2 1 = $ ,000 10

11 The Markowitz approach requires computation of the expected utility of the gamble as follows E[U(W)] = p i U(W i ) = 1 2 U(20,010) U(19,990) = 1 2 ln(20,010) + 1 ln(19,990) = W = e ln (W) = $19, Therefore we would pay a risk premium as large as $ The difference between Pratt Arrow and Markowitz is negligible. If we repeat this procedure for the second risk (2) Arrow Pratt measures are not even remotely close. The dollar difference between the two risk premiums is much larger, and is in fact over $100 dollars. Therefore the Markowitz works much better for larger gambles or where it is not actuarially neutral. However Arrow Pratt does help us distinguish between various types of concave utility functions. 11