Chapter Ge 1

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3 examples of GE:

pure exchange (Edgeworth box)

1 producer - 1 consumer

several producers

and an example illustrating the limits of the partial equilibriumapproach
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Individual Preferencesrepresented by a utility function ui

continuous (the representation of preferences by a utilityfunction requires transitive, complete, continuous

preferences) strictly quasi-concave (unique optimum)

strictly monotonic (stronger than locally non satiated)

Offer curve of i= optima of i (parameterized by p)
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Definition : a Walrasian equilibrium is (x, p) such that

1. individual optimality : i, xi solves

maxp.xp.i

ui(x) ,

2. market clearing

i

xi=

= intersection points of the two offer curves (other than theendowment point)GE determines the relative price only ( one defines a numeraire,

a good with price 1, without loss of generality)Uniqueness is not guaranteed

Examples : Cobb-Douglas, linear, Leontief preferences
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Two examples of non existence:

1. An important one: non convexity of one ui: no intersection ofthe offer curves because of a discontinuity

2. A more subtle one: non strict monotonicity of one ui:impossible to clear the markets by adjusting the prices
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Second example: 1 consumer + 1 producer

2 commodities: leisure (price w), consumption good (price p)

firm:

production function q=f(z) (f >0> f)

max pq wz

consumer:

utility u(l, x)

endowment (L, 0) owns the firm
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Definition: A Walrasian equilibrium is (l, x),(q, z),(w, p)

1. individual optimality: (q, z) solves

maxq=f(z)

pq wz

(l, x) solves

maxwl+pxwL+

u(l, x) , with =pq wz

2. market clearing

l

+z

=L and x

=q

In this example, equilibrium is unique.
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Illustration of the 2 Welfare Theorems

Th 1 : The (unique) equilibrium allocation is PO

Th 2 : The (unique) PO allocation is the equilibriumallocation (no transfer is needed in this example)

Without the convexity assumptions (preferences and productionset):

An equilibrium is still PO (Th 1 still holds)

A PO allocation may not be an equilibrium allocation, evenwith transfers (Th 2 does not hold)
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Remark: production function and production set

Definition of the production set Y:

y Y if and only ify= (y1,..., yL) is a technologicallyfeasible vector

convention sign: yl0whenever l is an output

For a technology defined by a production function f(the output isgood L, inputs are goods 1, ..., L 1):

the associated production set Y is

y IRL/yL f(y1, ...,yL1)

Y convex f concave
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Example: f(z) =Az

1 : IRTS (fconvex), no equilibrium
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Remark: Returns to Scale

For a technology defined by a production set Y:

decreasing (DRTS) y Y,a [0, 1] , ay Y

increasing (IRTS) y Y,a 1, ay Y

constant (CRTS) y Y,a 0, ay Y

For a technology defined by a production function f:

DRTS: z IRL1+ , a 1, f (az) af (z)

IRTS: z IRL1+ ,a 1, f(az) af(z)

CRTS: z IRL1+ ,a 0, f(az) =af(z) (f homogenous ofdegree 1)
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Third example: J producers

Jfirms use L inputs to produce one different output eachglobal inputs endowment z= (z1,..., zL) 0

technologies fj (C2,

fjzjl

>0 and D2fjnegative definite)

exogenous output prices p= (p1,...,pJ)

input prices w= (w1,...,wL)

Definition : An equilibrium is (z,w) IRJL+L+

j, zj maximizes pjfj(zj) w.zj

jzj = z
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GE versus partial equilibrium: a taxation example

Ntowns, 1 firm/town (production function f)Labor supply (inelastic) : Mworkers,Wage w, good = numeraire

At equilibrium, w=fMN

(from max profit : w=f (ln) and market clearing:

nln =M -

equilibrium is symmetric)

Introduction of a taxt

in town 1 :w

+t

=f

(l1)
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Partial equilibrium analysis in town 1:

workers freely move between towns + w in towns 2,...,N wremains constant in town 1

hence l1 determined by w+t=f

(l1) the profit decreases, not the wage

the firm bears the whole burden of the tax t
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GE Analysis:

w and l1, ..., lNdetermined by:

w+t = f (l1) and n 2,w=f (ln)

l1+ ... +lN = M

(hence l2 =... =lN= Ml1

N1)

Introduction of a small tax dt

dw+dt = f (l1) dl1 and dw=f (l) dl

dl1+ (N 1) dl = 0

(denote l=l2 =... = lN)
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Variation of the profit of a firm 1 =f (l1) (w+t) l1 and=f(l) wl

d1 = f (l1) dl1 (w+t) dl1 l1(dw+dt)

d = f (l) dl wdl ldw

And

d1 = f

M

N

dl1 wdl1

M

N (dw+dt)

d = fM

N

dl wdl

M

Ndw

with l1 =l= MN

at the no tax equilibrium (t= 0)
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The end of the chapter
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Chapter Ge 1