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講義 3：意思決定理論

上級ミクロ経済学^—財務省理論研修

安田 洋祐

政策研究大学院大学（GRIPS）

2012_年6_月7_日

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Preferences | 選好

To construct a model of individual choice, the notion of preferences (選好) plays a central role in economic theory, which specifies the form of consistency or inconsistency in the person’s choices.

We view preferences as the mental attitude of an individual toward alternatives independent of any actual choice.

◮ We require only that the individual make binary (二項の) comparisons, that is, that she only examine two choice alternatives at a time and make a decision regarding those two.

◮ For each pair of alternatives in the choice set X, the description of preferences should provide an answer to the question of how the agent compares the two alternatives.

◮ Considering questionnaire (_{アンケート}) R, we formulate the consistency requirements necessary to make the responses “preferences”.

Questionnaire R | アンケート ^R

R(x, y) for all x, y ∈ X, not necessarily distinct:

✄

✂Q Is x at least as preferred as y? Tick one and only one of the following two✁ options:

1. Yes (or, x is at least as good as y): x % y. 2. No (or, x is strictly worse than y): x y.

Def Preferences (R) on a set X is a binary relation % on X satisfying the following two axioms.

Axiom 1: Completeness (完備性) For any x, y ∈ X, x % y or y % x.

Axiom 2: Transitivity (推移性):

For any x, y, z ∈ X, if x % y and y % z, then x % z.

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Remarks on the Axioms | 公理に関する注意

Completeness formalizes the notion that the individual can make comparisons, that is, that she has the ability to discriminate and the necessary knowledge to evaluate alternatives. It says the individual can examine any two distinct alternatives.

Transitivity gives a very particular form to the requirement that the individual choices be consistent. Although we require only that she be capable of comparing two alternatives at a time, the axiom of transitivity requires that those pairwise comparisons be linked together in a consistent way.

✄

✂Rm The money pump argument when transitivity is violated.✁

→推移性を満たさない個人からはお金をいくらでも吸い取ることができる！

Utility Representation | 効用表現

✄

✂Rev Function U : X →✁ R represents (表現する) the preference % if for all x and y ∈ X, x % y if and only if U (x) ≥ U (y). If the function U represents the preference relation %, we refer to it as a utility function and we say that % has a utility representation (_効用表現).

✄

✂Q Under what conditions do utility representations exist?✁

Thm If % is a preference relation on a finite set X, then % has a utility representation with values being natural numbers.

Proof There is a minimal (resp. maximal) element (an element a ∈ X is minimal (resp. maximal) if a - x (resp. a % x) for any x ∈ X) in any finite set A⊂ X. We can construct a sequence of sets from the minimal to the maximal and can assign natural numbers according to their ordering.

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Continuous Preferences | 連続な選好

To guarantee the existence of a utility representation over consumption set, i.e., an infinite subset of_Rⁿ, we need some additional axiom.

Def A preference relation % on X is continuous (_連続, Axiom 3) if {xⁿ} (a sequence of consumption bundles) with limit x satisfies the following two conditions for all y ∈ X.

1. if x ≻ y, then for all n sufficiently large, xⁿ≻ y, and 2. if y ≻ x, then for all n sufficiently large, y ≻ xⁿ.

The equivalent definition of continuity is that the “at least as good as” and

“no better than” sets for each point x ∈ X are closed. This axiom rules out certain discontinuous (不連続な) behavior and guarantees that sudden preference reversals do not occur.

✞

✝

☎

Fg Figures 1.2 and 1.3 (see JR, pp.9)✆

Continuous Utility | 連続な効用

Given axioms 1-3, we can establish the existence of the (continuous) utility function.

Thm Assume that X is a convex subset ofRⁿ. If % is a continuous

preference relation on X, then % is represented by a continuous utility function.

Here are two remarks on continuity.

1. If % on X is represented by a continuous function U , then % must be continuous.

2. The lexicographic preferences (辞書的選好) are not continuous.

Thm The lexicographic preference relation %Lon [0, 1] × [0, 1], i.e., (a1, a2) %L(b1, b2) if a1> b1or both a1= b1and a2≥ b2, does not have a utility representation.

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Existence of Solutions to Consumer Problems | 消費者問題の解の存在

Thm If % is a continuous preference relation, then all consumer problems have a solution.

Proof Since the budget set is convex, we can apply the first theorem in the previous slide to establish that the preferences are represented by a continuous utility function.

Then, by the Weielstrass theorem, there exists a maximum (and minimum) value of continuous functions if the domain is a compact (that is, closed and bounded) set and a range isR. Since every budget set is compact and a utility function is continuous, there must exist a consumption bundle which gives a maximum utility value, a solution of the consumer problem.

Decision under Uncertainty | 不確実性下の意思決定

We have so far not distinguished between individual’s actions and consequences, but many choices made by agents take place under conditions of uncertainty. This lecture discusses such a decision under uncertainty, i.e., an environment in which the correspondence between actions and consequences is not

deterministic (確定的) but stochastic (確率的).

◮ To discuss a decision under uncertainty, we extend the domain of choice functions. The choice of an action is viewed as choosing a “lottery” (くじ) where the prizes are the consequences.

◮ An implicit assumption is that the decision maker does not care about the nature of the random factors but only about the distribution (_分布) of consequences.

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Lotteries | くじ (1)

We consider preferences and choices over the set of “lotteries.”

◮ Let S be a set of consequences (prizes). We assume that S is a finite set and the number of its elements (= |S|) is S.

◮ A lottery p is a function that assigns a nonnegative number to each prize s, where^P_s∈Sp(s) = 1 (here p(s) is the objective probability (客観確率) of obtaining the prize s given the lottery p).

◮ Let α ◦ x ⊕ (1 − α) ◦ y denote the lottery in which the prize x is realized with probability α and the prize y with 1 − α.

◮ Denote by L(S) the (infinite) space containing all lotteries with prizes in S. That is, {x ∈_R^S₊|P xs= 1}.

◮ We will discuss preferences over L(S).

Lotteries | くじ (2)

We impose the following three assumptions on the lotteries.

1. 1 ◦ x ⊕ (1 − 1) ◦ y ∼ x: Getting a prize with probability one is the same as getting the prize for certain.

2. α◦ x ⊕ (1 − α) ◦ y ∼ (1 − α) ◦ y ⊕ α ◦ x: The consumer does not care about the order in which the lottery is described.

3. β◦ (α ◦ x ⊕ (1 − α) ◦ y) ⊕ (1 − β) ◦ y ∼ (βα) ◦ x ⊕ (1 − βα) ◦ y: A consumer’s perception of a lottery depends only on the net probabilities of receiving the various prizes.

The first two assumptions appear to be innocuous.

The third assumption sometimes called “reduction of compound lotteries (_複 合くじ)” is somewhat suspect.

◮ There is some evidence to suggest that consumers treat compound lotteries different than one-shot lotteries.

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St Petersburg Paradox | セントペテルスブルグのパラドックス (1)

The most primitive way to evaluate a lottery is to calculate its mathematical expectation (数学的期待値), i.e., E[p] =^P_s∈Sp(s)s.

Daniel Bernoulli first doubt this approach in the 18th century when he examined the St Petersburg paradox (セントペテルスブルグのパラドックス^).

✄

✂Ex St Petersburg Paradox✁

A fair coin is tossed until it shows heads for the first time. If the first head appears on the k-th trial, a player wins $2^k. How much are you willing to pay to participate in this lottery?

✄

✂Rm The expected value of the lottery is infinite:✁ 2

2⁺ 2² 2^{2 +}

2³

2³ + · · · = 1 + 1 + 1 + · · · = ∞.

St Petersburg Paradox | セントペテルスブルグのパラドックス (2)

The St Petersburg paradox shows that maximizing your dollar expectation may not always be a good idea. It suggests that an agent in risky situation might want to maximize the expectation of some “utility function” with decreasing marginal utility:

E[u(x)] = u(2)¹ 2^{+ u(4)}

1 4^{+ u(8)}

1 8^{+ · · ·,} which can be a finite number.

✄

✂Q Under what kinds of conditions does a decision maker maximizes the✁ expectation of some “utility function”?

✄

✂Rm By utility theory, we know that for any preference relation defined on the✁ space of lotteries that satisfies continuity, there is a utility representation U : L(S) →R, continuous in the probabilities, such that p % q if and only if U(p) ≥ U (q).

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Expected Utility Theory | 期待効用理論 (1)

We will use the following two axioms to isolate a family of preference relations which have a representation by a more structured utility function.

◮ Independence Axiom (I, _{独立性公理}): For any p, q, r ∈ L(S) and any α∈ (0, 1), p % q ⇔ α ◦ p ⊕ (1 − α) ◦ r % α ◦ q ⊕ (1 − α) ◦ r.

◮ Continuity Axiom (C,_{連続性公理}): If p ≻ q ≻ r, then there exists α∈ (0, 1) such that

q∼ [α ◦ p ⊕ (1 − α) ◦ r].

Thm Let % be a preference relation over L(S) satisfying the I and C. There are numbers (v(s))s∈Ssuch that

p% q⇔ U (p) =^X

s∈S

p(s)v(s) ≥ U (q) =^X

s∈S

q(s)v(s).

Expected Utility Theory | 期待効用理論 (2)

Sketch of the proof Let M and m be a best and a worst certain lotteries in L(S). When M ∼ m, choosing v(s) = 0 for all s we have^P_s∈Sp(s)v(s) = 0 for all p ∈ L(S).

Consider the case that M ≻ m. By I and C, there must be a single number v(s) ∈ [0, 1] such that

v(s) ◦ M ⊕ (1 − v(s)) ◦ m ∼ [s]

where [s] is a certain lottery with prize s, i.e., [s] = 1 ◦ s. In particular, v(M ) = 1 and v(m) = 0. I implies that

p∼ (^X

s∈S

p(s)v(s)) ◦ M ⊕ (1 −^X

s∈S

p(s)v(s)) ◦ m.

Since M ≻ m, we can show that p% q⇔^X

s∈S

p(s)v(s) ≥^X

s∈S

q(s)v(s).

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vNM Utility Function | ノイマン - モルゲンシュテルン効用関数 (1)

Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function (ノイマン-モルゲンシュテルン効用関数).

✄

✂Q How can we construct the vNM utility function?✁

Let si(∈ S), i = 1, ..., K be a set of consequences and s1, sK be the best and the worst consequences. That is, for any i,

[s1] % [si] % [sK].

Then, construct a function v : S → [0, 1] in the following way: v(s1) = 1 and v(sK) = 0, and

[sj] ∼ v(sj) ◦ [s1] ⊕ (1 − v(sj) ◦ [sK] for all j.

By continuity axiom, we can find a unique value of v(sj) ∈ [0, 1].

vNM Utility Function | ノイマン - モルゲンシュテルン効用関数 (2)

✄

✂Q To what extent, vNM utility function is unique?✁

The vNM utilities are unique up to positive affine transformation (_{アフィン変} 換), i.e., multiplication by a positive number and adding any scalar, and are not invariant to arbitrary monotonic transformation.

Thm Suppose % is a preference relation defined over L(S) and let v(s) be the vNM utilities representing the preference relation. Then, defining w(s) = αv(s) + β for all s (for some α > 0 and some β), the utility function W(p) =^P_s∈Sp(s)w(s) also represents %.

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vNM Utility Function | ノイマン - モルゲンシュテルン効用関数 (3)

Proof For any lotteries p, q ∈ L(S), p % q if and only if X

s∈S

p(s)v(s) ≥^X

s∈S

q(s)v(s).

Now, the followings hold. X

s∈S

p(s)w(s) =^X

s∈S

p(s)(αv(s) + β) = α^X

s∈S

p(s)v(s) + β. X

s∈S

q(s)w(s) =^X

s∈S

q(s)(αv(s) + β) = α^X

s∈S

q(s)v(s) + β.

Thus,^P_s∈Sp(s)v(s) ≥^P_s∈Sq(s)v(s) holds if and only if P

s∈S^{p(s)w(s) ≥}

P

s∈Sq(s)w(s) (for α > 0).

Allais Paradox | アレのパラドックス (1)

Many experiments reveal systematic deviations from vNM assumptions. The most famous one is the Allais paradox (_{アレのパラドックス}).

✄

✂Ex Allais paradox✁ Choose first the between

L1= [3000] and L2= 0.8 ◦ [4000] ⊕ 0.2 ◦ [0] and then choose between

L3= 0.5 ◦ [3000] ⊕ 0.5 ◦ [0] and L4= 0.4 ◦ [4000] ⊕ 0.6 ◦ [0]. Note that L3= 0.5 ◦ L1⊕ 0.5 ◦ [0] and L4= 0.5 ◦ L3⊕ 0.5 ◦ [0]. Axiom I requires that the preference between L1and L2be the same as that between L3and L4. However, a majority of people express the preferences L1≻ L2and L3≺ L4, violating the axiom.

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Allais paradox | アレのパラドックス (2)

Assume L1≻ L2but α ◦ L ⊕ (1 − α) ◦ L1≺ α ◦ L ⊕ (1 − α) ◦ L2. (In our example of Allais paradox, α = 0.5 and L = [0].)

Then, we can perform the following trick on the decision maker:

1. Take α ◦ L ⊕ (1 − α) ◦ L1.

2. Take instead α ◦ L ⊕ (1 − α) ◦ L2, which you prefer (and you pay me something...).

3. Let us agree to replace L2 with L1 in case L2realizes (and you pay me something now...).

4. Note that you hold α ◦ L ⊕ (1 − α) ◦ L1. 5. Let us start from the beginning...

This argument may make the independence axiom looks somewhat reasonable (and Allais paradox unreasonable).

Risk Aversion | リスク回避 (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 (_単調), i.e., a > b implies [a] ≻ [b].

Def The individual is said to be

◮ risk averse (リスク回避的) if [E(p)] ≻ p or u(E(p)) > u(p)

◮ risk neutral (リスク中立的) if [E(p)] ∼ p or u(E(p)) = u(p)

◮ risk loving (リスク愛好的) if [E(p)] ≺ p or u(E(p)) < u(p) for any non-degenerated lottery p where u(p) =^P_s∈Spsu(s).

Thm 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 | リスク回避 (2)

◮ The certainty equivalent (確実性等価) of a lottery p, denoted by CE(p), is a prize satisfying [CE(p)] ∼ p.

◮ The risk premium (リスクプレミアム) of p is the difference P(p) := E(p) − CE(p).

◮ The preference relation %1is “more risk averse” than %2if 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 | 絶対的リスク回避度 (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 (正規化) is needed.

Depending on the way of normalization, there are at least two reasonable measures of risk aversion.

Def The first measure is called the (Arrow-Pratt measure of) absolute risk aversion (_{絶対的リスク回避度}), defined by

r(x) = −^u

′′(x) u^′(x)^.

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Absolute Risk Aversion | 絶対的リスク回避度 (2)

The next proposition gives a rationale for this measure.

Thm (Pratt’s Theorem) Let u1and 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(·) is the absolute risk aversion of ui.

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.

Constant Absolute Risk Aversion | 絶対的リスク回避度一定

Thm 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.

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Relative Risk Aversion | 相対的リスク回避度

Def The second measure is called the (Arrow-Pratt measure of) relative risk aversion (相対的リスク回避度), 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.

Thm 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(·) is the relative risk aversion of ui.

【補論】 Questionnaire P | アンケート ^P

✞

✝

☎

P(x, y) For all distinct x and y in the set X. How do you compare x and y?✆ Tick one and only one of the following three options:

1. I prefer x to y, or x is strictly preferred (_{強く選好される}) to y: x ≻ y 2. I prefer y to x, or y is strictly preferred to x: y ≻ x

3. I am indifferent, or x is indifferent (無差別である) to y: x ∼ y

Note that we implicitly assume that the elements in X are all comparable, and ignore the intensity of preferences.

A legal answer to the questionnaire P can be formulated as a function f which assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y. That is,

f(x, y) = 8

>>

><

>>

>: x≻ y y≻ x x∼ y

.

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【補論】 Preference P | 選好 ^P (1)

Preferences are characterized by axioms (_公理) that are intended to give formal mathematical expression to fundamental aspects of choice behavior and attitudes toward the objects of choice.

The following basic axioms are (almost) always imposed.

Def Preferences (P ) on a set X are a function f that assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y so that for any three different elements x, y and z in X, the following two properties hold:

1. No order effect: f (x, y) = f (y, x). 2. Transitivity:

2.1 if f (x, y) = x ≻ y and f (y, z) = y ≻ z, then f (x, z) = x ≻ z, and 2.2 if f (x, y) = x ∼ y and f (y, z) = y ∼ z, then f (x, z) = x ∼ z.

【補論】 Preference P | 選好 ^P (2)

The first property requires the answer to P (x, y) being identical to the answer to P (y, x), and the second requires that the answer to P (x, y) and P (y, z) are consistent with the answer to P (x, z) in a particular way.

✄

✂Ex Non-preference relation✁

For any x, y ∈R, f(x, y)(= f(y, x)) = x ≻ y if x ≥ y + 1 and f(x, y) = x ∼ y if |x − y| < 1. This is not a preference relation since transitivity is violated. For instance, suppose x = 1, y = 1.8, z = 2.6. Then,

f(x, y) = x ∼ y and f (y, z) = y ∼ z, but f (x, z) = z ≻ x, which violates transitivity (2-2).

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【補論】 Equivalence of the Two Preferences | 2 つの選好の同値性

We can translate one formulation of preferences to another by the following mapping (bijection). Note that completeness guarantees “x y and y  x” never happen.

◮ f(x, y) = x ≻ y ⇔ x % y and y x.

◮ f(x, y) = y ≻ x ⇔ y % x and x y.

◮ f(x, y) = x ∼ y ⇔ x % y and y % x.

In our lectures, we take the second definition, i.e., preference (R), and denote x≻ y when both x % y and y x, and x ∼ y, when x % y and y % x.

Def A preference (R) is called a preference relation (_選好関係).

【補論】 Revealed Preferences | 顕示選好

Important difference between choice (demand) and preferences or utility is that the former is observable while the latter cannot be.

We may want to develop the theory which is based on the observable choice behaviors, not on preferences or utility.

◮ We say that the preferences % (fully) rationalize the demand function x if for any (p, ω) the bundle x(p, ω) is the unique % best bundle within B(p, ω).

◮ We say that a is revealed to be better than b, if there is (p, ω) so that both a and b are in B(p, ω) and a = x(p, ω).

✄

✂Q What are general conditions guaranteeing that a demand function x(p, ω)✁ can be rationalized?

→ Present two axioms of revealed preferences.

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【補論】 Weak Axiom of Revealed Preferences | 顕示選好の弱公理

Def (Weak Axiom) The weak axiom of revealed preferences (WA) is a property of choice function which says that it is impossible that a be revealed to be better than b and b be revealed to be better than a. That is,

if px(p^′, ω^′) ≤ ω and x(p, ω) 6= x(p^′, ω^′), then p^′x(p, ω) > ω^′.

✞

✝

☎

Fg Figure 2.3 (see JR, pp.92)✆

✄

✂Rm Note that any choice function rationalized by some preference relation✁ must satisfy WA.

【補論】 Weak Axiom ⇒ Law of Demand | 弱公理 ⇒ 需要法則

Thm Let x(p, ω) be a choice function satisfying Walras’s Law and WA. Then,

1. x(·) is homogeneous of degree zero, and

2. if ω^′= p^′x(p, ω), then either x(p^′, ω^′) = x(p, ω) or (p^′− p)(x(p^′, ω^′) − x(p, ω)) < 0.

Proof The proof for 1 is left for the assignment. Assume that

x(p^′, ω^′) 6= x(p, ω). By Walras’s Law and the assumption that ω^′= p^′x(p, ω): (p^′− p)(x(p^′, ω^′) − x(p, ω))

= p^′x(p^′, ω^′) − p^′x(p, ω) − px(p^′, ω^′) + px(p, ω)

= ω^′− ω^′− px(p^′, ω^′) + ω

= ω − px(p^′, ω^′).

By WA the right hand side is less than 0.

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【補論】 Strong Axiom of Revealed Preferences | 顕示選好の強公理

The previous theorem implies that the compensated (Hicksian) demand function y(p^′) = x(p^′,p^′x(p, ω)) satisfies the law of demand (_需要法則), that is, yk is decreasing in pk.

WA is not a sufficient condition for extending the binary relation % (defined from the choice function) into a complete and transitive relation. The following strongercondition than WA is known to be necessary and sufficient.

Def (Strong Axiom) Choice function satisfies the strong axiom of revealed preferences (SA) if for every sequence of distinct bundles x⁰,x¹, ...,x^k, where x⁰is revealed preferred to x¹, and x¹is revealed preferred to x², ..., and x^k−1is revealed preferred to x^k, it is not the case that x^k is revealed preferred to x⁰.

【補論】 Properties on Concave vNM Function | 凹 vNM 関数の性質

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 (_{ジェンセンの不等式}): 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.

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【補問】 Proof of Pratt’s Theorem | プラットの定理の証明 (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 (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 (テイラー展開), u(x + ˆε) ≈ u(x) + ˆεu^′(x) + ^ˆε

2

2^u

′′(x), from which it follows that

E(u(x + ε)) ≈ u(x) +^σ

ε2

2^u

′′(x) (1)

where σ²εis the variance (分散) of random variable ε (note E(ε) = 0).

【補論】 Proof of Pratt’s Theorem | プラットの定理の証明 (2)

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

2^σ

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.✁

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【補論】 Relative Risk Aversion: Proof | 相対的リスク回避度：証明

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 2^x

2σε²

u^′′(x) u^′(x)

⇒ ˆP(ε) = −^σ

ε2

2 xu^′′(x)

u^′(x) ^{= −} σε²

2^rr(x).

✄

✂Rm If the individual has a constant relative risk aversion, then preferences✁ over proportional gambles will not be affected by x.