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講義 4：一般均衡理論（基礎編）

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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Market Economy | 市場経済

In previous lectures, we studied the behavior of individual consumers, describing optimal behavior when market prices were fixed and beyond the agent’s control. We begin to explore the consequences of that behavior when consumers (and firms) come together in markets. First, we consider price and quantity determination in a single market.

In a partial equilibrium (部分均衡) model,

◮ individual consumers and firms determine their demands and supplies for the good in question

◮ all prices other than the good in question are fixed

◮ the market price is adjusted to clear the market.

⇒ In the general equilibrium (一般均衡) model, prices of all goods vary and all markets clear at the same time.

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Perfectly Competitive Market | 完全競争市場

In (perfectly) competitive markets (_競争市場), buyers and sellers are sufficiently large in number to ensure that no single one of them, alone, has the power to influence market price.

⇒ Market price is outside of their control: they are price takers (価格受容者).

A Consumer’s Problem

maxx,z u(x, z) s.t. px + qz = ω

A Firm’s Problem maxy ^{py − c(y)}

where all prices other than p is assumed to remain fixed.

✄

✂Q How can we aggregate (✁ 集計する) individual demand or supply?

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Market Supply Function | 市場供給関数

The (individual firm) supply function (_{個別供給関数}) is straight-forwardly derived from the profit maximization problem. Its first order condition

p = c^′(y)

(and second order condition c^′′(y) ≥ 0) implies that supply function of a competitive firm coincides with its marginal cost curve (限界費用曲線).

The market (or industry) supply function (_{市場供給関数}) is simply the sum of the individual firm supply function. If yj(p) is firm j’s supply function in an industry with J firms, the market supply function is

Y (p) = XJ

j=1

yj(p).

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Aggregation across Consumers | 消費者間での集計

Suppose there are I consumers, each of whom has a demand function for n commodities; Consumer i’s demand function is

xi(p, ωi) = (x¹_i(p, ωi), · · · , xⁿ_i(p, ωi)).

Then, the aggregate demand function (_{集計需要関数}) is defined by

X(p, ω1, · · · , ωI) = XI i=1

xi(p, ωi).

The market (or industry) demand function (_{市場需要関数}) of good x is simply the sum of all consumers’ individual demand for the good:

X(p) = XI

i=1

xi(p)

assuming all the prices other than p as well as incomes fixed.

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Market Equilibrium | 市場均衡

The market supply function measures the total output supplied at any price; the market demand function measures the total output demanded at any price.

Def An equilibrium price (均衡価格) in a partial equilibrium is a price where the amount demanded equals the amount supplied:

XI i=1

xi(p) = XJ j=1

yj(p).

✄

✂Q Why does such a price deserve to be called an equilibrium?✁

✄

✂Rm At the equilibrium, no one would desire to change actions, while at any✁ other price some agent would find it in her interest to unilaterally change its behavior.

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Aggregation across Goods | 財に関する集計

When we analyze the demand for a single good (partial equilibrium study), it would be convenient to aggregate “all other goods”.

A Consumer’s Problem (again) maxx,z u(x, z) s.t. px + qz = ω

✄

✂Q Under what conditions can we study the demand problem for the z-goods✁ as a group, without worrying about how demand is divided among different components of the z-goods?

Suppose that the relative prices of the z-goods remain constant, so that q = P q⁰where P can be interpreted as some price index.

V (P, p, ω) = max

x,z u(x, z) s.t. px + P q⁰z = ω

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Composite Commodity | 複合材

Given the indirect utility function V , we can recover the demand function for the z-goods by Roy’s identity (and envelope theorem):

z(P, p, ω) = −

∂V(P,p,ω)

∂P

∂V(P,p,ω)

∂ω

= q⁰z(p, q, ω).

This shows that z(P, p, ω) is an appropriate quantity index for the z-goods consumption: such z is called composite commodity (_複合財).

Since all prices q move together (by assumption) and the demand function is homogeneous of degree 0, we can write

x = x(P, p, ω) = x(p/P, ω/P ),

which says that the demand for good x depends on the relative price of x to composite good and income divided by price index.

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From Partial to General Equilibrium | 部分から一般均衡へ

◮ Each agent determines his or her demands and supplies for the good in question, given that

◮ All prices of other goods are assumed to remain fixed.

◮ Equilibrium requires that the market in question clears.

General Equilibrium Model (一般均衡モデル)

◮ Each agent determines his or her demands and supplies for all the goods simultaneously, where

◮ _All prices are subject to change.

◮ Equilibrium requires that all markets clear at the same time.

For simplicity, let us consider a pure exchange economy (_{純粋交換経済}): the special case of the general equilibrium model where all of the economic agents are consumers, i.e., there is no production.

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Competitive Equilibrium | 競争均衡 (1)

The allocation with the price vector constitute a competitive/Walrasian equilibrium (_競争/_{ワルラス均衡}) if every consumer maximizes her utility and all markets clear, i.e., supply equal to demand for every good. To state this formally, let us define the excess demand function.

Def The (aggregate) excess demand function (_{超過需要関数}) for good j is the real-valued function,

zj(p) =^X

i∈I

xⁱj(p, p · eⁱ) −^X

i∈I

eⁱj.

The excess demand function is the vector-valued function, z(p) = (z1(p), ..., zN(p)).

◮ _{When zj}(p) > 0, the aggregate demand for good j exceeds its aggregate endowment; there is excess demand for good j.

◮ _{When zj}(p) < 0, there is excess supply of good j.

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Competitive Equilibrium | 競争均衡 (2)

Lemma If ui satisfies the assumptions in the theorem below, then for all p ≫ 0, the consumer’s problem has a unique solution xⁱ(p, p · eⁱ). Moreover, xⁱ(p, p · eⁱ) is continuous in p on Rⁿ++^.

Thm Suppose that utility function ui is continuous, strongly increasing, and strictly quasiconcave for all i ∈ I. Then, for all p ≫ 0, the excess demand function satisfies,

1. Continuity (_連続性): z(·) is continuous at p. 2. Homogeneity (同次性): z(λp) = z(p) for all λ > 0. 3. Walras’ law (ワルラス法則): p · z(p) = 0.

Def An allocation-price pair (x, p) where p ≫ 0 is called a

competitive/Walrasian equilibrium (_競争/_{ワルラス均衡}) if z(p) = 0.

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Walras’ Law | ワルラス法則

Walras’ law states that the value of aggregate excess demand is identically (_恒 等的に) zero at any set of positive prices.

Proof When ui is strongly increasing, each consumer’s budget constraint holds with equality, i.e., pxⁱ(p, p · eⁱ) = peⁱ. Then,

pz(p) = p ^X

i∈I

xⁱ(p, p · eⁱ) −^X

i∈I

eⁱ

!

=^X

i∈I

“pxⁱ(p, p · eⁱ) − pe^i”= 0.

Moreover, if at some set of prices n − 1 markets are in equilibrium, Walras’ law ensures the nth market is also in equilibrium.

Corollary If demand equals supply in n − 1 markets, and pn> 0, then demand must equal supply in the nth market.

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Competitive Equilibrium: Example | 競争均衡：例

✄

✂Ex Consider two agents (1 and 2) and two goods (x and y) exchange✁ economy. Suppose that agents’ utility functions and initial endowments are given as follows:

u1(x1, y1) = x^a₁y₁^1−a, u2(x2, y2) = x^b₂y^1−b₂ ω1= (1, 0), ω2= (0, 1)

Solve a competitive equilibrium price ratio py/px.

Answer Let us first derive the demand function xifor i = 1, 2. Since utility functions are both Cobb-Douglas, we obtain

pxx1= aω1= apx⇒ x1= a pxx2= bω2= bpy⇒ x2= b^py

px

.

Since the excess demand for good x must be 0, x1+ x2= a + b^py

px

= 1 ⇒ ^py px

=^{1 − a} b ^.

Note that by Walras’ law, the excess demand for good y is also 0.

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First Theorem on Welfare Economics | 厚生経済学の第 1 定理 (1)

Lemma Let (x, p) be a competitive equilibrium. If ui(yi) > ui(xi) for some bundles yi, then

p · yi> p · xi.

If i has an increasing utility function and ui(yi) ≥ ui(xi) for some bundle yi, then

p · yi≥ p · xi.

The following theorem which shows the efficiency of competitive market is called the first fundamental theorem of welfare economics (厚生経済学の第 1基本定理), or first welfare theorem.

Thm If utility function uiis increasing for all i ∈ I and (x, p) is a competitive equilibrium, then x is in the core.

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First Theorem on Welfare Economics | 厚生経済学の第 1 定理 (2)

Proof Suppose, contrary to the theorem, that x is not in the core. Then some coalition S ⊂ I can block x. That is, there exist reallocation y among S 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. By Lemma, we obtain

p · yi≥ p · xifor all i ∈ S, and p · yi> p · xifor at least one i ∈ S.

Now adding these inequalities over all individuals in S, X

i∈S

p · yⁱ>^X

i∈S

p · xⁱ=^X

i∈S

p · eⁱ.

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First Theorem on Welfare Economics | 厚生経済学の第 1 定理 (3)

This inequality can be rewritten as X

i∈S

p · yⁱ= p ·^X

i∈S

yⁱ>^X

i∈S

p · eⁱ= p ·^X

i∈S

eⁱ

⇒ p · ^X

i∈S

yⁱ−^X

i∈S

eⁱ

!

> 0,

which contradicts to^P_i∈Syⁱ=^P_i∈Seⁱ.

Thm If utility function uiis increasing for all i ∈ I and (x, p) is a competitive equilibrium, then x is Pareto efficient.

✞

✝

☎

Fg Figure 5.4 (see JR, pp.213)✆

The first welfare theorem claims that a competitive equilibrium allocation is in the core, and is Pareto efficient.

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Second Theorem on Welfare Economics | 厚生経済学の第 2 定理 (1)

Thm Consider an exchange economy with^P_i∈Ieⁱ≫ 0, and assume that utility function uiis continuous, strongly increasing, and strictly quasiconcave for all i ∈ I. Then, any Pareto efficient allocation x is a competitive

equilibrium allocation when endowments are redistributed to be equal to x.

Corollary Under the assumptions of the preceding theorem, ifex is Pareto efficient, thenex is a competitive equilibrium allocation for the price vector p after redistribution of initial endowments to any feasible allocationee, such that

p ·_eeⁱ= p ·_exⁱfor all i ∈ I.

The existence of equilibrium price vector supporting x is guaranteed by separating hyperplane theorem (_{分離超平面定理}). In order to apply this theorem, we need a convex environment, which is much more restrictive than the one needed for the first welfare theorem.

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Second Theorem on Welfare Economics | 厚生経済学の第 2 定理 (2)

The second theorem may look more difficult to understand intuitively than the first theorem, but it is indeed fundamental to interpret decentralized planning in a private property economy (私有経済).

◮ Pareto efficiency of the competitive equilibrium (First theorem) is satisfactory with respect to the efficiency criterion, but it may lead to undesirable income distributions.

◮ The second theorem states: whichever Pareto efficient allocation we wish to decentralize, it is possible to decentralize (分権化する) this allocation as a competitive equilibrium so long as the incomes of the agents are chosen appropriately.

◮ That is, a private property economy can achieve any Pareto efficient allocation so long as the appropriate lump-sup transfers are made.

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【補論】 Properties of Aggregate Demand Function | 集計需要関数の性質

◮ The aggregate demand function clearly satisfies homogeneous of degree 0 in all prices and the vector of buyers’ incomes.

◮ It also becomes continuous whenever individual demand functions are continuous.

Unfortunately, the aggregate demand function in general possesses no interesting properties other than homogeneity and continuity.

◮ For example, aggregate version of Slutsky equation or strong axiom of revealed preference cannot hold.

◮ The properties of aggregate demand function are completely different from those of individual demand function.

✄

✂Q Under what condition the aggregate demand may look as though it were✁ generated by a single “representative (代表的な)” consumer?

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【補論】 Representative Consumer | 代表的消費者 (1)

The next theorem shows that the aggregate demand function possesses nice properties when the consumers’ indirect utility functions take a specific functional form.

Thm The aggregated demand function can be generated by a representative consumer if and only if indirect utility functions of all individual consumers take the following Gorman form (ゴーマン型):

vi(p, ωi) = ai(p) + b(p)ωi.

◮ Note that the ai(p) term can differ across consumers, but the b(p) term is assumed to be identical for all consumers.

◮ Gorman form is imposed on an indirect utility function, not on a (direct) utility function.

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【補論】 Representative Consumer | 代表的消費者 (2)

Proof of if part (⇐) By Roy’s identity, the demand function for good j of consumer i will also take the Gorman form

x^j_i(p, ωi) = α^j_i(p) + β^j(p)ωi

where

α^j_i(p) = −

∂a_i(p)

∂p_j

b(p) ^{, β}

j_{(p) = −}

∂b(p)

∂p_j

b(p)^.

Note that marginal propensity to consume good j

„

=^∂x

j i^(p,ωi)

∂ω_i

« is independent of the income level and constant across consumers.

In other words, income effect is proportional to consumer’s income level, which makes possible to aggregate individual incomes.

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【補論】 Representative Consumer | 代表的消費者 (3)

The aggregate demand for good j then take the form

X^j(p, ω1, · · · , ωI) = − 0

@ XI i=1

∂a_i(p)

∂p_j

b(p) ⁺

∂b(p)

∂p_j

b(p) XI

i=1

ωi

1 A .

This (market) demand function can be generated by the following representative indirect utility function

v(p, ω) = XI

i=1

ai(p) + b(p)ω

where ω =^PI_i=1ωishows the aggregate income of consumers.

✄

✂Q When does indirect utility function takes Gorman form?✁

⇒ A utility function is homothetic (_{ホモセティック}) or quasilinear (_準線形).

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【補論】 Homogeneity and Relative Prices | 同次性と相対価格

Aggregate excess demand function is homogeneous of degree 0.

⇒ Let us normalize (_正規化) prices and express demands in terms of the following relative prices:

pi=Pn^pˆi j=1^pˆj

where ˆpiis the original (absolute) price of good i.

Since the normalized prices pimust sum up to 1, i.e.,^Pn_i=1pi= 1, we can restrict our attention to price vectors belonging to the n − 1 dimensional unit simplex:

Sⁿ⁻¹= (

p ∈ Rⁿ+| Xn

i=1

pi= 1 )

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【補論】 Existence of Competitive Equilibrium | 競争均衡の存在 (1)

Thm If z : Sⁿ⁻¹_{→ R}ⁿis a continuous function that satisfies Walras’ law, pz = 0 for all p ∈ Sⁿ⁻¹, then there exists some p^∗in Sⁿ⁻¹such that z(p^∗) = 0.

To prove the theorem, let us define a function g : Sⁿ⁻¹→ Sⁿ⁻¹by gi(p) = ^{pi+ max(0, zi(p))}

1 +^Pn_j=1max(0, zj(p)) for i = 1, · · · , n. Note that this function gi(p)

◮ is continuous, since z and max function are continuous.

◮ takes a value on Sⁿ⁻¹, since^Pn_i=1gi(p) = 1.

◮ has an economic interpretation: if there is excess demand in some market, then the relative price of that good is increased.

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【補論】 Existence of Competitive Equilibrium | 競争均衡の存在 (2)

Applying Brouwer fixed-point theorem (ブラウワーの不動点定理), we obtain:

Lemma g : Sⁿ⁻¹→ Sⁿ⁻¹has a fixed point, p^∗such that p^∗= g(p^∗). That is,

p^∗i = ^p

∗

i+ max(0, zi(p^∗))

1 +^Pn_j=1max(0, zj(p^∗)) for i = 1, · · · , n.

We now have to show that p^∗is a competitive equilibrium.

◮ Using Warlas’ law, we can show that zi(p^∗) = 0 for all i.

◮ It is common to show the existence of equilibrium by applying a version of fixed-point theorems in Economics.

✄

✂Q Can we establish the existence without assuming continuity of z (especially✁ at the boundary points where pj= 0 for some j)?

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【補論】 Existence of Competitive Equilibrium | 競争均衡の存在 (3)

The excess demand function may not even be well defined on the boundary of the price simplex. However, this discontinuity can be handled by slightly more complicated mathematical argument.

Thm Consider an exchange economy with^P_i∈Ieⁱ≫ 0, and assume that utility function uiis continuous, strongly increasing, and strictly quasiconcave for all i ∈ I. Then, there exists at least one price vector p^∗≫ 0 such that z(p^∗) = 0.

✄

✂Rm The assumption of strongly increasing utilities is somewhat restrictive,✁ but it allows us to keep the analysis relatively simple.

◮ Cobb-Douglas functional form of utility is not strongly increasing on Rⁿ+^.

◮ Competitice equilibrium with Cobb-Douglas preferences is nonetheless guaranteed.

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