1. 12
Uncertainty

2. Uncertainty is Pervasive
What is uncertain in economic systems?
tomorrow’s prices
future wealth
future availability of commodities
present and future actions of other people.

3. Uncertainty In this lecture we introduce
Different attitudes towards risk
Indifference curves for a risk-averse consumer
State-contingent budget constraint
Optimal choice of insurance

4. Preferences Under Uncertainty
Different attitudes towards risk
Risk-aversion
Risk-neutrality
Risk-loving

5. Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2.
U($90) = 12, U($0) = 2.
Expected utility is

6. Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2.
U($90) = 12, U($0) = 2.
Expected utility is

7. Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win $0 with probability 1/2.
Expected money value of the lottery is

8. Preferences Under Uncertainty
EU = 7 and EM = $45.
U($45) > 7  $45 for sure is preferred to the lottery  risk-aversion.
U($45) < 7  the lottery is preferred to $45 for sure  risk-loving.
U($45) = 7  the lottery is preferred equally to $45 for sure  risk-neutrality.

9. Preferences Under Uncertainty12
EU=7 2 $0
$45
$90
Wealth

10. Preferences Under Uncertainty
U($45) > EU  risk-aversion. 12
U($45)
EU=7 2 $0
$45
$90
Wealth

11. Preferences Under Uncertainty
U($45) > EU  risk-aversion. 12
MU declines as wealth
rises.
U($45)
EU=7 2 $0
$45
$90
Wealth

12. Preferences Under Uncertainty12
EU=7 2 $0
$45
$90
Wealth

13. Preferences Under Uncertainty
U($45) < EU  risk-loving. 12
EU=7
U($45) 2 $0
$45
$90
Wealth

14. Preferences Under Uncertainty
U($45) < EU  risk-loving. 12
MU rises as wealth
rises.
EU=7
U($45) 2 $0
$45
$90
Wealth

15. Preferences Under Uncertainty12
EU=7 2 $0
$45
$90
Wealth

16. Preferences Under Uncertainty
U($45) = EU  risk-neutrality. 12
U($45)= EU=7 2 $0
$45
$90
Wealth

17. Preferences Under Uncertainty
U($45) = EU  risk-neutrality. 12
MU constant as wealth
rises.
U($45)= EU=7 2 $0
$45
$90
Wealth

18. Preferences Under Uncertainty
State-contingent consumption plans that give equal expected utility are equally preferred.
Indifference curve for contingent consumption
Now, we have two possible states: “accident” and “no accident”
Contingent consumptions: & Ca
Cna

19. Preferences Under Uncertainty
Cna
Indifference curves: EU1 < EU2 < EU3
EU3
EU2
EU1 Ca

20. Preferences Under Uncertainty
What is the MRS of an indifference curve?
Get consumption c1 with prob. 1 and c2 with prob. 2 (1 + 2 = 1).
EU = 1U(c1) + 2U(c2).
For constant EU, dEU = 0.

21. Preferences Under Uncertainty

22. Preferences Under Uncertainty

23. Preferences Under Uncertainty

24. Preferences Under Uncertainty

25. Preferences Under Uncertainty

26. Preferences Under Uncertainty
Cna
Indifference curves EU1 < EU2 < EU3
EU3
EU2
EU1 Ca
Slope:

27. Uncertainty
QUESTION 1
Convex preferences shown above apply for risk-averse consumer.
Can you show why?

28. Uncertainty
Preferences are convex when for any X, Y and Z where
X Z Y Z
And for every 𝜽 ϵ[𝟎,𝟏]:
𝜽𝑿+(𝟏−𝜽)Y≽𝒁 ~ f ~ f

29. Uncertainty For risk aversion person utility is concave.
Utility is concave when:
𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y)
For all X and Y and for every 𝜽 ϵ[𝟎,𝟏]:

30. Uncertainty f f ~ ~ Now we have X,Y,Z X Z and Y Z
And concave utility U() that represents preferences:
𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y)
≥𝜽𝑼 𝒁 +(𝟏−𝜽)U(Z) ≥𝑼 𝒁
For all X and Y and for every 𝜽 ϵ[𝟎,𝟏]: ~ f ~ f

31. Uncertainty 𝐔(𝜽𝑿+(𝟏−𝜽)Y)≥𝜽𝑼 𝑿 +(𝟏−𝜽)U(Y) ≥𝜽𝑼 𝒁 +(𝟏−𝜽)U(Z) ≥𝑼 𝒁
Means that:
𝑼(𝜽𝑿+(𝟏−𝜽)Y)≥𝑼 𝒁
Where follows:
𝜽𝑿+(𝟏−𝜽)Y≽𝒁
 That means preferences are convex!

32. Insurance
State contingent budget constraint

33. States of Nature Possible states of Nature: “car accident” (a)
“no car accident” (na).
Accident occurs with probability a, does not with probability na ; a + na = 1.
Accident causes a loss of $L.

34. Contingencies
A contract implemented only when a particular state of Nature occurs is state-contingent.
E.g. the insurer pays only if there is an accident.

35. State-Contingent Budget Constraints
Each $1 of accident insurance costs .
Consumer has $m of wealth.
Cna is consumption value in the no-accident state.
Ca is consumption value in the accident state.

36. State-Contingent Budget Constraints
Without insurance,
Ca = m - L
Cna = m.

37. State-Contingent Budget Constraints
Cna
The endowment bundle. m Ca

38. State-Contingent Budget Constraints
Buy $K of accident insurance.
Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K.

39. State-Contingent Budget Constraints
Buy $K of accident insurance.
Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K.
So K = (Ca - m + L)/(1- )

40. State-Contingent Budget Constraints
Buy $K of accident insurance.
Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K.
So K = (Ca - m + L)/(1- )
And Cna = m -  (Ca - m + L)/(1- )

41. State-Contingent Budget Constraints
Buy $K of accident insurance.
Cna = m - K.
Ca = m - L - K + K = m - L + (1- )K.
So K = (Ca - m + L)/(1- )
And Cna = m -  (Ca - m + L)/(1- )
I.e.

42. State-Contingent Budget Constraints
Cna
The endowment bundle. m Ca

43. State-Contingent Budget Constraints
Cna
The endowment bundle. m Ca

44. State-Contingent Budget Constraints
Cna
The endowment bundle. m
Where is the
most preferred state-contingent consumption plan? Ca

45. Choice Under Uncertainty
Q: How is a rational choice made under uncertainty?
A: Choose the most preferred affordable state-contingent consumption plan.

46. State-Contingent Budget Constraints
Cna
The endowment bundle. m
Where is the
most preferred state-contingent consumption plan?
Affordable plans Ca

47. State-Contingent Budget Constraints
Cna
More preferred m
Where is the
most preferred state-contingent consumption plan? Ca

48. State-Contingent Budget Constraints
Cna
Most preferred affordable plan m Ca

49. State-Contingent Budget Constraints
Cna
Most preferred affordable plan m Ca

50. State-Contingent Budget Constraints
Cna
Most preferred affordable plan m
MRS = slope of budget constraint Ca

51. State-Contingent Budget Constraints
Cna
Most preferred affordable plan m
MRS = slope of budget constraint; i.e. Ca

52. Competitive Insurance
Suppose entry to the insurance industry is free.
Expected economic profit = 0.
I.e. K - aK - (1 - a)0 = ( - a)K = 0.
I.e. free entry   = a.
If price of $1 insurance = accident probability, then insurance is fair.

53. Competitive Insurance
When insurance is fair, rational insurance choices satisfy

54. Competitive Insurance
When insurance is fair, rational insurance choices satisfy
I.e.

55. Competitive Insurance
When insurance is fair, rational insurance choices satisfy
I.e.
Marginal utility of income must be the same in both states.

56. Competitive Insurance
How much fair insurance does a risk-averse consumer buy?

57. Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
Risk-aversion  MU(c)  as c .

58. Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
Risk-aversion  MU(c)  as c .
Hence

59. Competitive Insurance
How much fair insurance does a risk-averse consumer buy?
Risk-aversion  MU(c)  as c .
Hence
I.e. full-insurance.

60. “Unfair” Insurance
Suppose insurers make positive expected economic profit.
I.e. K - aK - (1 - a)0 = ( - a)K > 0.

61. “Unfair” Insurance
Suppose insurers make positive expected economic profit.
I.e. K - aK - (1 - a)0 = ( - a)K > 0.
Then   > a 

62. “Unfair” Insurance
Rational choice requires

63. “Unfair” Insurance
Rational choice requires
Since

64. “Unfair” Insurance Rational choice requires Since
Hence for a risk-averter.

65. “Unfair” Insurance Rational choice requires Since
Hence for a risk-averter.
I.e. a risk-averter buys less than full “unfair” insurance.