Lec12 最近の更新履歴 yyasuda's website

Lecture 12: General Equilibrium

Advanced Microeconomics I

Yosuke YASUDA

National Graduate Institute for Policy Studies

November 19, 2013

From Partial to General Equilibrium

Partial Equilibrium Model

◮ 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

◮ 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: the special case of the general equilibrium model where all of the economic agents are consumers, i.e., there is no production.

2 / 15

Competitive Equilibrium (1)

The allocation with the price vector constitute a competitive (Walrasian) equilibrium 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 for good j is the real-valued function,

zj(p) =^X

i∈I

xⁱ_j(p, p · eⁱ) −^X

i∈I

eⁱ_j.

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

Competitive Equilibrium (2)

Lemma If ui satisfies the asumptions 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 u_i is continuous, strongly increasing, and strictly quasiconcave for all i ∈ I. Then, for all p≫ 0, the excess demand function satisfies,

1. Continuity: z(·) is continuous at p. 2. Homogeneity: z(λp) = z(p) for all λ > 0. 3. Walras’ law: p · z(p) = 0.

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

4 / 15

Walras’ Law

Walras’ law states that the value of aggregate excess demand is identically 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) − peⁱ
= 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.

Competitive Equilibrium: Example

✝

☎

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^b2y¹⁻₂ ^b ω1 = (1, 0), ω2 = (0, 1)

Solve a competitive equilibrium price ratio p_y/p_x.

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

p_xx1= aω1 = ap_x ⇒ x1 = a pxx2= bω2= bpy ⇒ x2 = b^py

px

.

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

p_x ^{= 1 ⇒} py

p_x ⁼ 1 − a

b ^.

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

6 / 15

Homogeneity and Relative Prices

Aggregate excess demand function is homogeneous of degree 0.

⇒ Let us normalize prices and express demands in terms of the following relative prices:

p_i = _Pn^pˆi

j=1^pˆj

where ˆpi is the original (absolute) price of good i.

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

Sⁿ⁻¹= (

p_{∈ R}ⁿ₊| Xn

i=1

pi = 1 )

Existence of Competitive Equilibrium (1)

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

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

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

8 / 15

Existence of Competitive Equilibrium (2)

Applying Brouwer fixed-point theorem, we obtain the following. Lemma g : Sⁿ⁻¹ → Sⁿ⁻¹ has a fixed point, p^∗ such that

p^∗ = g(p^∗). That is, p^∗_i = ^p

∗

i ^{+ max(0, zi(p}

∗))

1 +^Pn_j=1max(0, z_j(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 p_j = 0 for some j)?

Existence of Competitive Equilibrium (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 ui is 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.

10 / 15

First Welfare Theorem (1)

Lemma Let (x, p) be a competitive equilibrium. If u_i(y_i) > u_i(x_i) for some bundles y_i, then

p· y_i > p · x_i.

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

p· yi ≥ p · xi.

The following theorem which shows the efficiency of competitive market is called the first fundamental theorem of welfare economics, or first welfare theorem.

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

First Welfare Theorem (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 · xi for all i ∈ S, and p· yi > p · xi for 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.

12 / 15

First Welfare Theorem (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 ui is 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.

Second Welfare Theorem (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, if e^xis Pareto efficient, then e^x is a competitive equilibrium allocation for the price vector p after redistribution of initial endowments to any feasible allocation ee, such that

p_{· ee}ⁱ _{= p · e}xⁱ for all i ∈ I.

The existence of equilibrium price vector supporting x is

guaranteed by separating hyperplane theorem. 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.

14 / 15

Second Welfare Theorem (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.

◮ 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 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.