Lecture 2: Risk Aversion
Advanced Microeconomics II
Yosuke YASUDA
Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp
November 25, 2014
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Properties on Concave vNM Utility Function
Risk aversion is closely related to the concavity of the vNM utility function. Let us recall some basic properties of concave functions:
1 An increasing and concave function must be continuous (but not necessarily differentiable).
2 Jensen Inequality: If u is concave, then for any finite
sequence of α1, α2, ..., αK of positive numbers that sum up to 1, the following inequality must hold:
u
K
X
k=1
αkxk
!
≥
K
X
k=1
αku(xk).
3 If u is twice differentiable, then for any a < c, u′(a) ≥ u′(c), and thus u′′(x) ≤ 0 for all x.
Risk Aversion (1)
We continue to assume that a decision maker satisfies vNM assumptions and that the space of prizes S is a set of real numbers.
s ∈ S is interpreted as “receiving s dollars.”
assume % is monotone, i.e., a > b implies [a] ≻ [b]. Definition 1
The individual is said to be
risk averse if [E(p)] ≻ p or u(E(p)) > u(p) risk neutral if [E(p)] ∼ p or u(E(p)) = u(p) risk loving if [E(p)] ≺ p or u(E(p)) < u(p)
for any non-degenerated lottery p where u(p) =Ps∈Spsu(s). Theorem 2
Let% be a preference on L(Z) represented by the vNM utility function u. The preference relation% is risk averse if and only if u is strictly concave.
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Risk Aversion (2)
The certainty equivalent of a lottery p, denoted by CE(p), is a prize satisfying [CE(p)] ∼ p.
The risk premium of p is the difference P(p) := E(p) − CE(p).
The preference relation %1 is “more risk averse” than %2 if CE1(p) ≤ CE2(p) for all p.
✞
✝
☎
Rm The individual is risk averse if and only if P (p) > 0 for all p.✆
✞
✝
☎
Fg Figures 9.2 and 9.3 (see Rubinstein, pp.112-113).✆ It is often convenient to have a measure of risk aversion. Intuitively, the more concave the expected utility function, the more risk averse the consumer.
Absolute Risk Aversion (1)
Thus, one might think risk aversion could be measured by the second derivative of the expected utility function.
However, the second derivative is not invariant to the positive linear transformation of the expected utility function.
Therefore, some normalization is needed.
Depending on the way of normalization, there are at least two reasonable measures of risk aversion.
Definition 3
The first measure is called the (Arrow-Pratt measure of) absolute risk aversion, defined by
r(x) = −u
′′(x)
u′(x).
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Absolute Risk Aversion (2)
The next proposition gives a rationale for this measure. Theorem 4 (Pratt’s Theorem)
Let u1 and u2 be twice differentiable, increasing, and strictly concave vNM utility functions. Then, the following properties are equivalent.
(i) CE1(p) ≤ CE2(p) for all p.
(ii) the function ϕ, defined by u1(t) = ϕ(u2(t)), is concave. (iii) r1(x) ≥ r2(x) for all x, where ri(x) is the absolute risk
aversion of ui at x.
Note that (i) is the definition of “more risk averse”. The
equivalence between (i) and (iii) means that a decision maker has higher absolute risk aversion if and only if she is more risk-averse.
Proof of Pratt’s Theorem (1)
Sketch of the Proof.
To establish (i) ⇔ (iii), it is enough to show that P is positively related to r. Let ε be a “small” random variable with expectation of zero, i.e., E(ε) = 0. The risk premium P (ε) (at initial wealth x) is defined by
E(u(x + ε)) = u(CE(x + ε)) = u(x − P (ε)).
For any value ˆεof ε (by the second order Taylor series expansion), u(x + ˆε) ≈ u(x) + ˆεu′(x) + εˆ
2
2u
′′(x),
from which it follows that E(u(x + ε)) ≈ u(x) + σ
2 ε
2 u
′′(x) (1)
where σ2ε is the variance of random variable ε (note E(ε) = 0).
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Proof of Pratt’s Theorem (2)
Continued.
On the other hand, the following approximation holds
u(x − P (ε)) ≈ u(x) − P (ε)u′(x), (2)
since P (ε) is small due to ε being “small”. By (1) and (2), P(ε) = −1
2σ
2 ε
u′′(x) u′(x) =
σε2
2 r(x).
That is, the (coefficient of) absolute risk aversion at the level of wealth x, r(x) = −u
′′(x)
u′(x), is twice of the risk premium per unit of variance for small risk ε.
✞
✝
☎
Rm r(x) can serve as a local measure of risk aversion.✆
Constant Absolute Risk Aversion
Definition 5
We say that preference relation % exhibits invariance to wealth if (x + p1) % (x + p2) is true or false independent of x.
Theorem 6
If u is a vNM continuous utility function representing preferences that are monotonic and exhibit both risk aversion and invariance to wealth, then u must be exponential,
u(x) = −ce−θx+ d for some c, θ > 0 and d.
r(x) becomes θ and is therefore constant, i.e., independent of x: invariance to wealth ⇔ constant absolute risk aversion. However, it is commonly observed that each person becomes less risk averse when she has more wealth.
That is, absolute risk aversion is decreasing function of x.
The second measure fixes this problem to some extent. 9 / 14
Relative Risk Aversion (1)
Definition 7
The second measure is called the (Arrow-Pratt measure of) relative risk aversion, defined by
rr(x) = −u
′′(x)x
u′(x) .
This measure turns out to be appropriate measure of evaluating the risk attitude towards the following type of proportional risk: with probability p, a consumer with wealth x will receive a times of her current wealth x
with probability 1 − p she will receive b times of x.
Relative Risk Aversion (2)
Theorem 8
Assume that the assumptions of Pratt’s Theorem holds. Then, for any proportional risk, the decision maker 1 is more risk averse than 2 if and only if rr1(x) ≥ rr2(x) for all x, where rri(x) is the relative risk aversion of ui at x.
If rr(x) is constant, i.e., the individual has a constant relative risk aversion, then preferences over proportional gambles will not be affected by wealth level x.
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Relative Risk Aversion (3)
Sketch of the Proof.
Let ˆP(ε) be a relative risk premium for any proportional risk ε (at the wealth level x), defined by
E(u(x(1 + ε))) = u(x(1 − ˆP(ε))) = u(x − x ˆP(ε)). Note that, by definition of the (absolute) risk premium P (ε),
E(u(x(1 + ε))) = E(u(x + xε)) = u(x − P (xε)). where we set E(xε) = 0. Therefore,
x ˆP(ε) = P (xε) = −1 2x
2σε2
u′′(x) u′(x)
⇒ ˆP(ε) = −σ
2 ε
2
xu′′(x) u′(x) = −
σ2ε
2 rr(x).
Further Properties on vNM Utility Function (1)
Definition 9
We say that p first-order stochastically dominates (FOSD) q, denoted by pD1q, if p % q for any % on L(S) satisfying vNM assumptions as well as monotonicity in money. That is, pD1q if Eu(p) ≥ Eu(q) for all increasing u.
For any lottery p and a number x,
Let G(p, x) =Ps≥xp(s), the probability that the lottery p yields a prize at least as high as x.
Let F (p, x) denote the cumulative distribution function of p, that is F (p, x) =Ps≤xp(s).
The next theorem characterizes FOSD. Theorem 10
pD1q if and only if for all x, G(p, x) ≥ G(q, x), or alternatively, F(p, x) ≤ F (q, x).
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Further Properties on vNM Utility Function (2)
Proof.
We only show (⇐): Let x0< x1 < · · · < xK be the prizes in the union of the supports of p and q. Then, the expected utility attached to p is written by
Eu(p) =
K
X
k=0
p(xk)u(xk) = u(x0)+
K
X
k=1
G(p, xk)(u(xk)−u(xk−1)).
Now, if G(p, xk) ≥ G(q, xk) for all k, then for all increasing u,
Eu(p) = u(x0) +
K
X
k=1
G(p, xk)(u(xk) − u(xk−1))
≥u(x0) +
K
X
k=1
G(q, xk)(u(xk) − u(xk−1)) = Eu(q).
