1. Frank Cowell: Microeconomics Risk MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Consumption and Uncertainty Almost essential Consumption and Uncertainty Prerequisites November 2006

2. Frank Cowell: Microeconomics Risk and uncertainty In dealing with uncertainty a lot can be done without introducing probability. In dealing with uncertainty a lot can be done without introducing probability. Now we introduce a specific probability model 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 Facilitates discussion of risk Introduces new way of modelling preferences Introduces new way of modelling preferences

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

4. Frank Cowell: Microeconomics Probability What type of probability model? What type of probability model? A number of reasonable versions: A number of reasonable versions:  Public observable  Public announced  Private objective  Private subjective Need a way of appropriately representing probabilities in economic models. Need a way of appropriately representing probabilities in economic models. Lottery government policy? coin flip emerges from structure of preferences.

5. Frank Cowell: Microeconomics Ingredients of a probability model We need to define the support of the distribution We need to define the support of the distribution  The 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 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: Take some particular cases: a collection of examples

6. Frank Cowell: Microeconomics Some examples Begin with two cases of discrete distributions 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 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 Finally an example of continuous distribution with unbounded support

7. Frank Cowell: Microeconomics 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 .   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 .

8. Frank Cowell: Microeconomics 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

9. Frank Cowell: Microeconomics “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)

10. Frank Cowell: Microeconomics 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)

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

12. Frank Cowell: Microeconomics Lognormal distribution function x 012345678910 1

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

14. Frank Cowell: Microeconomics Risk aversion and the function u With a probability model it makes sense to discuss risk attitudes in terms of gambles 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 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 can capture this in more than one way We will investigate the standard approaches... We will investigate the standard approaches......and then introduce two useful definitions...and then introduce two useful definitions

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

16. Frank Cowell: Microeconomics Attitudes to risk  u(x)u(x) x BLUE x x RED ExEx Risk-loving  u(x)u(x) x BLUE x x RED ExEx Risk-neutral u(x)u(x) x BLUE x x RED ExEx  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

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

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

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

20. Frank Cowell: Microeconomics Change the u-function u x BLUE x x RED u(x BLUE ) u(x RED ) ExEx    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

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

22. Frank Cowell: Microeconomics Absolute risk aversion Definition of absolute risk aversion for scalar payoffs. Definition of absolute risk aversion for scalar payoffs. u xx (x) u xx (x)  (x) :=   u x (x) u x (x) For risk-averse individuals  is positive. For risk-averse individuals  is positive. For risk-neutral individuals  is zero. For risk-neutral individuals  is zero. Definition ensures that  is independent of the scale and the origin of u. 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.

23. Frank Cowell: Microeconomics Relative risk aversion Definition of relative risk aversion for scalar payoffs: Definition of relative risk aversion for scalar payoffs: u xx (x) u xx (x)  (x) :=  x  u x (x) u x (x) Some basic properties of  are similar to those of  : 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 : Obvious relation with absolute risk aversion :   (x) = x  (x)

24. Frank Cowell: Microeconomics 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)

25. Frank Cowell: Microeconomics Interpreting  and  Think of  as a measure of the concavity of u. Think of  as a measure of the concavity of u. Risk premium is approximately ½  (x) var(x). Risk premium is approximately ½  (x) var(x). Likewise think of  as the elasticity of marginal u. Likewise think of  as the elasticity of marginal u. In both interpretations an increase in the “curvature” of u increases measured risk aversion. 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(uφ  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… An increase in  or  also associated with increased curvature of IC…

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

27. Frank Cowell: Microeconomics Overview... Probability Risk comparisons Special Cases Lotteries Risk CARA and CRRA

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

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

30. Frank Cowell: Microeconomics Constant Absolute RA   Case where  = ½   Slope of IC the same along 45° ray (standard vNM)   But, for CARA, slope of IC the same along any 45° line x BLUE x RED O

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

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

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

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

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

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

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

38. Frank Cowell: Microeconomics 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

39. Frank Cowell: Microeconomics 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

40. Frank Cowell: Microeconomics 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)

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

42. Frank Cowell: Microeconomics Axioms on preferences Transitivity over lotteries Transitivity over lotteries  If ° and "...  If  ° <  ' and  ' <  "... ...then °". ...then  ° <  ". Independence of lotteries Independence of lotteries  If ° and (0,1)...  If  ° <  ' and (0,1)... ...then ° ]" ] " ...then °  ]  " < '  ]  " Continuity over lotteries Continuity over lotteries  If °" then there are numbers and  such that  If  ° Â  ' Â  " then there are numbers and  such that  °  ]"  °  ]  " Â  '   °  ]"   ' Â  °  ]  "

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

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

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