General Equilibrium Theory Getting Started

General Equilibrium Theory

Getting Started

Main objectives of today’s session:

. Examples of GE models.

• Partial vs. general equilibrium

– definition & tax policy example

• A Robinson Crusoe GE-model

– centralized and decentralized allocations

– normative question: efficiency

– positive question: existence

• Two consumers, two goods (Edgeworth box)

– Pareto efficient-, bargaining-, market allocations

• 2x2x2 GE Model

– technical-, Pareto-, product mix efficiency

– 1st FUN Theorem

• Bonus stuff: Existence of equilibrium

c© Ronald Wendner GE-Intro-1 v3.2

1 Partial vs. General Equilibrium

• competitive (market) equilibrium:

price taking assumption

– partial equilibrium model

– general equilibrium model

• partial equilibrium model

– 2 simplifications: change in price in some market

∗ has no impact on prices in other markets

∗ has no wealth effects for a consumer

∗ considers single market in isolation

• general equilibrium model

– change in price in some market

∗ typically affects prices in other markets

∗ has impact on wealth of (some) consumers

∗ considers market interactions

• often: results from partial equilibrium theory

invalidated by general equilibrium analysis


• consider some change in economic environment (tax,...)

– partial equilibrium analysis: primary effect

ceteris paribus assumption:

“other things being equal...”

– general equilibrium analysis:

also feedback effects

But how can we know that feedback effects are negli-

gible as compared to primary effects?

My argument is: “Generally, we cannot know.”

Convincing Example. Rise in wage tax (→ class)

• partial equilibrium: firms’ owners bear all of the tax

burden

• general equilibrium: workers bear all of the tax

burden


• L commodities (markets): l ∈ {1, ..., L}price vector p = (p1, p2, ..., pL)

demand of commodity l: xlsupply of commodity l: yl

• general equilibrium: yl(p) ≥ xl(p), pl ≥ 0,

[yl(p)− xl(p)] pl = 0 for all l = 1, ...L

→ functions of all prices, not just pl

→ free goods

• prime questions

– existence

– uniqueness

– stability

– efficiency

– relation b/w strategic bargaining equilibrium

and price taking equilibrium


2 An Elementary GE-Model (1x1)

• consumer (Robinson): u(.) = u(x1, x2),

x1 leisure, x2 consumption good (oysters)

u1 > 0, u2 > 0, u11 < 0, u22 < 0, u12 > 0

endowment of time: ω1 (168 hours/week)

• firm: y2 = f (y1)

f ′ > 0, f ′′ < 0, f ′(0)→∞

• find (x1, x2, y1, y2) such that

markets clear: x1 + y1 = ω1, x2 = y2

utility is maximized

• allocation

• centralized allocation

• decentralized allocation


• centralized allocation (social planner)

maxu(x1, x2)

s.t. resource constraint

x2 = y2 = f (y1) = f (ω1 − x1)

→ u1/u2 = f ′ ⇔ MRSx1,x2 = MRT−y1,y2

→ Pareto efficiency

@ other feasible allocation with higher u(.)

for consumer

→ no prices involved

• decentralized allocation

agents optimize independently, as price takers

prices adjust until all markets clear

→ Walrasian equilibrium

(price-, market equilibrium)

- output price = 1 (numeraire)


A. firm’s problem (PMP)

max π = f (y1)− w y1→ [f ′(y1) = w] and [y2 = f (y1)] ⇒ (y∗1, y

∗2)

B. hh’s problem (UMP): hh is price and profit taker

maxu(x1, x2) s.t. x2 + w x1 = wω1 + π∗

maxu(x1, w(ω1 − x1) + π∗)

→ u1/u2 = w ⇒ x∗1

budget: x∗2 = w(ω1 − x∗1) + π∗

C. Results

→ budget line = isoprofit line (→ class)

→ price coordinates firm’s and household decisions (→class)


→ Walras law holds (→ class)

→ market equilibrium is efficient (→ class)

→ existence of equilibrium (→ class)

3 A 2x2 GE model; Edgeworth box

so far: no allocation b/w consumers (i = 1, 2)

• 3 types of allocations

– Pareto efficient (PE) allocations

– bilateral bargaining allocations (BA)

– Walrasian (or market) equilibrium allocations (WE)

• modeling device: Edgeworth box (EB)

initial endowments: generic notation ωli

– ωi = (ω1i, ω2i), i = 1, 2

– aggregate endowments: ωl = ωl1 + ωl2, l = 1, 2

– Edgeworth box: rectangular area

lengths: ω1 = ω11 + ω12, ω2 = ω21 + ω22

– consumption sets of individuals 1,2

– origin of i = 1: lower left angle

– origin of i = 2: upper right angle


• allocations

– initial endowment allocation: ω = (ω11, ω21, ω12, ω22)

– allocation x = (x11, x21, x12, x22)

– feasibility∑2i=1 xli ≤

∑2i=1 ωli, l = 1, 2

– nonwastefulness∑2i=1 xli =

∑2i=1 ωli, l = 1, 2

• indifference curves

– ω & individual rationality

– preferred trades of i: ui(x1i, x2i) ≥ ui(ω1i, ω2i)

– bilateral bargaining

– best bargaining allocations


• Pareto efficient allocations

– definition and ∼i-curves

– PE and prices

– Pareto set and contract curve

– Pareto efficiency vs. equity

→ Let x be a BA. Then x ∈ contract curve.

x satisfies individual rationality + PE

– calculating a PE allocation

maxx11,x21,x12,x22

u1(x11, x21)

s.t.

u2(x12, x22) ≥ u2 (P.1)

x11 + x12 = ω1

x21 + x22 = ω2

Query. Characterize PE allocations for (P1).

Query. Calculate PE allocations for:

ui(.) = x1ix2i, i = 1, 2, ω1 = 2, ω2 = 4.


• Walrasian (market) equilibrium

– (x1, x2, p)

i max ui s.t. Bi, i = 1, 2

both markets clear: xl1 + xl2 ≤ ωl1 + ωl2,

l = 1, 2

– i are price takers

– i’s wealth: wi = p1ω1i + p2ω2i

→ i’s budget line in Edgeworth box

• %i max consumption plans


• price mechanism & WE (x1, x2, p)

→ Let x be a WE allocation. Then x ∈ set of BA.

Specifically, x is PE: MRS11,2 = MRS2

1,2.

Query. Calculate the WE for:

ui(.) = x1ix2i, i = 1, 2, ω1 = (1, 0), ω2 = (1, 4).


4 A 2x2x2 GE model

2 factors – 2 commodities – 2 households

• The economy

– L = 4, outputs l = 1, 2, factors l = 3, 4: yl

– I = 2, initial endowment vectors ωi, i = 1, 2

aggregate endowments: ωl =∑

i ωli, l = 3, 4

– J = 2 firms, each producing one output

firm 1: y1 = f (y3, y4)

firm 2: y2 = g(ω3 − y3, ω4 − y4)π1 = p1 f (y3, y4)− p3y3 − p4y4π2 = p2 g(ω3 − y3, ω4 − y4)− p3(ω3 − y3)−p4(ω4 − y4)

– households

preferences ui(x1i, x2i)

wealth Mi =∑

l plωli +∑

j αijπj

– functions ui, f, g

strictly increasing, strictly concave

partial derivatives w.r.t. i−th or j−th

argument: fi, or gj


4.1 Technical efficiency (TE)

maxy3,y4

f (y3, y4)

s.t. g(ω3 − y3, ω4 − y4) ≥ y2 (1)

f1 = λg1f2 = λg2

}f1f2

=g1g2

(2)

Query.

(i) Set up the Lagrangian and derive (2).

(ii) Give an economic interpretation to (2).

(iii) Why is MRTSf3,4 > MRTSg3,4 not TE? Give a

graphical interpretation using an Edgeworth box.

(1), (2) implicit functions in (y3, y4), with exogenous

variable y2:

H1(y3, y4; y2) = 0 , H2(y3, y4; y2) = 0 . (3)

→ ∃ y3 = h1(y2), y4 = h2(y2)

if Jacobian of (3) exists and its determinant 6= 0

by Implicit function theorem


4.2 PE ⇒ TE

in contrast to 2x2 economy, (y1, y2) not given

L(x11, x21, y3, y4, λ) = u1(x11, x21)

+ λ{u2[f (y3, y4)− x11, g(ω3 − y3, ω4 − y4)− x21]− u2}

∂L∂x11

= 0 ,∂L∂x21

= 0 ⇒ u11u12

=u21u22

Query. Show that λ > 0 and give an interpretation.

∂L∂y3

= 0 ,∂L∂y4

= 0 ⇒ f1f2

=g1g2

Query. Why is TE a necessary condition for PE?

4.3 PE ⇒ Product mix efficiency (PME)

∂L∂y3

= 0 ,∂L∂y4

= 0 ⇒ u21u22

=g1f1

=g2f2

(4)

Claim. g1f1

= g2f2

= MRT1,2

Then, by (4), MRSi1,2 = MRT1,2 (PME).


Product mix efficiency

MRT1,2 = −d y2d y1

along production possibility frontier

as y2 = g(ω3 − y3, ω4 − y4):d y2 = g1 (−1)d y3 + g2 (−1)d y4 ⇒−(d y2)/g2 = [(g1/g2) d y3 + d y4] (*)

as y1 = f (y3, y4):

d y1 = f1 d y3 + f2 d y4 ⇒(d y1)/f2 = [(f1/f2) d y3 + d y4] (**)

from (TE), (f1/f2) = (g1/g2) ⇒ RHS (*) = RHS (**)


−d y2g2

=d y1f2⇔ g2

f2= −d y2

d y1= MRT1,2 (5)

(4) in (5):

MRSi1,2 =g2f2

= −d y2d y1

= MRT1,2 . (6)

Query. PE ⇒ PME (MRSi1,2 = MRT1,2). Why is an

allocation for which MRSi1,2 > MRT1,2 not PE?

4.4 A WE allocation is PE

• utility maximization

MRSi1,2 = p1/p2 ⇒ MRS11,2 = MRS2

1,2 (7)

• profit maximization

∂π1∂y3

= p1 f1 − p3 = 0 (8)

∂π1∂y4

= p1 f2 − p4 = 0 (9)

∂π2∂y3

= −p2 g1 + p3 = 0 (10)

∂π2∂y4

= −p2 g2 + p4 = 0 (11)


• (8)+(9) and (10)+(11) yield:

f1f2

=p3p4

=g1g2

(TE)

• (8)+(10) and (9)+(11) yield:

MRT12 =g1f1

=p1p2

=g2f2

• considering (7):

MRT12 =p1p2

= MRSi1,2 (PME)

→ First fundamental theorem of welfare economics

• BA ⊂ contract curve (→ PE)

• WE allocation ⊂ BA (→ PE)


5 Bonus stuff: Existence of equilibrium

• Setup

L commodities (l = 1, ..., L), p = (p1, ..., pL)

I consumers (i = 1, ..., I)

J firms (j = 1, ..., J)

generic notation: xli, ylj

aggregate excess demand:

zl(p) =∑I

i=1 xli(p)−∑J

j=1 ylj(p)−∑I

i=1 ωli,

l = 1, ..., L

• Assumptions

(1) Walras law:∑L

l=1 pl zl(p) = 0

(2) zl(p) continuous and HD0 in p

– diminishing MRS, MRT

(quasiconcavity of f (.), u(.))

• Price simplex

P = {p ∈ RL | pl ≥ 0,∑L

l=1 pl = 1}- discuss Walras law, L = 2, L = 3


• Corollary to Brower’s fixed-point theorem:

Let f (p) : P → P be continuous. Then there exists

p∗ ∈ P such that: f (p∗) = p∗. That is, p∗ is a fixed-

point.

• Walrasian equilibrium (WE): price vector and allo-

cation: (p∗, x, y) such that:

(i) hh maximize utility,

(ii) firms maximize profits,

(iii) all markets clear:

zl(p∗) ≤ 0, p∗l ≥ 0, zl(p

∗) p∗l = 0

Theorem 1 Suppose z(p) is well-defined for all p ∈P . If assumptions (1) and (2) are satisfied, then there

exists a Walrasian equilibrium.

Proof (sketch). Price adjustment function:

fk(p) =max[0, pk + zk(p)]∑Ll=1 max[0, pl + zl(p)]

Notice: (1)f (p) cont., (2) f (p) : P → P thus: ∃ p∗:

fk(p∗) = p∗k =

max[0, p∗k + zk(p∗)]∑L

l=1 max[0, p∗l + zl(p∗)]≥ 0


• Case 1: p∗k = 0. Then zk(p∗) ≤ 0 → WE p∗

• Case 2: p∗k > 0. Let

λ ≡ 1∑Ll=1 max[0, p∗l + zl(p∗)]

> 0 :

p∗k = fk(p∗) = λ [p∗k + zk(p

∗)]

⇔(1− λ) p∗k = λ zk(p∗)

⇔(1− λ) p∗k zk(p∗) = λ zk(p

∗)2

⇔(1− λ)∑

k∈Case2

p∗k zk(p∗) = λ

∑k∈Case2

zk(p∗)2 (∗)

Notice: for Case 1: p∗k zk(p∗) = 0.

Walras law:

0 =∑

k∈Case1

p∗k zk(p∗) +

∑k∈Case2

p∗k zk(p∗)

⇒∑

k∈Case2

p∗k zk(p∗) = 0 .

Thus RHS of (*) = 0⇒ zk(p∗) = 0 for all k ∈ Case 2. ||


General Equilibrium Theory Getting Started

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