1. Frank Cowell : Risk RISK MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Consumption and Uncertainty Almost essential Consumption and Uncertainty Prerequisites July 2015 1

2. Frank Cowell : Risk Risk and uncertainty  In dealing with uncertainty a lot can be done without introducing probability  Now we introduce a specific probability model This could be some kind of exogenous mechanism Could just involve individual’s perceptions  Facilitates discussion of risk  Introduces new way of modelling preferences July 2015 2

3. Frank Cowell : Risk Overview… Probability Risk comparisons Special Cases Lotteries Risk An explicit tool for model building July 2015 3

4. Frank Cowell : Risk Probability  What type of probability model?  A number of reasonable versions: public observable public announced private objective private subjective  Need a way of appropriately representing probabilities in economic models July 2015 4 Lottery government policy? coin flip emerges from structure of preferences

5. Frank Cowell : Risk Ingredients of a probability model  We need to define the support of the distribution smallest closed set whose complement has probability zero convenient way of specifying what is logically feasible (points in the support) and infeasible (other points)  Distribution function F represents probability in a convenient and general way encompass both discrete and continuous distributions discrete distributions can be represented as a vector continuous distribution – usually specify density function  Take some particular cases: a collection of examples July 2015 5

6. Frank Cowell : Risk Some examples  Begin with two cases of discrete distributions #  = 2. Probability  of value x 0 ; probability 1–  of value x 1 #  = 5. Probability  i of value x i, i = 0,…,4  Then a simple example of continuous distribution with bounded support The rectangular distribution – uniform density over an interval  Finally an example of continuous distribution with unbounded support July 2015 6

7. Frank Cowell : Risk Discrete distribution: Example 1 x  Below x 0 probability is 0  Probability of x ≤ x 0 is  x1x1 x0x0 1   Probability of x ≤ x 1 is 1  Suppose of x 0 and x 1 are the only possible values F(x)  Probability of x ≥ x 0 but less than x 1 is  July 2015 7

8. Frank Cowell : Risk Discrete distribution: Example 2 x  Below x 0 probability is 0  Probability of x ≤ x 0 is   x1x1 x0x0 1 00  Probability of x ≤ x 1 is   +    There are five possible values: x 0,…, x 4 F(x)  0  1  0  1  2  3  Probability of x ≤ x 2 is   +   +   x4x4 x2x2 x3x3  0  1  2  Probability of x ≤ x 3 is   +   +   +    Probability of x ≤ x 4 is 1    +   +   +   +   = 1 July 2015 8

9. Frank Cowell : Risk “Rectangular” : density function x  Below x 0 probability is 0 x1x1 x0x0  Suppose values are uniformly distributed between x 0 and x 1 f(x) July 2015 9

10. Frank Cowell : Risk Rectangular distribution x  Below x 0 probability is 0  Probability of x ≥ x 0 but less than x 1 is [x  x 0 ] / [x 1  x 0 ] x1x1 x0x0 1  Probability of x ≤ x 1 is   Values are uniformly distributed over the interval [x 0, x 1 ] F(x) July 2015 10

11. Frank Cowell : Risk Lognormal density x 012345678910  Support is unbounded above  The density function with parameters  = 1,  = 0.5  The mean July 2015 11

12. Frank Cowell : Risk Lognormal distribution function x 012345678910 1 July 2015 12

13. Frank Cowell : Risk Overview… Probability Risk comparisons Special Cases Lotteries Risk Shape of the u-function and attitude to risk July 2015 13

14. Frank Cowell : Risk Risk aversion and the function u  With a probability model it makes sense to discuss risk attitudes in terms of gambles  Can do this in terms of properties of “felicity” or “cardinal utility” function u Scale and origin of u  are irrelevant But the curvature of u  is important  We can capture this in more than one way  We will investigate the standard approaches…  …and then introduce two useful definitions July 2015 14

15. Frank Cowell : Risk Risk aversion and choice  Imagine a simple gamble  Two payoffs with known probabilities: x RED with probability  RED x BLUE with probability  BLUE Expected value E x =  RED x RED +  BLUE x BLUE  A “fair gamble”: stake money is exactly E x  Would the person accept all fair gambles?  Compare E u(x) with u( E x) depends on shape of u July 2015 15

16. Frank Cowell : Risk Attitudes to risk  u(x)u(x) x BLUE x x RED E x Risk-loving  u(x)u(x) x BLUE x x RED E x Risk-neutral u(x)u(x) x BLUE x x RED E x  Risk-averse  Shape of u associated with risk attitude  Neutrality: will just accept a fair gamble  Aversion: will reject some better-than-fair gambles  Loving: will accept some unfair gambles July 2015 16

17. Frank Cowell : Risk Risk premium and risk aversion x BLUE x RED O  RED – _____  BLUE  RED – _____  BLUE  Certainty equivalent income  A given income prospect  Slope gives probability ratio E x   Mean income  The risk premium  P 0  P  Risk premium:  Amount that you would sacrifice to eliminate the risk  Useful additional way of characterising risk attitude – example July 2015 17

18. Frank Cowell : Risk An example…  Two-state model  Subjective probabilities (0.25, 0.75)  Single-commodity payoff in each case July 2015 18

19. Frank Cowell : Risk Risk premium: an example u u(x)u(x) x BLUE x x RED u(x BLUE ) u(x RED ) E x u( E x)  E u(x ) amount you would sacrifice to eliminate the risk u( E x) E x   Expected payoff & U of expected payoff  Expected utility and certainty-equivalent  The risk premium again  Utility values of two payoffs E u(x )  July 2015 19

20. Frank Cowell : Risk Change the u-function u x BLUE x x RED u(x BLUE ) u(x RED ) E x   The utility function and risk premium as before  Now let the utility function become “flatter”… u(x BLUE )   Making the u-function less curved reduces the risk premium…  …and vice versa  More of this later July 2015 20

21. Frank Cowell : Risk An index of risk aversion?  Risk aversion associated with shape of u second derivative or “curvature”  But could we summarise it in a simple index or measure?  Then we could easily characterise one person as more/less risk averse than another  There is more than one way of doing this July 2015 21

22. Frank Cowell : Risk Absolute risk aversion  Definition of absolute risk aversion for scalar payoffs u xx (x)  (x) :=   u x (x)  For risk-averse individuals  is positive  For risk-neutral individuals  is zero  Definition ensures that  is independent of the scale and the origin of u Multiply u by a positive constant… …add any other constant…  remains unchanged July 2015 22

23. Frank Cowell : Risk Relative risk aversion  Definition of relative risk aversion for scalar payoffs: u xx (x)  (x) :=  x  u x (x)  Some basic properties of  are similar to those of  : positive for risk-averse individuals zero for risk-neutrality independent of the scale and the origin of u  Obvious relation with absolute risk aversion :  (x) = x  (x) July 2015 23

24. Frank Cowell : Risk Concavity and risk aversion  u(x)u(x) payoff utility x û(x)û(x)  Draw the function u again  Change preferences: φ is a concave function of u  Risk aversion increases More concave u implies higher risk aversion now to the interpretations lower risk aversion higher risk aversion û = φ(u) July 2015 24

25. Frank Cowell : Risk Interpreting  and   Think of  as a measure of the concavity of u  Risk premium is approximately ½  (x) var(x)  Likewise think of  as the elasticity of marginal u  In both interpretations an increase in the “curvature” of u increases measured risk aversion Suppose risk preferences change… u is replaced by û, where û = φ(u) and φ is strictly concave Then both  (x) and  (x) increase for all x  An increase in  or  also associated with increased curvature of IC… July 2015 25

26. Frank Cowell : Risk Another look at indifference curves  u  and  determine the shape of IC  Alf and Bill differ in risk aversion x BLUE x RED O x BLUE x RED O  Alf, Charlie differ in subj probability Bill Alf Charlie Same us but different  s Same  s but different us July 2015 26

27. Frank Cowell : Risk Overview… Probability Risk comparisons Special Cases Lotteries Risk CARA and CRRA July 2015 27

28. Frank Cowell : Risk Special utility functions?  Sometimes convenient to use special assumptions about risk Constant ARA Constant RRA  By definition  (x) = x  (x)  Differentiate w.r.t. x: d  (x) d  (x)  =  (x) + x  dx dx  So one could have, for example: constant ARA and increasing RRA constant RRA and decreasing ARA or, of course, decreasing ARA and increasing RRA July 2015 28

29. Frank Cowell : Risk Special case 1: CARA  We take a special case of risk preferences  Assume that  (x) =  for all x  Felicity function must take the form 1 u(x) :=   e  x   Constant Absolute Risk Aversion  This induces a distinctive pattern of indifference curves… July 2015 29

30. Frank Cowell : Risk Constant Absolute RA  Case where  = ½  Slope of IC is same along 45° ray (standard vNM)  For CARA slope of IC is same along any 45° line x BLUE x RED O July 2015 30

31. Frank Cowell : Risk CARA: changing   Case where  = ½ (as before)  Change ARA to  = 2 x BLUE x RED O July 2015 31

32. Frank Cowell : Risk Special case 2: CRRA  Another important special case of risk preferences  Assume that  (x) =  for all   Felicity function must take the form 1 u(x) :=  x 1   1    Constant Relative Risk Aversion  Again induces a distinctive pattern of indifference curves… July 2015 32

33. Frank Cowell : Risk Constant Relative RA  Case where  = 2  Slope of IC is same along 45° ray (standard vNM)  For CRRA slope of IC is same along any ray  ICs are homothetic x BLUE x RED O July 2015 33

34. Frank Cowell : Risk CRRA: changing  x BLUE x RED O  Case where  = 2 (as before)  Change RRA to  = ½ July 2015 34

35. Frank Cowell : Risk CRRA: changing  x BLUE x RED O  Case where  = 2 (as before)  Increase probability of state RED July 2015 35

36. Frank Cowell : Risk Overview… Probability Risk comparisons Special Cases Lotteries Risk Probability distributions as objects of choice July 2015 36

37. Frank Cowell : Risk Lotteries  Consider lottery as a particular type of uncertain prospect  Take an explicit probability model  Assume a finite number  of states-of-the-world  Associated with each state  are: A known payoff x , A known probability   ≥ 0  The lottery is the probability distribution over the “prizes” x ,  =1,2,…,  The probability distribution is just the vector  := (  ,   ,…,    ) Of course,   +   +…+   = 1  What of preferences? July 2015 37

38. Frank Cowell : Risk The probability diagram: #  =2  BLUE  RED (1,0) (0,1)  Cases where 0 <  < 1  Probability of state BLUE  Cases of perfect certainty  Probability of state RED  RED  BLUE   The case (0.75, 0.25) (0, 0.25) (0.75, 0)  Only points on the purple line make sense  This is an 1-dimensional example of a simplex July 2015 38

39. Frank Cowell : Risk The probability diagram: #  =3 0  BLUE  RED  GREEN  Third axis corresponds to probability of state GREEN (1,0,0) (0,0,1) (0,1,0)  There are now three cases of perfect certainty  Cases where 0 <  < 1  RED  GREEN  BLUE  (0, 0, 0.25) (0.5, 0, 0) (0, 0.25, 0)  The case (0.5, 0.25, 0.25)  Only points on the purple triangle make sense  This is a 2-dimensional example of a simplex July 2015 39

40. Frank Cowell : Risk Probability diagram #  =3 (contd.) (1,0,0) (0,0,1) (0,1,0). (0.5, 0.25, 0.25)  All the essential information is in the simplex  Display as a plane diagram  The equi-probable case  The case (0.5, 0.25, 0.25) (1/3,1/3,1/3) July 2015 40

41. Frank Cowell : Risk Preferences over lotteries  Take the probability distributions as objects of choice  Imagine a set of lotteries  °,  ',  ",…  Each lottery  has same payoff structure State-of-the-world  has payoff x  … and probability   ° or   ' or   " … depending on which lottery  We need an alternative axiomatisation for choice amongst lotteries  July 2015 41

42. Frank Cowell : Risk Axioms on preferences July 2015 42

43. Frank Cowell : Risk Basic result  Take the axioms transitivity, independence, continuity  Imply that preferences must be representable in the form of a von Neumann-Morgenstern utility function:    u  x      or equivalently:        where    u  x    So we can also see the EU model as a weighted sum of  s July 2015 43

44. Frank Cowell : Risk  -indifference curves  Indifference curves over probabilities  Effect of an increase in the size of  BLUE (1,0,0) (0,0,1) (0,1,0). July 2015 44

45. Frank Cowell : Risk What next?  Simple trading model under uncertainty  Consumer choice under uncertainty  Models of asset holding  Models of insurance  This is in the presentation Risk Taking July 2015 45