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講義 2:双対性と厚生評価

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)

201264

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Connection between UMP and EMP | UMP と EMP の関係

There is a strong link between the utility maximization problem (UMP,効用最 大化問題) and the expenditure minimization problem (EMP,支出最小化問題).

Let us first consider the following practice question.

✂Q A consumer has the following indirect utility function:✁ v(p1, p2, ω) = ω

2

2p1p2

.

1. What is the consumer’s Marshallian demand for good 1? 2. What is the expenditure function?

3. What is the consumer’s Hicksian demand for good 1?

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Answers to the Question | 問題への答え

1. Using Roy’s identity, we obtain

x1(p, ω) = −

∂v(p,ω)

∂p1

∂v(p,ω)

∂ω

= −

2pω22 1p2 ω p1p2

= ω

2p1

.

2. By duality (双対性, explained formally later), the indirect utility function can be translated into the following expenditure function:

u=e(p, u)

2

2p1p2

⇔ e(p, u) =p2p1p2u. 3. Using Shephard’s lemma, we obtain

xh1(p, u) = ∂e(p, u)

∂p1

=r p2u 2p1

.

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Dual Problem - Theory | 双対問題 - 理論 (1)

Applying the duality idea to the consumer problem, we can establish the close relationship between the indirect utility and expenditure functions, and between the Marshallian and Hicksian demand functions.

Let v(p, ω) and e(p, u) be the indirect utility function and expenditure function. Then, by definition, the following property must hold:

e(p, v(p, ω)) ≤ ω for all (p, ω) ≫ 0. v(p, e(p, u)) ≥ u for all (p, u) ∈ Rn++× R.

The next theorem demonstrates that under certain conditions on preferences, both of these inequalities must be equalities.

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Dual Problem - Theory | 双対問題 - 理論 (2)

Thm Suppose the consumer’s preference satisfy continuity and monotonicity. Then for all p ≫ 0, ω ≥ 0 and u ∈ R:

e(p, v(p, ω)) = ω (1)

and

v(p, e(p, u)) = u. (2)

Note that, holding prices in both functions constant, we can invert the indirect utility function (note this is strictly increasing in ω) in its income variable. Applying the inverse function, denoted by v−1(p : ·), to both sides of (2), we obtain

e(p, u) = v1(p : u).

Similarly, applying the inverse function (逆関数) of the expenditure function, denoted by e1(p : ·), to both sides of (1), we obtain

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Dual Problem - Theory | 双対問題 - 理論 (3)

Thm Suppose the consumer’s preference is continuous, monotone and strictly convex. Then, we have the following relations between the Hicksian and Marshallian demand functions for p ≫ 0, ω ≥ 0 and u ∈ R, and i = 1, 2, ..., n:

xi(p, ω) = xhi(p, v(p, ω)) and

xhi(p, u) = xi(p, e(p, u)).

1. Marshallian demand at prices p and income ω is equal to the Hicksian demand at those prices and the maximum utility level that can be achieved at those prices and income ω.

2. Hicksian demand at any prices p and utility level u is the same as the Marshallian demand at those prices and an income level equal to the minimum expenditure necessary at those prices to achieve utility level u.

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Slutsky Equation | スルツキー方程式 (1)

When the price of a good declines, there are two conceptually separate reactions: The consumer is expected to substitute the relatively cheaper good for the now relatively more expensive good (= substitution effect,代替効果), and to arrange her purchases of all goods due to the expansion of her effective income, i.e., the budget set (= income effect,所得効果).

Thm Suppose the consumer’s preference is continuous, monotone and strictly convex, and all the relevant functions are differentiable. Let ube the level of utility the consumer achieves at prices p and income ω. Then, for i, j= 1, ..., n,

∂xi(p, ω)

∂pj TE

= ∂x

h i(p, u)

∂pj SE

−xj(p, ω)∂xi(p, ω

)

IE ∂ω

.

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Slutsky Equation | スルツキー方程式 (2)

By duality, xhi(p, u) = xi(p, e(p, u)).

Since this equality holds for all p ≫ 0, differentiating both sides with respect to pj preserves the equality.

∂xhi(p, u)

∂pj

=∂xi(p, e(p, u

))

∂pj

+∂xi(p, e(p, u

))

∂ω

∂e(p, u)

∂pj

.

By duality and Shephard’s lemma, e(p, u) = e(p, v(p, ω)) = ω

∂e(p, u)

∂pj

= xhj(p, u) = xhj(p, v(p, ω)) = xj(p, ω). Substituting these relations into the second equation,

∂xhi(p, u)

∂pj

=∂xi(p, ω

)

∂pj

+∂xi(p, ω

)

∂ω xj(p, ω

).

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Slutsky Equation | スルツキー方程式 (3)

The above equation is called the Slutsky equation (スルツキー方程式), sometimes called the “Fundamental Equation of Demand Theory”, which provides neat analytical expressions for substitution and income effects. When j = i, the Slutsky equation shows the response of the Marshallian demand to a change in own price.

∂xi(p, ω)

∂pi

=∂x

h i(p, u)

∂pi

− xi(p, ω)∂xi(p, ω)

∂ω .

Although substitution effects are not observable, demand theory can provide some strong properties on own-price effects and cross-substitution effects. The first claim says that own-price effects can never be positive.

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Substitution Effects | 代替効果 (1)

Thm Suppose e(p, u) is twice continuously differentiable in p. Then, for i= 1, ..., n,

∂xhi(p, u)

∂pi

≤ 0.

Proof By Shephard’s lemma,

xhi(p, u) = ∂e(p, u)

∂pi

.

Differentiating again with respect to pi, we obtain

∂xhi(p, u)

∂pi

=

2e(p, u)

∂p2i .

The right hand side must be non-positive since the expenditure function is a concave function of p.

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Substitution Effects | 代替効果 (2)

The non-positive own-price effects give us some implication to the response of the Marshallian demand as well.

A good is called normal (正常財) (resp. inferior (下級財)) if consumption of it increases (resp. declines) as income increases, holding prices constant.

A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decline in quantity

demanded (known as Giffen’s paradox), the good must be inferior.

The next theorem says that “cross-substitution effects” are symmetric.

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Substitution Effects | 代替効果 (3)

Thm Suppose that e(p, u) is twice continuously differentiable in p. Then, for i, j= 1, ..., n,

∂xhi(p, u)

∂pj

= ∂x

h j(p, u)

∂pi

.

Proof By Shephard’s lemma,

∂xhi(p, u)

∂pj

=

∂pj

(∂e(p, u)

∂pi

) =

2e(p, u)

∂pj∂pi

, and

∂xhj(p, u)

∂pi

=

∂pi

(∂e(p, u)

∂pj

) =

2e(p, u)

∂pi∂pj

.

By Yong’s theorem (ヤングの定理), we complete the proof:

2e(p, u)

∂pj∂pi

=

2e(p, u)

∂pi∂pj

.

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How to Measure Welfare Change | 厚生の変化をどうはかるか?

When the economic environment or market outcome changes, a consumer may be made better off (改善) or worse off (悪化). Economists often want to measure how consumers are affected by these changes, and have developed several tools for the assessment of welfare (厚生).

The obvious measure of the welfare change involved in moving from (p0, ω0) to (p1, ω1) is just the difference in indirect utility:

v(p1, ω1) − v(p0, ω0).

If the utility difference is positive, then the policy change is worth doing, at least as far as this consumer is concerned.

If it is negative, the policy change is not worth doing.

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Consumers’ Surplus | 消費者余剰

Suppose that the price of some good moves from p0to p1while the prices of other goods and initial wealth remain unchanged.

Def The classical measure of welfare change is consumers’ surplus (CS,消費 者余剰), which is the area below the Marshallian demand curve and above market price. The change of CS is defined as

∆CS := CS(p0, ω) − CS(p1, ω) = Z p0

p1

x(p, ω)dp.

This is simply the area to the left of the Marshallian demand curve between p0 and p1.

Although CS is intuitive and simple, it is an exact measure of welfare change only in special circumstances.

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Beyond Consumers’ Surplus | 消費者余剰を超えて

Depending on how to quantify utility changes, we have two different measures which are better than CS.

Def The compensating variation (CV,補償変分) and equivalent variation (EV,等価変分) are defined as follows:

v(p1, ω+ CV ) = v(p0, ω), v(p0, ω− EV ) = v(p1, ω).

EV uses the current prices as the base and asks what income change at the current prices would be equivalent to the proposed change in terms of its impact on utility.

CV uses the new prices as the base and asks what income change would be necessary to compensate the consumer.

That is, EV (resp. CV ) requires to keep a consumer’s utility constant before (resp. as a result of) a price change.

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Compensating Variation | 補償変分

Using the definitions of CV and expenditure function, CV can be written by e(p1, v(p0, ω)) = e(p1, v(p1, ω+ CV ))

= ω + CV

⇒ CV = e(p1, v(p0, ω)) − e(p0, v(p0, ω)) By Shepard’s lemma, we obtain

CV = e(p1, v(p0, ω)) − e(p0, v(p0, ω))

= Zp1

p0

∂e(p, v(p0, ω))

∂p dp=

Z p1 p0

qh(p, v(p0, ω))dp.

This is simply the area to the left of the Hicksian demand curve between p0 and p1, when the target utility level is v(p0, ω).

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Equivalent Variation | 等価変分

Similarly, EV can be expressed by

e(p0, v(p1, ω)) = e(p0, v(p0, ω− EV )) = ω − EV

⇒ EV = e(p1, v(p1, ω)) − e(p0, v(p1, ω))

= Z p1

p0

qh(p, v(p1, ω))dp.

This is simply the area to the left of the Hicksian demand curve between p0 and p1, when the target utility level is v(p1, ω).

The absolute value of ∆CS is always between that of CV and EV . These three measures coincide if and only if there is no income effect, for instance, when the utility function is quasi-linear (準線形): u(x1, x2) = f (x1) + x2.

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Exchange Economy | 交換経済

Next, we introduce the welfare measures that do not rely on any quantitative assessment.

Consider an exchange economy (交換経済) with I people and n goods where all of the economic agents are consumers and production is absent.

Let e = (e1, ...,eI) denote the economy’s initial endowment (初期保有) vector, where ei= (ei1, ..., ein) denotes i’s initial endowment.

Define an allocation (配分) as a vector, x = (x1, ...,xI), where xi= (xi1, ..., xin) denotes i’s consumption bundle according to the allocation.

The set of feasible allocations is this economy is given by F(e) = {x |X

i∈I

xi=X

i∈I

ei}.

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Pareto Efficiency | パレート効率性 (1)

A situation is called Pareto efficient (パレート効率的) if there is no way to make someone better off without making someone else worse off.

That is, there is no way to make all agents better off.

To put it differently, each agent is as well off as possible, given the utilities of the other agents.

This central welfare notion in Economics is formally defined as follows.

Def A feasible allocation, x ∈ F (e), is (strongly) Pareto efficient if there is no other feasible allocation, y ∈ F (e) such that

ui(yi) ≥ ui(xi) for all i ∈ I, and ui(yi) > ui(xi) for at least one i ∈ I.

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Pareto Efficiency | パレート効率性 (2)

Def A feasible allocation, x ∈ F (e), is weakly Pareto efficient if there is no other feasible allocation, y ∈ F (e) such that

ui(yi) > ui(xi) for all i ∈ I

It is straightforward that an allocation that is (strongly) Pareto efficient is also weakly Pareto efficient.

In general, the reverse is not true. However, under some additional weak assumptions, the reverse implication is true.

Thm Suppose that preference relations are continuous and monotonic. Then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient.

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Block and Core | ブロックとコア

Let S ⊂ I denote a coalition (subset) of consumers.

Def A coalition S blocks (ブロックする) x ∈ F (e), if there is an allocation (among S) y such that

X

i∈S

yi=X

i∈S

ei,

ui(yi) ≥ ui(xi) for all i ∈ S, and ui(yi) > ui(xi) for at least one i ∈ S.

That is, an allocation x is blocked whenever some group can do better than they do under x by simply going it alone.

Def The core (コア) of an exchange economy C(e) is the set of all feasible allocations which cannot be blocked by any coalition.

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Edgeworth Box | エッジワース・ボックス

The most useful example of an exchange economy is one in which there are two people and two goods. This economy’s set of allocations can be illustrated in an Edgeworth box (エッジワース・ボックス) diagram.

The length of the horizontal axis measures the total amount of good 1.

The height of the vertical axis measures the total amount of good 2.

Each point in this box is a feasible allocation.

Fg Figures 5.1 and 5.2 (see JR, pp.196-197)✆

✂Q How will agents trade their goods in voluntary exchange?✁

⇒ If they trade both efficiently and in mutually beneficial way, then the allocation must be on the contract curve (契約曲線).

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【補論】 Dual Problem - Example | 双対問題 - 例 (1)

✂Ex A dual turtle problem.✁

1. The maximal distance a turtle can travel in 1 day is 1 km. 2. The minimal time it takes a turtle to travel 1 km is 1 day.

Let M (t) be the maximal distance the turtle can travel in time t.

✂Q What kinds of conditions (on M (t)) are needed for the above statements✁ to be equivalent?

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【補論】 Dual Problem - Example | 双対問題 - 例 (2)

Assume M is “strictly increasing” and “continuous.” Then,

The maximal distance a turtle can travel in tis x is equivalent to

The minimal time it takes a turtle to travel xis t

Proof (⇒): If the maximal distance that the turtle can pass within tis x, and if the minimal time to cover the distance xis strictly less than t, then by strict monotonicity the turtle would cover a distance strictly larger than x. (⇐): If it takes tfor the turtle to cover the distance xand if the turtle passes a larger distance than xin t, then by continuity the turtle will be beyond the distance with strictly less time than t.

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【補論】 General Environmental Changes | 一般的な環境変化

Consider the move from (p0, ω0) to (p1, ω1). Then,

Def The compensating variation (CV) and equivalent variation (EV) are defined as follows:

v(p1, ω1+ CV ) = v(p0, ω0), v(p0, ω0− EV ) = v(p1, ω1).

✂Rm CV and EV can be expressed as follows:✁

e(p1, v(p0, ω0)) = e(p1, v(p1, ω1+ CV )) = ω1+ CV

⇒ CV = e(p1, v(p0, ω0)) − ω1, e(p0, v(p1, ω1)) = e(p0, v(p0, ω0− EV )) = ω0− EV

⇒ EV = ω0− e(p0, v(p1, ω1)).

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【補論】 Pareto Efficiency and Calculus | パレート効率性と計算

Thm A feasible allocation xis Pareto efficient if and only if xsolves the following maximization problems for i = 1, ..., I:

maxx ui(xi) s.t. I

X

h=1

xkh≤ ek k= 1, ..., n uj(xj) ≤ uj(xj) for all j 6= i.

Proof (⇐) Suppose xsolves all maximization problems but xis not Pareto efficient. This means that there is some allocation x where someone i is strictly better off. But then xcannot solve the problem for i, a contradiction. (⇒) Suppose xis Pareto efficient, but it does not solve one of the problems. Instead, let x solve that particular problem. Then x makes one of the agents strictly better off without hurting any other agents, a contradiction.

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【補論】 Social Welfare Function | 社会厚生関数 (1)

Def A social welfare function (社会厚生関数) W : RI→ R is a hypothetical scheme for ranking potential allocations of resources based on the private utilities they provide to individuals:

Social Welfare = W (u1(x1), · · · , uI(xI))

Assume that W is increasing in each of its arguments. Then we immediately obtain the following theorem.

Thm If xsolves the following (social welfare) maximization problem, then xis Pareto efficient.

maxx W(u1(x1), · · · , uI(xI))

s.t.

I

X

h=1

xkh≤ ek for k = 1, ..., n.

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【補論】 Social Welfare Function | 社会厚生関数 (2)

Imposing additional assumptions, we can completely characterize Pareto efficient allocations by the maximization problem of weighted average of individual utilities.

Thm If ui is an increasing and concave function for all i ∈ I. Then, x is Pareto efficient if and only if xsolves the following maximization problem for some (λ1,· · · , λI) ∈ RI+\ {0}.

maxx I

X

i=1

λiui(xi)

s.t.

I

X

h=1

xkh≤ ek for k = 1, ..., n.

✂Q What happens if a consumer’s utility function is not concave?✁

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