2. Equlibrium and Efficiency 　　　　（均衡と効率性）

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2.1 Introduction

  competition and efficiency（競争と効率性）   Smith’s invisible hand   model of competitive economy（競争経済）

  combine independent decision-making of consumers and firms into a complete model of the economy

  existence of competitive equilibrium   competitive equilibrium（競争均衡）=efficient   coordinating role played by prices

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2.2 Economic Models   Why employ models to make predictions?

  experiments（実験） --> difficult   economic system is too large and too complex   formal model = logical consistency（論理的整合性）

  Many different models partial or general （部分と一般）   trade-off between specified and general

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2.3 Competitive Economies

  competitive（競争的）   individual agents do not consider their actions to

have any effect on prices   prices（価格） = fixed or parametric   negligible in size relative to the market

  prices   values --> relative values of different goods   signals --> coordinating the decisions of agents

  economic efficiency（経済的効率性）

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  secondary feature in the model   symmetric information（対称的情報）   uncertainty（不確実性） = all agents are equally

uninformed   any agent has no informational advantage

  two forms of competitive model   exchange economy（交換経済）   production economy（生産経済）

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2.3.1 Exchange Economy   trade of commodities between two parties for

mutual advantage（相互利益）   two-consumer, two-good exchange economy → all results extend to large numbers   endowment = initial stock of the goods 　（初期保有）↓　　　(constant) exchange (market prices = exchange rates ) 　　　↓　　competitive = price-taking（価格受容） 　　　　　　　　　　　　　　　　　　　　　 　　　　　　　　　　　　　　　behavior desirable consumption plans（消費計画）

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consumer h   endowment

: h’s initial stock of good i   prices p1, p2

: h’s consumption plan   budget constraint（予算制約）

  preference（選好） = utility function（効用関数）

  marginal utility（限界効用） is positive for both goods   indifferent curve（無差別曲線） has convex（凸） shape

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Edgeworth box

  feasible plan（実行可能計画）

  rectangle（長方形）   length   southwest corner: consumer 1’s origin（原点）   northeast corner: consumer 2’s origin

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2

1

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Equilibrium   budget constraints（予算制約）

  consumer 1: relative to the southwest corner   consumer 2: relative to the northeast corner   same endowment point（初期保有点）   gradient（傾き）

  equilibrium prices（均衡価格）   utility maximization for both consumers 　　（効用最大化）   supply = demand（供給=需要）

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2

1

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2

1

good 1: excess demand （超過需要） p1 goes up

good 2: excess supply （超過供給） p2 goes down

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2

1

good 1: excess demand （超過供給） p1 goes down

good 2: excess supply （超過需要） p2 goes up

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2

1

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equilibrium （均衡）

consumers’ demand functions   max of utility subject to budget constraint   demand   demand = supply (market clearing：市場清算)

  equilibrium   the prices lead to a budget line on which the

indifference curves has a point of common tangency（共通の接点）

  number of prices, number of independent equations（独立の方程式）

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  only relative prices（相対価格） matter   budget constraint always passes through the

endowment point（初期保有点）   gradient（傾き） is determined by the price ratio   demand and supply depend on the relative

pricebut not on the absolute values

homogeneous of degree 0（０次同次）   any scaling up or down will also be equilibrium

prices   price normalization（価格の正規化）   numeraire（基準財） = unit of account

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  Walras’s law   interdependence（相互依存） between the two

equilibrium equations   excess demand （超過需要）

  total value of excess demands, for any prices

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  Walras’s law (continued)（ワルラス法則）   if Z1=0, then Z2=0   if Z2=0, then Z1=0 since p1Z1+p2Z2=0   equilibrium in one market implies equilibrium in

the other   choose good 1 as the numeraire (p1=1) and plot

the excess demand for good 2 as a function of p2   if the ED of 2 is zero, then the ED of 1 is zero ---> equilibrium

  extension of many consumers and goods   number of equilibrium conditions

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2.3.2 Production and Exchange   addition of production to exchange economy

  initial endowments (e.g. labor)（初期保有）   consumption goods produced from ini. end.   intermediates（中間財）: produced from ini. end.

and input for another   Arrow-Debreu economy

  model of competitive economy（競争経済）   producer (firm): production technology（生産技術）   consumer: preference, initial endowments shareholdings（株式保有） of the firms

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firm (producer)   production set（生産集合）

  available production technology = knowledge inputs ---> outputs　　　　　　　　　　（知識）   Yj : firm j’s production set   input: negative（負≦0） = subtraction from the

stock of that good available for consumption   output: positive（正≧0）   production plan:   prices:   profit:

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-2

3

Good 2

Good 1

Yj

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Process of profit maximization   isoprofit curve（等利潤線） 0 inner product > 0 (acute angle)   π

Process of profit maximization

  profit maximization（利潤最大化）   y* : tangency point between the highest

attainable isoprofit curve and production set   the higher the isoprofit curve, the bigger the

profit (northeast shift gives more profit)

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Good 2

Good 1

π > 0

π = 0

π < 0

p2

p1

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economy with n goods

  m firms（企業）   production plan（生産計画）

  chosen to maximize profits s.t. production set   supply function 　　（供給関数）   level of profit

  aggregate supply negative or positive

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private ownership economy

  H consumers（消費者）:   initial endowments（初期保有） ω h　　

  shareholdings in firms 　　（株式保有率）   budget constraint（予算制約）

  change in prices ---> endowment dividend income（配当所得）

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  demand function（需要関数）   aggregate demand 　　（集計された需要）   equilibrium demand = supply (not fixed)   excess demand=0

  demand (supply) is determined by relative prices ---> Zi is homogeneous of degree zero

  normalization, numeraire（正規化、基準財）

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2.4 Efficiency of Competition   efficiency（効率性）

  if more cannot be achieved, then the outcome is efficient

2.4.1 Single Consumer   individual preferences = social preferences   best outcome = first-best

  no constraint on policy choice   single firm & single consumer diagram = Robinson Crusoe economy

  Robinson acts as a firm carrying out production and as a consumer of the product of the firm

  1) Robinson = social planner for coordination   2) profit maximization & utility maximization 　　　　（利潤最大化）　　　　　　（効用最大化）

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Good 2

Good 1

1

p1

π

good 2 =numeraire

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Good 2

Good 1

1

p1

π

good 2 =numeraire

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  budget constraint（予算制約）

or where is net consumption（純消費）   utility maximization --->   equilibrium for the economy --->

  net consumption = supply   profit maximization = utility maximization   efficient（効率的）

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Good 2

Good 1

1

p1

π

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Good 2

Good 1

1

p1

π

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Good 2

Good 1

1

p1

π

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characterization of first-best allocation   tangency point（接点） between the two

curves   marginal rate of substitution（限界代替率）

MRS12=U1/U2   marginal rate of transformation（限界変形率）

MRT12(=MC1/MC2)   MRS12=MRT12   marginal value（限界価値） = marginal cost（限界費用） at the first-best

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  the market achieves efficiency 　　（市場は効率性を達成する）

  utility maximization（効用最大化） ---> MRS12=p1/p2   profit maximization（利潤最大化） ---> MRT12=p1/p2   competitive equilibrium（競争均衡） MRS12=p1/p2 = MRT12

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constant return to scale（規模に関して収穫一定）   efficient production frontier（生産フロンティア） =straight line through the origin   only equilibrium : zero profit（ゼロ利潤）

  positive ---> infinitely large profit   isoprofit curve with π =0 equal to efficient PF   two further implications

  equilibrium price ratio is determined by the zero-profit condition and is independent of demand

  since the profit income of the consumer is zero, the budget constraint passes through the origin

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Good 2

Good 1

1

p1

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2.4.2 Pareto-Efficiency 　　　（パレート効率性）   more than one consumer   Pareto-efficiency

  no unexploited economic gains   consider whether it is possible to undertake a

reallocation of resources（資源の再配分） that can benefit at least one consumer without harming any other

  if possible, there would exist unexploited gains   if not, the initial position is Pareto-efficient   efficient allocation of resources

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  feasibility（実行可能性）: consumption, production, ini. endowment x = y + ω

  Definition 1: A feasible consp. allocation is Pareto-efficient if there does not exist an alternative feasible allocation such that

i. Allocation gives all consumers at least as much utility as .

ii. Allocation gives at least one consumer more utility than .   is Pareto-efficient if there is no that can make

someone better off without making anyone worse off

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2.4.3 Efficiency in an Exchange Economy   Two Theorems of Welfare Economics   First Theorem（厚生経済学の第1定理）

  a competitive equilibrium is Pareto efficient   Second Theorem（厚生経済学の第2定理）

  any Pareto-efficient allocation can be decentralized as a competitive equilibrium

  encouragement of competition   Edgeworth box diagram

  a is not a Pareto-efficient allocation   b is Pareto-preferred to a

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1

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  Pareto-efficient allocation c   beginning at c, any change in the allocation must

lower the utility of at least one of the consumers   tangency between the two indifference curves   MRS121=MRS122

  Pareto-efficient allocation（パレート効率的配分） is not unique

---> contract curve（契約曲線）

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1 contract curve （契約曲線）

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  competitive equilibrium（競争均衡）   a price line through the initial endowment point ω

that is tangential to both indifference curves（無差別曲線） at the same point

  competitive equilibrium = Pareto-efficient   utility maximization ---> MRS12h=p1/p2   competitive equilibrium MRS121=p1/p2 = MRS122

  Theorem 1 (First Theorem of Welfare Economics) The allocation of commodities at a competitive equilibrium is Pareto-efficient.

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2

1

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  Second Theorem   whether any chosen Pareto-efficient allocation can

be made into CE by choosing suitable location for the initial endowment

  possible if indifference curves are convex（凸）   a budget line through the Pareto-ef.allo. exists   consumers’ potimal choice   decentralization（分権化）

  Theorem 2 (Second Theorem of Welfare Economics) With convex preferences, any Pareto-efficient allocation can be made a competitive equilibrium.

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2.4.4 Extension to Production   production ---> supply is variable consumer’s income differs   Robinson Crusoe economy: same   more than one consumer

  utility maximization ---> MRSijh=pi/pj   profit maximization ---> MRTijm=pi/pj   competitive equilibrium efficiency in consumption MRSijh=pi/pj = MRSijh’ efficiency in production MRTijm=pi/pj = MRTijm’   MRSijh= MRTijm --->First Theorem

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  Second Theorem   W: feasible output plans w=y+ω

  Z: quantities for Pareto-improvement   if both are convex, a common tangent exists   total consumption

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Price line

Feasible set

W Z

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2.5 Lump-Sum Taxation   decentralization mechanism Second Theorem   lump-sum transfers（一括移転） 　　（一括固定税、一括補助金）   lump sum

  no change in a consumer’s behavior can affect the size of the transfer

  differentiated from income or commodity taxes（所得税、物品税）

  redistributive instrument（再分配の手段）

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2

1

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  initial endowment   Pareto efficient e   equilibrium prices   income at the initial point   budget constraint   lump-sum taxes（一括（固定）税）

  consumer 1 pays a tax   consumer 2 pays a negative tax   pair of taxes = transfer of endowment   efficiency in achieving distributional objectives

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