1. Risk
Chapter 11

2. 11. Risk 11.1 Expected Utility 11.2 Attitudes towards Risk
11.3  Stochastic Dominance◦
11.4  Pareto Efficient Risk Sharing

3. Definition
Risk is a situation where the uncertainty can be captured by objective probabilities.
Measuring probabilities by:
Statistical basis
experimental basis
analytical basis
The most basic decision theory under risk is the expected utility hypothesis developed by John von Neumann and Oskar Morgenstern
the use of the expected utility hypothesis is widespread in game theory as we will see in Chapter 12

4. History of vNM and its Claims
In decision theory, the von Neumann- Morgenstern utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future.
This function is known as the von Neumann- Morgenstern utility function. The theorem is the basis for expected utility theory.
von Neumann
Morgenstern

5. …History of vNM and its Claims
In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function;[1] 
They proved that an agent is (VNM-)rational if and only if there exists a real-valued function u defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of u, which can then be defined as the agent's VNM-utility (it is unique up to adding a constant and multiplying by a positive scalar).
No claim is made that the agent has a "conscious desire" to maximize u, only that u exists.

6. …History of vNM and its Claims
Any individual whose preferences violate von Neumann and Morgenstern's axioms would agree to a Dutch book, which is a set of bets that necessarily leads to a loss.
Therefore, it is arguable that any individual who violates the axioms is irrational.
The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as "rationality is VNM- rationality".p

7. 11.1 Expected Utility 11.1.1 Commodity space
For decision-making under risk the commodity space has to be redefined
Commodity space
Prizes: X = {x1,x2,x3}
Lottery: p = (p1,p2,p3) where p1 + p2 + p3 = 1
The consumer’s decision problem is to choose between different lotteries.
Commodity Space (Δ) is the set of all possible lotteries consisting of any (p1,p2,p3) so long as p1 + p2 + p3 = 1

8. Figure 11.1 Commodity space of lotteries

9. Figure 11.2 The Marschak triangle
(0, 0, 1)
(0.2, 0.5, 0.3)
(1, 0, 0)
Figure 11.2 The Marschak triangle

10. Preferences
We assume that a consumer’s preferences over lotteries, ≿ , satisfy certain assumptions (called ‘axioms’):
vNM1 Reflexivity: for any lottery p in Δ, p ≿ p.
vNM2 Totality: for any two lotteries p and q in Δ, either p ≿ q, or q ≿ p, or both.
vNM3 Transitivity: for any three lotteries p, q and r in Δ, if p≿ q and q ≿ r, then p ≿ r.

11. Preferences
vNM4 Independence: for any three lotteries p, q and r in Δ, if p ≿ q, then for any t in the range 0 < t ≤ 1,
tp + (1 − t)r ≿ tq + (1 − t)r.
vNM5 Continuity: for any three lotteries p, q and r in Δ, if p ≿ q and q ≿ r, then there is some t in the range 0 ≤ t ≤ 1 for which
q ∼ tp + (1 − t)r.

12. Preferences
The first three axioms should be familiar from Chapter 3 as the requirement for regular preferences
The idea of continuity too is fairly intuitive: if p ≿ q ≿ r, then there is some average of the best and worst lotteries that is indifferent to the one in the middle.
The idea of independence is that the ranking between p and q is independent of any third lottery r when this lottery is averaged with p and q in the same manner.

13. Preferences
Example:
p = (0.2, 0.5, 0.3) is at least as good as q = (0.5, 0.3, 0.2), and r = (0.4, 0.4, 0.2). Let t = 1 .
Independence requires that:
1 2 p r = (0.2, 0.5, 0.3) (0.4, 0.4, 0.2) = (0.3, 0.45, 0.25)
be at least as good as
1 2 q r = 1 2 (0.5,0.3.,0.2) (0.4,0.4,0.2) = (0.45,0.35,0.2)

14. Expected Utility Theorem
If a consumer’s preferences over lotteries satisfy these axioms, then:
(a) there is a von Neumann-Morgenstern (vNM) utility function u(x) over the set of prizes, X, and
(b) that the consumer’s preferences over lotteries can be represented by an expected utility (EU) function V, where her utility from lottery p is given by
V(p) = p1u(x1) + p2u(x2) + p3u(x3)

15. ... Expected Utility Theorem
FIRST: This theorem holds for any positive monotonic transformation v of the vNM utility u, so long as v = au + b where a > 0.
SECOND: Because the u(xi) terms are multiplicative constants, the EU function is linear in the probabilities. This implies that the consumer’s preferences over lotteries generate linear indifference curves.

16. ... Expected Utility Theorem
Example:
Given that a consumer’s preferences over lotteries can be represented by the EU function V(p)
Since: p1 + p2 + p3 = 1, we substitute: p2 = 1 – p1 – p3, then:
V(p) = p1u(x1) + (1 − p1 − p3)u(x2) + p3u(x3)
= −p1 [u(x2) − u(x1)] + u(x2) + p3[u(x3) − u(x2)]
To draw an indifference curve, fix the utility level V(p) at 𝑉
𝑝 3 = 𝑉 −𝑢( 𝑥 2 ) 𝑢 𝑥 3 −𝑢( 𝑥 2 ) + 𝑢( 𝑥 2 )−𝑢( 𝑥 1 ) 𝑢 𝑥 3 −𝑢( 𝑥 2 ) 𝑝 1
vertical intercept
slope

17. ... Expected Utility Theorem
The slope of any indifference curve is a constant that depends only the vNM utilities, not on the probabilities of the prizes
Figure 11.3 EU indifference curves

18. 11.2 Attitudes towards Risk
Assume that the set X consists of monetary prizes with:
x1 = $6400, x2 = $9100 and x3 = $10,000.
Initial wealth: x3 = $10,000 but there is a 25 percent chance of a fire which would reduce her wealth by $3600 to x1 = $6400.
We can write this lottery as p = (0.25, 0, 0.75): she will either have $6400 with probability 0.25 or she will be left with $10,000 with probability 0.75.
Then the expected value of her wealth under the lottery p is:
0.75×10, × ×6400 = $9100.

19. ... 11.2 Attitudes towards Risk
Now consider a second lottery q = (0, 1, 0) which yields $9,100 for sure, the same as the expected value of her wealth under lottery p.
When two lotteries yield the same expected value of wealth, we say that they are actuarially fair, so p and q are actuarially fair.
Definition
Risk-averse: A consumer is risk-averse if, given a choice between a lottery p or receiving a the expected value of p for sure, she preferes the latter.

20. ... 11.2 Attitudes towards Risk
If a consumer is risk-averse then her vNM utility fuction is concave and vice versa.
V(q) > V(p) ⇒ u(x2) > 0.25u(x1) u(x3) ≡ 𝑢
Risk-averse
Risk-lover

21. 11.3 Stochastic Dominance
If one lottery is better than another in some sense, we say that the first stochastically dominates the other
There are two important types of stochastic dominance:
first order, and
second order

22. 11.3.1 First order stochastic dominance
A lottery r first order stochastically dominates (FOSD) q if one of the following conditions hold:
r1 <q1 andr1+r2 =q1+q2,(i.e.,r2 >q2);
r1 <q1 andr1+r2 <q1+q2,(i.e.,r3 >q3);or
r1 =q1 andr1+r2 <q1+q2,(i.e.,r3 >q3).
Figure 11.5 First order stochastically dominant lotteries

23. FOSD
The notion of first order stochastic dominance captures the idea that lottery r is better than lottery q because the probability mass shifts from lower-ranking prize(s) towards the better prize(s)
FOSD q
q = (0.3, 0.5, 0.2)
q FOSD

24. 11.3.2 Second order stochastic dominance
Here the set of prizes must be monetary with x3 > x2 > x1
Monetary prizes mean that we can now calculate the expected value of a lottery
A lottery q is a mean-preserving spread of r if both lotteries have the same expected value but q1 > r1 and q3 > r3
SOSD captures the idea that lottery q has the same expected value but is a riskier lottery than r.
A lottery r second order stochastically dominates (SOSD) q if q is a mean-preserving spread of r.

25. Figure 11.7 Second order stochastically dominant lottery
SOSD
x1 = $0,
x2 = $50,
x3 = $100
r = (0.2, 0.6, 0.2)
q = (0.3, 0.4, 0.3)
Figure 11.7 Second order stochastically dominant lottery

26. ...11.3.2 SOSD (in Marschak triangle)
Denote the expected value of the lottery q by x ̄
𝑥 = q1x1 + q2x2 + q3x3
Substituting q2 with 1 − q1 − q3 and simplifying, we obtain
𝑞 3 = 𝑥 𝑥 3 − 𝑥 𝑥 2 − 𝑥 1 𝑥 3 − 𝑥 2 𝑞 1
This is the equation for the iso-expected value line which shows all combinations of q1 and q3 that yield the same expected value of 𝑥

27. ...11.3.2 SOSD (in Marschak triangle)
iso-expected value line
Figure 11.8 Second order stochastic dominance in the Marschak triangle

28. SOSD (Risk aversion)
Risk aversion requires that the vNM utility function u(x) be strictly concave
Rearranging we obtain
slope of the consumer’s EU
slope of iso-expected value lines
Figure 11.9 Strictly concave vNM utility

29. Figure 11.10 Risk-averse EU and second order stochastic dominance
SOSD
When r SOSD q, a risk-averse consumer prefers lottery r over q if, and only if, he EU indifference curves are steeper than the iso-expected value line
EU indifference curves
iso-expected value line
Figure Risk-averse EU and second order stochastic dominance

30. 11.4 Pareto Efficient Risk Sharing
State-contingent claims bundle, (xH, xL), is a lottery over prizes xH and xL with probabilities p and 1 − p.
line of certainty
𝑉 𝑎 𝑥 𝐻 𝑎 , 𝑥 𝐿 𝑎 =𝑝 𝑥 𝐻 𝑎 +(1−𝑝) 𝑥 𝐿 𝑎
If 𝜔a = (676, 196), then:
u(𝜔a)= 𝑝 −𝑝 =𝑝 −𝑝 14 =12𝑝+14
If p=1/2 then u(𝜔a) = 20
𝑀𝑅𝑆 𝑎 = 𝑝 1−𝑝 . 𝑀𝑈 𝐻 𝑎 𝑀𝑈 𝐿 𝑎 = 𝑝 1−𝑝 . 𝑥 𝐿 𝑎 𝑥 𝐻 𝑎
certainty equivalent
Figure State-contingent claims

31. Edgeworth-box (reminder)
Consumer 1
Consumer 2

32. Edgeworth-box (reminder)
Consumer 2
Pareto Efficient Allocations
MRS1 = MRS2
Consumer 1

33. 11.4.1 Risk-averse and Risk-neutral
Consumer a is risk-avers with 𝑢 𝑎 𝑥 𝑎 = 𝑥 𝑎 , so its EU is:
𝑉 𝑎 𝑥 𝐻 𝑎 , 𝑥 𝐿 𝑎 =𝑝 𝑥 𝐻 𝑎 +(1−𝑝) 𝑥 𝐿 𝑎
⇒ 𝑴𝑹𝑺 𝒂 = 𝒑 𝟏−𝒑 . 𝒙 𝑳 𝒂 𝒙 𝑯 𝒂
Consumer b is risk-neutral with 𝑢 𝑏 𝑥 𝑏 = 𝑥 𝑏 , so its EU is:
𝑉 𝑏 𝑥 𝐻 𝑏 , 𝑥 𝐿 𝑏 =𝑝 𝑥 𝐻 𝑏 +(1−𝑝) 𝑥 𝐻 𝑏
⇒ MRSb = p/(1 - p)
Interior contract curve is found by setting MRSa to MRSb
⇒ 𝑥 𝐿 𝑎 = 𝑥 𝐻 𝑎 i.e line of certainty, Ca

34. ...11.4.1 Risk-averse and Risk-neutral
Pareto Efficient
A: Consumer a is as well off as at ω but consumer b is better off.
B: Consumer b is as well off as at ω, but a is better off.
In principle, the consumers could agree to any insurance contract at any point between A and B
Figure Full insurance for consumer a

35. 11.4.2 Two risk-averse consumers
Pareto Efficient
Both consumers are risk-averse
Bilateral negotiations would lead to an allocation where a agrees to transfer some good x to b in the high state in exchange for some transfer from b to a in the low state.
Each consumer is only partially insured
Figure Partial insurance