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講義 5：一般均衡理論（発展編）

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11_日

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From Pure Exchange to Production Economy | 交換経済から生産経済へ

exchange economy where all agents are consumers.

Now we expand our description of the economy to include production as well as consumption.

✄

✂Rm We will find that most of the important properties of competitive market✁ systems uncovered earlier continue to hold.

In a general equilibrium model with production:

◮ _{効用最大化} Consumers, 1, 2, ..., I, act to maximize utility subject to their budget constraints.

◮ _{利潤最大化} Firms, 1, 2, ..., J , seek to maximize profit.

◮ _競争市場 Both consumers and firms are price takers.

◮ _私有経済 Firms’ profits are shared among individual consumers.

Firm Behavior | 企業行動 (1)

Def Let Y denote the aggregate production possibilities set (_{集計された生} 産可能集合), defined as the sum of the individual production possibility sets:

Y =^X

j∈J

Yj= {y | y =^X

j∈J

yjwhere yj∈ Yj}.

Let y(p) be the aggregate net supply function (_{集計された純供給関数}) defined as the sum of the individual net supply functions:

y(p) =^X

j∈J

yj(p).

where yj(p) associates to each vector p the profit-maximizing net output vector at those prices.

✄

✂Rm Y represents all production plans that can be achieved by some✁ distribution of production among J individual firms.

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Firm Behavior | 企業行動 (2)

The next theorem says that if each firm maximizes profits, then aggregate profits must be maximized. Conversely, if aggregate profits are maximized, then each firm’s profits must be maximized.

Thm An aggregate production plan y maximizes aggregate profit, if and only if each firm’s production plan yjmaximizes its individual profit for all j ∈ J. The theorem implies that there are two equivalent ways to construct the aggregate net supply function:

1. Add up the individual firms’ net supply functions.

2. Add up the individual firms’ production sets and then determine the net supply function that maximizes profits on this aggregate production set.

→_{証明は補論を参照．}

Consumer Behavior | 消費者行動 (1)

Def Consumer i’s share of the profits of firm j is represented by 0 ≤ θij≤ 1 for all i ∈ I and j ∈ J

where X

i∈I

θij= 1 for all j ∈ J.

That is, each firm is completely owned by individual consumers. Then, the budget constraint of each consumer i becomes:

px_i≤ pe_i+^X

j∈J

θijpy_j(p).

✄

✂^{Rm Given e}✁ ^{iand θij}, the budget set (予算集合) is characterized by p alone. Hence, consumer i’s demand function can be written as a function of p, denoted by xi(p).

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Consumer Behavior | 消費者行動 (2)

Def Let x(p) =^P_i∈Ixi(p) be the aggregate (consumer) demand function (_{集計された需要関数}), the sum of the individual demand functions.

Def The aggregate excess demand function (集計された超過需要関数^{) is} z(p) = x(p) − y(p) − e

where e is the aggregate supply from consumers, e =^P_i∈Iei.

◮ Walras’ law (_{ワルラス法則}) holds in the production economy for the same reason that it holds in the pure exchange economy.

◮ Each consumer satisfies her budget constraint, so the economy as a whole has to satisfy an aggregate budget constraint (_{集計された予算制約}).

Thm (Walras’ law) If uiis strictly increasing for all i ∈ I, then pz(p) = 0 must hold for all p.

Competitive Equilibrium | 競争均衡

The production economy is represented by (ui, ei, θij, Yj)i∈I,j∈J.

Def An allocation-price pair (x, y, p) where p ≫ 0 is called a competitive (Walrasian) equilibrium (_競争均衡), if z(p) = 0.

The next theorem guarantees the existence of equilibrium (均衡の存在). Thm Consider a production economy (ui, ei, θij, Yj)i∈I,j∈J. Suppose that the following conditions are satisfied:

◮ Utility function uiis continuous, strongly increasing, and strictly quasiconcave for all i ∈ I.

◮ _{0 ∈ Yj⊆ R}ⁿ_{, Yj}is closed, bounded and strongly convex.

◮ _{y +}^P

i∈I^ei≫ 0 for some aggregate production vector y ∈ Y .

Then, there exists at least one price vector p^∗≫ 0 such that z(p^∗) = 0. That is, the competitive equilibrium price exists.

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Competitive Equilibrium: Example | 競争均衡：例 (1)

✄

✂Ex Consider a Robinson Crusoe economy (✁ ロビンソン・クルーソー経済⁾ where a consumer has the following Cobb-Douglas utility function for consumption x and leisure R and initial endowments e:

u(x, R) = x^aR^1−a

The consumer is endowed with one unit of labor (労働)/leisure (余暇) and the firm has a production function (生産関数) x = L^1/2. Let the price of x be normalized by 1. Then, solve a competitive equilibrium price of labor, w^∗.

Answer Let us first solve the profit maximization problem: maxL ^L

1/2_{− wL}

From the first order condition, ¹₂L^−1/2− w = 0, we obtain L(w) = ¹

(2w)^{2, xs(w) =} 1

2w^{, π(w) =} 1 4w^.

Competitive Equilibrium: Example | 競争均衡：例 (2)

Since the firm’s profits are distributed to the consumer, she solves: maxx,R^x

a_R1−as.t. x + wR = w + ¹ 4w

By the property of the Cobb-Douglas utility function, we obtain xd(w) = a(w + ¹

4w^{), R(w) =} 1 − a

w ^{w +} 1 4w^.

In a competitive equilibrium, the supply and demand for x coincide, xs(w^∗) = xd(w^∗) ⇔ ¹

2w^{∗ = a}

„ w^∗+ ¹

4w

«

⇒ w^∗=^{„ 2 − a} 4a

«1/2

.

Note that, by Walras’ law, the labor market also clears.

◮ _{財市場の均衡 ⇐⇒ 労働市場の均衡}

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First Welfare Theorem | 厚生経済学の第一定理

Thm If each uiis strictly increasing on Rⁿ+, then every competitive equilibrium is Pareto efficient (パレート効率的).

Proof Suppose not, and let (x^′, y^′) be a Pareto dominating allocation. Then, since consumers are maximizing utility,

px^′_i> pe_i+^X

j∈J

θijpy_j

must hold for all i ∈ I. Summing over consumers,

p^X

i∈I

x^′_i>^X

i∈I

pe_i+^X

j∈J

py_j.

Feasibility of x^′implies

p ^X

j∈J

y^′j+^X

i∈I

ei

!

>^X

i∈I

pe_i+^X

i∈J

py_j⇔^X

j∈J

py^′_j>^X

j∈J

py_j,

which contradicts profit maximization by firms.

Second Welfare Theorem | 厚生経済学の第二定理

Thm Suppose the conditions stated in the existence theorem are satisfied. Let (x^∗, y^∗) be a feasible Pareto efficient allocation. Then, there are income transfers (所得移転), T1, ..., TI, satisfying^P_i∈ITi= 0, and a price vector p such that for all j ∈ J and for all i ∈ I.

1. y^∗_j maximizes py_js.t. yj∈ Yj.

2. x^∗_i maximizes ui(xi) s.t. px_i≤ pe_i+^P_j∈Jθijpy_j+ Ti.

✄

✂Rm The transfer T✁ i must be set equal to Ti= px^∗_i − pe_i+^X

j∈J

θijpy^∗_j

!

Note that by feasibility of (x^∗, y^∗), X

i∈I

Ti= p^X

i∈I

(x^∗i− ei) − p^X

j∈J

y^∗j= p(x^∗− (e + y^∗)) = 0.

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Interpretation of General Equilibrium Model | 一般均衡モデルの解釈

(i) at a single (_単一の) date ⇒ timeless?

(ii) in a given certain (_確実な) environment ⇒ no uncertainty?

Make the markets for goods (i) over time or (ii) under uncertainty look just like those in the general equilibrium model.

◮ All we have to do is re-interpret the commodity space, R^n. Then, the same formal results as before will follow:

(i) Establish an intertemporal (_通時的な) equilibrium and intertemporally efficient allocation.

(ii) Establish an equilibrium for goods across uncertain events and an efficient allocation of risk bearing (_{リスク負担}).

Incorporate Time and Uncertainty | 時間と不確実性を取り込む

To incorporate Time Let xktdenote the amount of good k consumed at date t. If there are two goods k = 1, 2, and two dates t = 1, 2, then,

◮ A consumption bundle (x11, x12, x21, x22) is a vector of four numbers. That is, there are four distinct goods.

◮ _{⇒ x12}is the amount of good k = 1 consumed at date t = 2.

◮

✄

✂Ex k = {apple, orange} and t = {today, tomorrow}.✁

To incorporate Uncertainty Let xks denote the amount of good k consumed at state (_状態) of nature (/the world) s.

◮ _{⇒ x12}is the amount of good k = 1 consumed at state s = 2.

◮

✄

✂Ex k = {umbrellas, sunscreen} and s = {sunny, rainy}.✁

⇒ Incorporate both via contingent commodities (条件付き財).

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Overlapping Generations Model | 世代重複モデル (1)

Consider an economy with the following with the following structure called overlapping generations (OLG) model (世代重複モデル).

◮ Each period, n agents are born; each lives for two periods.

◮ At any time after the first period, 2n agents are alive: n young agents and n old agents.

◮ Each agent has an endowment of 2 units of consumption when she is born, and

◮ Indifferent between consumption when she is young and old.

Thm There is a competitive equilibrium in this OLG model where pt is non-decreasing in t and every agent consumes all of her endowment when she is young, which is not Pareto efficient.

⇒ Is this violating the first welfare theorem (厚生経済学の第一定理)??

Overlapping Generations Model | 世代重複モデル (2)

Proof Suppose that each member of generation t + 1 transfers one unit of its endowment to generation t. Now generation 1 is better off since it receives 3 unit of consumption in its lifetime. None of the other generations are worse off.

⇒ Pareto improvement (_{パレート改善}) on the original equilibrium!

✄

✂Q Why does the first welfare theorem fail?✁

✄

✂A There are an infinite (✁ 無限の) number of goods:

◮ The value of both the aggregate consumption stream and the aggregate endowment become infinite.

◮ The contradiction in the last step of the proof of the first welfare theorem no longer holds.

⇒ One should be very careful in extrapolating results of models with finite horizons (_有限期間) to models with infinite horizons (_無限期間).

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Uncertainty in Edgeworth Box | エッジワース・ボックス (1)

Consider an economy consisting of two consumers and one good, wheat. The endowments of the agents in wheat depend on the weather: Agent 1 (/2) has an endowment of

◮ _w¹

1 ^(/w21) if the weather is nice (_良い天気) ⇒ good 1.

◮ _w¹

2 ^(/w22) if the weather is bad (悪い天気) ⇒ good 2.

Suppose that consumer i has preferences over the contingent consumption plans that satisfy expected utility hypothesis (期待効用仮説):

Ui(xⁱ1, xⁱ2) = π1ui(xⁱ1) + π2ui(xⁱ2)

where π1(π2) is the objective probability of nice (bad) weather.

Assume that consumers are risk averse (_{リスク回避}), i.e., ui is concave.

✞

✝

☎

Fg Draw the Edgeworth Box. What if an agent is risk neutral (✆ _{リスク中立})?

Uncertainty in Edgeworth Box | エッジワース・ボックス (2)

Analyze the competitive equilibrium in this exchange economy.

✄

✂Rm The marginal rate of substitution (✁ 限界代替率) is constant along the 45 degree line (45度線), i.e., xⁱ1= xⁱ2, since

∂Ui/∂xⁱ₂

∂Ui/∂xⁱ₁ ⁼ π2

π1

dui/∂xⁱ₂ dui/∂xⁱ₁ ⁼

π2

π1

.

When there is no macroeconomic risk, i.e., w₁¹+ w²₁= w₂¹+ w²₂:

◮ The equilibrium must be an allocation on the diagonal (対角線) in which agents’ consumption is independent of the weather.

◮ Both consumers are perfectly insured: xⁱ₁= xⁱ₂.

◮ The price ratio is equal to the ratio of probabilities: ^p_p¹

2 ⁼

π1 π2^.

When there is macroeconomic risk, i.e., w¹₁+ w₁²> w¹₂+ w₂²:

◮ ^p1 p2 ^<

π1

π2 must hold and insurance is imperfect for both agents.

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Contingent Commodities | 条件付き財

Each (contingent) commodity is characterized by

1. _財自体 what it is (its description) 2. _場所 where it is available (its location) 3. 時間 when it is available (its date)

4. _状態 under what condition it is deliverable (the state of the world)

Def A contingent commodity (条件付き財) xktsis a promise (約束) of delivery of a particular good or service k at a particular date t if an uncertain event s actually occurs.

◮ The price is not necessary the price of a definite consumption.

◮ Instead, the price of a contingent commodity, a specific good deliverable if a specified event occurs.

Arrow-Debreu Equilibrium | アロー・デブリュー均衡 (1)

In principle, time/date can be incorporated in the state of nature. Consider an exchange economy with I agents and K goods:

◮ Distinguish two dates: date 0 (ex ante,事前), date 1 (ex post,事後).

◮ There are S mutually exclusive (背反な) state of nature.

◮ At date 0, the future is uncertain. At date 1, each agent observes the realized state and consumes accordingly.

◮ _{Let x}ⁱ

ks be the quantity of good k ∈ K consumed in state s ∈ S by agent i ∈ I.

◮ _{Let w}ⁱ

ks be the initial endowment of good k ∈ K for agent i ∈ I in state s ∈ S.

If a market for each good k and every state s is created at date 0, we have a system of (complete) Arrow-Debreu markets (アロー・デブリュー市場^).

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Arrow-Debreu Equilibrium | アロー・デブリュー均衡 (2)

Let pksbe the system of prices for k ∈ K and s ∈ S. Each consumer i solves the following problem:

max Ui(xⁱ) =^X

s∈S

πsui(xⁱ_s)

s.t. ^X

s∈S

X

k∈K

pksxⁱks≤^X

s∈S

X

k∈K

pkswⁱks

Def An Arrow-Debreu equilibrium (アロー・デブリュー均衡) is a system of prices p^∗_{∈ R}^KS₊ and an allocation (x^∗1, ..., x^∗I) such that

1. Optimization x^∗i is a solution to the above maximization problem for p^∗ for any i = 1, ..., I.

2. Feasibility ^P_i∈Ix^∗i=^P_i∈Iwⁱ.

✄

✂Rm The existence of equilibrium and the fundamental theorems holds under✁ the assumptions similar to the standard model.

Arrow-Debreu Equilibrium | アロー・デブリュー均衡 (3)

Def An allocation (x^∗1, ..., x^∗I) is called

1. ex ante Pareto efficient if there exists no feasible allocation x such that Ui(xⁱ) ≥ Ui(x^∗i) for all i ∈ I, and

Uj(x^j) > Uj(x^∗j) for some j ∈ I.

2. ex post Pareto efficient if there exists no state s and no feasible allocation xs such that

ui(xⁱ_s) ≥ ui(x^∗i_s ) for all i ∈ I, and uj(x^js) > uj(x^∗s^j) for some j ∈ I.

Thm An ex ante Pareto efficient allocation is always ex post Pareto efficient, but the converse is not true.

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Arrow-Debreu Securities | アロー・デブリュー証券

✄

✂Rm The previous theorem implies that re-opening the (slot) markets after the✁ realization of the state s would not lead to any trade since the allocation is ex post Pareto optimal as well.

Def An Arrow-Debreu security (アロー・デブリュー証券) is a contract (契 約) that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state of nature occurs and pays zero in all the other states.

Suppose there exist no market for contingent commodities, but Arrow-Debreu securities for all state s ∈ S are exchanged at t = 0.

◮ Each agent can reallocate her resources among different states of nature by using the securities markets (証券市場).

◮ If the expectations of spot market prices at t = 1 are all correct, the Arrow-Debreu equilibrium allocation is achieved.

◮ Given this perfect foresight (_完全予見), we can organize the economy with S + K markets rather than S × K contingent markets.

【補論】 Firm Behavior: Proof | 企業行動：証明

Proof We only show “⇐”, since the converse is straightforward.

Let {yj}j∈J be a set of profit-maximizing production plans for the individual firms. Suppose that y =^P_j∈Jyjis not (aggregate) profit-maximizing at prices p. This implies that there is some other production plan y^′=^P_j∈Jy_j^′ with y^′_j in Yjthat has higher profits:

X

j∈J

py^′_j= p^X

j∈J

y^′_j> p^X

j∈J

yj=^X

j∈J

py_j.

By inspecting the sums on each side of this inequality, we see that some individual firm j must have higher profits at y^′j than at yj.

✄

✂Rm In a capitalist economy (✁ _{資本主義経済}) or private ownership economy (_私 有経済), consumers own firms and are entitled to a share of the profits.

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【補論】 The Proof of Walras’ Law | ワルラス法則の証明

Proof We expand z(p) according to its definition. pz(p) = p(x(p) − y(p) − e)

= p ^X

i∈I

xi(p) −^X

j∈J

yj(p) −^X

i∈I

ei

!

=^X

i∈I

pxi(p) −^X

j∈J

pyj(p) −^X

i∈I

pei.

Since the budget constraint of every consumer holds with equality, px_i(p) = pe_i+^X

j∈J

θijpy_j(p).

Substituting it into the above equation, we obtain pz(p) =^X

i∈I

pe_i+^X

i∈I

X

j∈J

θijpy_j(p) −^X

j∈J

pyj(p) −^X

i∈I

pei

=^X

j∈J

pyj(p) −^X

j∈J

pyj(p) = 0.

【補論】 Intertemporal Preferences | 通時的な選好

We assume that the consumer has preferences over streams of consumption over time. The following simplifications are common:

Additively Separable (_加法分離) Form

U (c1, ..., cT) =

T

X

t=1

ut(ct)

This form allows different utility function in different period, ut.

The next form assumes the same utility function in each period, but with (exponential) discounting by a discount factor (_割引因子) δ^t.

Time-Stationary (定常的) Form

U (c1, ..., cT) =

T

X

t=1

δ^tu(ct)

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