General Equilibrium Theory: Examples
3 examples of GE:
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pure exchange (Edgeworth box)
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1 producer - 1 consumer
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several producers
and an example illustrating the limits of the partial equilibrium
approach
First example: Edgeworth Box
A pure exchange economy (no production possibilities):
2 consumers i = A, B
2 commodities l = 1, 2
individual endowments ωi = (ωi1 , ωi2 )
global endowment $ = ωA + ωB
allocation x = (xA , xB ) with xi = (xi1 , xi2 ), xi ≥ 0
price p = (p1 , p2 )
Edgeworth box = allocations such that xA + xB = $
Endogenous wealth: wi = p.ωi
The budget line splits the box into the 2 budget sets
Individual Preferences
represented by a utility function ui
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continuous (the representation of preferences by a utility
function ”requires” transitive, complete, continuous
preferences)
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strictly quasi-concave (unique optimum)
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strictly monotonic (stronger than locally non satiated)
Offer curve of i = optima of i (parameterized by p)
Definition : a Walrasian equilibrium is (x ∗ , p) such that
1. individual optimality : ∀i, xi∗ solves
max ui (x) ,
p.x≤p.ωi
2. market clearing
X
xi = $
i
= intersection points of the two offer curves (other than the
endowment 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
Two examples of non existence:
1. An important one: non convexity of one ui : no intersection of
the 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
Illustration of the 2 Welfare Theorems
Th 1 : The allocation x ∗ of an equilibrium (x ∗ , p) is Pareto-optimal
Definition : Equilibrium with nominal transfers = (x ∗ , p, tA , tB ) s.t.
1. tA + tB = 0 (tA , tB ∈ IR )
2. ∀i, xi∗ maximizes ui (xi ) under p.xi ≤ p.ωi + ti
3. xA + xB = $
Th 2 : If x is a PO allocation, then x is the allocation of an
equilibrium with transfers
(compute the relative price p2 /p1 = MRS, then define the transfer:
ti = p.xi∗ − p.ωi )
Th 2 requires the convexity of preferences (not Th 1)
real transfers can be considered as well (example:
p.xi ≤ p.ωi + p1 ti , transfer of good 1)
Second example: 1 consumer + 1 producer
2 commodities: leisure (price w ), consumption good (price p)
firm:
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production function q = f (z) (f 0 > 0 > f 00 )
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max pq − wz
consumer:
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utility u (l, x)
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endowment (L, 0)
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owns the firm
Definition: A Walrasian equilibrium is (l ∗ , x ∗ ),(q ∗ , z ∗ ),(w , p)
1. individual optimality: (q ∗ , z ∗ ) solves
max pq − wz
q=f (z)
(l ∗ , x ∗ ) solves
max
wl+px≤wL+π
u (l, x) , with π = pq ∗ − wz ∗
2. market clearing
l ∗ + z ∗ = L and x ∗ = q ∗
In this example, equilibrium is unique.
Illustration of the 2 Welfare Theorems
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Th 1 : The (unique) equilibrium allocation is PO
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Th 2 : The (unique) PO allocation is the equilibrium
allocation (no transfer is needed in this example)
Without the convexity assumptions (preferences and production
set):
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An equilibrium is still PO (”Th 1 still holds”)
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A PO allocation may not be an equilibrium allocation, even
with transfers (”Th 2 does not hold”)
Remark: production function and production set
Definition of the production set Y :
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y ∈ Y if and only if y = (y1 , ..., yL ) is a technologically
feasible vector
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convention sign: yl < 0 whenever l is an input, yl > 0
whenever l is an output
For a technology defined by a production function f (the output is
good L, inputs are goods 1, ..., L − 1):
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the associated production set Y is
n
o
y ∈ IR L /yL ≤ f (−y1 , ..., −yL−1 )
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Y convex ⇔ f concave
Example: f (z) = Az α
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α < 1 : DRTS (f concave), ”everything is OK”
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α = 1 : CRTS (f linear), technology determines the relative
prices, π = 0, the production level is determined by demand
(the supply is infinitely elastic)
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α > 1 : IRTS (f convex), no equilibrium
Remark: Returns to Scale
For a technology defined by a production set Y :
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decreasing (DRTS) ∀y ∈ Y , ∀a ∈ [0, 1] , ay ∈ Y
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increasing (IRTS) ∀y ∈ Y , ∀a ≥ 1, ay ∈ Y
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constant (CRTS) ∀y ∈ Y , ∀a ≥ 0, ay ∈ Y
For a technology defined by a production function f :
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L−1
DRTS: ∀z ∈ IR+
, ∀a ≥ 1, f (az) ≤ af (z)
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L−1
IRTS: ∀z ∈ IR+
, ∀a ≥ 1, f (az) ≥ af (z)
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L−1
CRTS: ∀z ∈ IR+
, ∀a ≥ 0, f (az) = af (z) (f homogenous of
degree 1)
Third example: J producers
J firms use L inputs to produce one different output each
global inputs endowment z̄ = (z̄1 , ..., z̄L ) 0
∂f
technologies fj (C 2 , ∂zjlj > 0 and D 2 fj negative definite)
exogenous output prices p = (p1 , ..., pJ )
input prices w = (w1 , ..., wL )
JL+L
Definition : An equilibrium is (z ∗ , w ) ∈ IR+
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∀j, zj∗ maximizes pj fj (zj ) − w .zj
P ∗
j zj = z̄
1st Order Conditions (for an interior equilibrium only, ∀j, zj 0)
= a system of equations with unknown (z, w ) characterizes the
equilibrium
∂fj
(zj ) = wl ,
∂zjl
X
∀l,
zjl = z̄l .
∀l, ∀j, pj
j
Illustration of Th 1: an equilibrium allocation z ∗ maximizes the
production value:
X
pj fj (zj )
Pmax
j
zj =z̄
j
Proof:
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P
P
the joint profit j (pj fj (zj ) − w .zj ) is j pj fj (zj ) − w .z̄.
Hence, the max of joint profit and the max of the prod value
have the same solution.
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the FOC of max joint profit (under the constraint j zj = z̄)
are the same as the FOC of equilibrium
∂fj
(zj ) = wl ,
∂zjl
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∀l,
zjl = z̄l .
∀l, ∀j, pj
j
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hence z ∗ maximizes the joint profit.
Producers (but not consumers) can be aggregated: a unique firm
with J technologies make the same decisions (and gets the same
profit) as J independent firms with one technology each.
GE versus partial equilibrium: a taxation example
N towns, 1 firm/town (production function f )
Labor supply (inelastic) : M workers,
Wage w , good = numeraire
At equilibrium, w = f 0 M
N
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(from max profit : w = f 0 (ln ) and market clearing: n ln = M equilibrium is symmetric)
Introduction of a tax t in town 1 : w + t = f 0 (l1 )
Partial equilibrium analysis in town 1:
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workers freely move between towns + w in towns 2,...,N ⇒ w
remains constant in town 1
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hence l1 determined by w + t = f 0 (l1 )
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the profit decreases, not the wage
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the firm bears the whole burden of the tax t
GE Analysis:
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w and l1 , ..., lN determined by:
w + t = f 0 (l1 ) and ∀n ≥ 2, w = f 0 (ln )
l1 + ... + lN
= M
(hence l2 = ... = lN =
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M−l1
N−1 )
Introduction of a small tax dt
dw + dt = f 00 (l1 ) dl1 and dw = f 00 (l) dl
dl1 + (N − 1) dl
= 0
(denote l = l2 = ... = lN )
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Variation of the profit of a firm π1 = f (l1 ) − (w + t) l1 and
π = f (l) − wl
dπ1 = f 0 (l1 ) dl1 − (w + t) dl1 − l1 (dw + dt)
dπ = f 0 (l) dl − wdl − ldw
And
M
M
dl1 − wdl1 −
(dw + dt)
dπ1 = f
N
N
M
0 M
dπ = f
dl − wdl − dw
N
N
0
with l1 = l =
M
N
at the no tax equilibrium (t = 0)
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Variation of the aggregate profit dπ1 + (N − 1) dπ
0 M
− w (dl1 + (N − 1) dl)
= f
N
M
M
− (dw + dt) − (N − 1) dw
N
N
= 0
since
dw + dt = f
dl1 + (N − 1) dl
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00
M
N
dl1 and dw = f
00
M
N
dl
= 0
the workers bear the whole burden of the tax (w decreases,
not the profit)
The end of the chapter