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Microeconomic Analysis (L1D007)

Microeconomic Analysis (L1D007). School of Economics. Dr. M. Montero. Choice under Risk: Expected Utility Theory. Choice under uncertainty/ risk. So far, the consumer knows the exact consequences of his choices This is rarely the case in practice Examples: Degree/module choice

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Microeconomic Analysis (L1D007)

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  1. Microeconomic Analysis (L1D007) School of Economics Dr. M. Montero Choice under Risk: Expected Utility Theory

  2. Choice under uncertainty/risk So far, the consumer knows the exact consequences of his choices This is rarely the case in practice Examples: • Degree/module choice How much will I like it? What mark can I expect? • Investment What will be the return?

  3. Difference between risk and uncertainty In both cases, several outcomes are possible Risk: Probabilities are determined Uncertainty: Probabilities are undetermined We will only discuss risk. If the objective probabilities are not known, we assume subjective probabilities.

  4. Lotteries A (simple) lottery is a collection of outcomes with their corresponding probabilities Outcome Probability Outcome Probability £ 35,000 £ 50,000 0.5 0.9 0.5 0.1 £ 25,000 £ 15,000 Probabilities are determined (objectively or subjectively)

  5. Theories of choice under risk Several theories varying in sophistication The simplest one assumes that people look only at expected monetary payoff They choose the option that yields more money on average

  6. Evaluating the lottery “Employee” xp(x) x (outcome) p(x) (probability) 35,000 17,500 0.5 25,000 0.5 12,500 Total 30,000

  7. Evaluating the lottery “Self-employed” xp(x) x (outcome) p(x) (probability) 50,000 45,000 0.9 15,000 0.1 1,500 Total 46,500

  8. Everybody should be self-employed! Is “employee” really such a bad choice? It may give less on average… …. but there is less risk involved! It seems reasonable to assume that people care also about risk

  9. Example 1: a fair gamble Consider a choice between: • 10,000 pounds for sure • An even chance of getting 0 or 20,000 Most people would much prefer the safe option… A minority will prefer the risky option… But hardly anybody will be indifferent! Nevertheless, the theory predicts indifference because on average they both yield 10,000

  10. Example 2: the sadistic philanthropist(Schotter, p. 544) The individual needs 20,000 right now for a life-saving operation Outcome Probability Outcome Probability 10,000 0 0.5 0.99 0.5 0.01 15,000 20,000 Expected monetary value: 12,500 Expected monetary value: 200 Most people would choose gamble B

  11. Choice under risk so far • Each choice leads to a probability distribution over outcomes 2. Choice on basis of average monetary payoff 3. Problems with this approach • Risk is not taken into account • Utility is ignored (the sadistic philanthropist) • New theory: choice on basis of expected (= average) utility

  12. Expected utility • 1. People look at average utility • 2.So we cannot make any predictions without • knowing the utility function. • 3. Expected utility allows us to account for • different risk preferences • existence of insurance • 4. It also allows us to have a theory of choice when the outcomes are not money

  13. Utility functions A utility function assigns a number to each possible outcome u(x) is a measure of the satisfaction that the individual receives from outcome x If the outcome x is a number (typically an amount of money) we will usually assume that the corresponding utility can be computed systematically by using a formula Examples:u(x) = x, u(x) = x2, u(x) = x

  14. How to calculate expected utility Probability p(x) Outcome x Utility u(x) = x2 p(x)u(x) 100 1 0.01 10 1 1 0.25 0.25 0 0 0 0.74 1.25 Expected utility

  15. We can evaluate all kinds of choices Outcome x Probability p(x) p(x)u(x) Utility u(x) DVD player 100 1 0.01 teddy bear 10 2.5 0.25 nothing 0 0 0.74 3.5 Expected utility

  16. The case of the sadistic philanthropist now explained! u(x) p(x)u(x) 0 0 A 0 0 0 Expected utility u(x) p(x)u(x) 0 0 B 1 0.01 0.01 Expected utility

  17. Different risk attitudes now explained! Employee or self-employed? Different people may make different choices Employee: £ 25,000 with probability 0.5 and 35,000 with probability 0.5 Self-employed: £50,000 with probability 0.9 and £ 0 with probability 0.1 If u(x) = x0.1 we choose Employee If u(x) = x we choose Self-employed

  18. A few remarks • We need more information now: u(x) • The old theory is a special case withu(x) = x • Cardinal and ordinal utility If expected utility is going to make sense, we need u(x) to becardinal

  19. Ordinal and Cardinal Utility Preferences are represented by utility functions A utility function can be ordinal or cardinal, depending on how much information it contains An ordinal utility function respects the direction of the preferences i.e, most preferred options are assigned a higher value It says nothing about the intensity of preferences

  20. Suppose the consumer has the following preferences: Examples of ordinal utility functions Utility function u has u(x)=10, u(y)=9, u(z)=1 Utility function v has v(x)=100, v(y)=1, v(z)=0 Both are ordinal utility functionsrepresenting these preferences because they respect the individual’s ranking

  21. Cardinal utility functions contain more information • they respect the ranking of the alternatives • they contain information about the intensity • of preferences Cardinal utility functions • In the previous example, if we interpret u and v as cardinal, they cannot refer to the same individual • U implies that a and b are quite close • V implies that a and b are far apart Ordinal representation:almost anything will do Cardinal representation:how constrained are we?

  22. Analogy: the measurement of temperature Celsius Fahrenheit 212 100 100 units 180 units z y 32 0

  23. C F 212 100 y z 32 0 • Scales may differ in the location of the origin or/and the size of the units • These are the only arbitrary elements: knowing them is enough to determine temperature • z = 1.8y + 32 • utility functions: u(x) = a v(x) + b with a > 0 • We can make meaningful statements about • distances between points

  24. Consider three alternatives, x y and z, with Why do we need cardinal utility? How does y compare to an equal chance of x and z? u(x)=10, u(y)=9, u(z)=1 v(x)=100, u(y)=1, u(z)=0 Both u and v are ordinal representations According to u: 0.5u(x)+0.5u(z)= 5.5 < 9 = u(y) According to v: 0.5v(x)+0.5v(z)= 50 > 1 = v(y) No coherent answer to the question!

  25. U(x) U(y) U(z) Consider three alternatives, x y and z, with Constructing our own utility function • Suppose you can choose between • y for sure • x with probability p, z with probability 1-p Which value of p makes you indifferent? Suppose this is p = 1/3 Then, if we set u(x) = 1 and u(z) = 0, U(y)=1/3u(x)+2/3u(z)=1/3 See also Schotter, p.549

  26. The theory of expected utility assumes that people evaluate gambles according to their average utility Conclusion If x is an amount of money, the previous theory of expected monetary value is obtained as a particular case with u(x) = x

  27. Choice under uncertainty so far 1. Each choice leads to a probability distribution 2. Choice on basis of average monetary payoff 3. Problems with this approach • Risk is not taken into account • Utility is ignored (the sadistic philanthropist)

  28. Alternative: expected utility 1. People look at average utility 2. Addresses the two shortcomings 3. Intensity of preferences becomes important 4. Next: how the shape of the utility function determines the attitude towards risk

  29. Different attitudes toward risk Suppose we have two alternative lotteries: A. 75 with probability 1/2, otherwise 25 B. 50 for sure • Note: B is the average money from A for sure • Risk-neutral: indifferent between A and B • Risk-averse: strictly prefers B • Risk-preferring: strictly prefers A

  30. u(x) x u(x) x u(x) x Can we infer risk attitudes from the shape of u(x) ? • Distinguish three cases: • Decreasingmarginal utility • Increasingmarginal utility • Constantmarginal utility

  31. A way to view the lotteries • We can see lottery A as a situation in which • we start from 50 (lottery B!) and then: • With probability 1/2 we win 25 • This generates a utility gain • With probability 1/2 we lose 25 • This generates a utility loss If gain > loss, then the consumer prefers lottery A

  32. Decreasing marginal utility (concave u) u(x) u(75) gain u(50) loss u(25) 0 25 50 75 100 x The sure outcome is better!

  33. 0 25 50 75 100 Increasing marginal utility (convex u) u(x) u(75) gain u(50) loss u(25) x The sure outcome is worse!

  34. 0 25 50 75 100 Constant marginal utility (linear u) u(x) u(75) gain u(50) loss u(25) x It doesn’t matter!

  35. The previous demonstration is only valid when we are choosing between a sure outcome and an equal chance of two other outcomes What if the choice was the following: A. 40 with probability 1/3, otherwise 10 B. 20 for sure There are still utility gains and losses but we cannot compare them directly because they occur with different probability We can still check that risk attitude is determined by the shape of u

  36. Decreasing marginal utility (concave u) u(x) u(40) u(20) u(A) u(10) 0 10 20 40 x u(A) = 2/3 u(10) + 1/3 u(40) As expected, the sure outcome is better!

  37. Increasing marginal utility (convex u) u(x) u(40) u(A) u(20) u(10) 0 10 20 40 x Again u(A) = 2/3 u(10) + 1/3 u(40) As expected, the sure outcome is not preferred

  38. Relation between u and risk attitude • Risk averse Concave u (u’’(.) < 0) • Risk preferring (u’’(.) > 0) Convex u • Risk neutral Linear u (u’’(.) = 0)

  39. The certainty equivalent of a lottery The certainty equivalent of lotteryA is the amount of moneyxe(A) such that the decision maker is indifferent between receiving xe(A) for sure and receiving A

  40. Example (and notation) Lottery C: 25 with probability 0.4 0 with probability 0.6 We can denote C as (25, 0.4; 0, 0.6) The expected monetary value of C, denoted by E(C), is found as

  41. In order to find the certainty equivalent we need the utility function • Suppose u(x)=√x • Then the expected utility of lottery C is • The certainty equivalent is the value x such that u(x) = U(C), or √x = 2. Thus xe(C) = 4

  42. Lottery A: (40,1/3;10,2/3) u(x) u(40) u(A) u(10) 0 10 15 20 40 x Expected monetary value of A is 20 The certainty equivalent of A is 15

  43. Risk averse consumer: xe(A) < E(A) • Risk preferring consumer: xe(A) > E(A) • Risk neutralconsumer:xe(A) = E(A)

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