General Equilibrium
PURE EXCHANGE ECONOMY

Let x = { x^{1}, ...,
x^{L} }
represent an allocation of goods to L consumers.
(Each element of the above list is itself a vector of n goods.)

Let w = {
w^{1}, ..., w^{L} }
represent the initial endowments of the L consumers.
 Let p = ( p_{1}, ...,
p_{n} ) represent a vector of n prices.
A competitive equilibrium (or Walrasian equilibrium) is
represented by a list (x^{*},
p^{*}) such that
 x^{*l} is preferred to
x^{l} for all
x^{l} that satisfies the budget
constraint
p^{*} x^{l} ≤
p^{*} w^{l}
That is, all consumers maximize their utility.
 Demand does not exceed supply for each good. That is,
Σ_{l} ( x^{*l} 
w^{l} )
≤ 0
Define an aggregate excess demand function z(p) =
∑_{l} x^{*l}(p) 
w^{l}.
 If each consumer has strictly increasing and strictly
convex preferences, then z(p) is continuous.
 z(p) is homogeneous of degree 0.
 Walras' law holds: p z(p) = 0.
Proof of Walras' law.
Since utility functions are strictly increasing, the budget constraint
holds as an equality. The budget constraint for consumer l is
p x^{l} = p w^{l}
Sum over all l, we get
p Σ_{l} x^{l}
= p Σ_{l} w^{l}
which is precisely Walras' law.
The practical significance of Walras' law is that if
z_{1} = z_{2} = ... =
z_{n1} =
0 and if p_{n} > 0,
then z_{n} must be zero.
This says that in computing the
competitive equilibrium, we just have to make sure that n1 of the markets
clear. Once this is satisfied, the nth market also clears.
To prove the existence of a Walrasian equilibrium, we need a
fixedpoint theorem:
Brouwer's fixedpoint theorem. If f: S → S is a continuous
mapping from a compact and convex set S to itself, then there exists
some x in S such that x=f(x).
Proof of existence of Walrasian equilibrium. Since excess
demand functions are homogeneous of degree 0, whenever
z(p^{*}) ≤ 0, we have
z(tp^{*}) ≤ 0. In other words, whenever
(x^{*}, p^{*}) is a
competitive equilibrium, (x^{*},
tp^{*}) is also a competitive equilibrium. We
normalize prices in such a way that they always sum to 1. Hence, we
restrict our attention to prices that belong to the n1 dimensional
unit simplex:
S = { p in R_{+}^{n} :
&sum_{i=1}^{n}
p_{i} = 1 }
Define the mapping g: S → S by
g_{i}(p) = [ p_{i} +
max{0,z_{i}(p)} ] / [ 1 +
∑_{j}
max{0,z_{j}(p)} ]
Note that g is continuous and the range of g is in S because
∑_{i}
g_{i} = 1. So g is a continuous mapping from S
to S. By Brouwer's fixedpoint theorem, there exists a
p^{*} such that p^{*} =
g(p^{*}).
We want to show that this p^{*} is a Walrasian
equilibrium. From the fixedpoint property of
p^{*}, we have, for i=1, ..., n,
p_{i}^{*} = [
p_{i}^{*} +
max{0,z_{i}(p^{*})} ] / [ 1 +
∑_{j}
max{0,z_{j}(p^{*})} ]
Crossmultiply to get:
p_{i}^{*}
∑_{j}
max{0,z_{j}(p^{*})}
= max{0,z_{i}(p^{*})}
Multiply the ith equation
z_{i}(p^{*}) and sum
over all the n equations:
∑_{i}
p_{i}^{*}
z_{i}(p^{*})
∑_{j}
max{0,z_{j}(p^{*})}
= ∑_{i}
z_{i}(p^{*})
max{0,z_{i}(p^{*})}
By Walras' law, the left side of the above equation is 0, so
∑_{i}
z_{i}(p^{*})
max{0,z_{i}(p^{*})} =
0
But each of the n terms in this sum is nonnegative. So for the sum
to be equal to 0, we must have
z_{i}(p^{*}) ≤ 0 for
each i.
Note that a Walrasian equilibrium need not be unique.
First Welfare Theorem. If an allocation (x, p) is a Walrasian
equilibrium, then x is a Pareto efficient allocation.
Proof. Let x' be a feasible allocation that everyone prefers to
x. Then, for every consumer l, the bundle x'^{l} must
be beyond l's budget:
p w^{l} < p x'^{l}
Sum over all l and using the fact that x' is feasible, we arrive at a
contradiction:
p ∑_{l} w^{l} <
p ∑_{l} x'^{l} =
p ∑_{l} w^{l}
Second Welfare Theorem. Suppose x^{*}
is a Pareto efficient
allocation in an economy with endowment vector w. Assume that
preferences are convex. If the endowments are redistributed so that
the new endowment vector is x^{*}. Then
x^{*} is a competitive equilibrium allocation
associated with this economy with endowment vector
x^{*}.
Proof. Since preferences are convex, the aggregate
excess demand function
for the economy with endowment vector x^{*}
is continuous, so a Walrasian equilibrium exists.
Let (x^{~}, p^{~}) be a
Walrasian equilibrium for this economy. We want to show that
(x^{*}, p^{~}) is a
Walrasian equilibrium.
Since in a Walrasian equilibrium
everyone prefers the equilibrium bundle to his endowment bundle,
we must have
u^{l}(x^{~l}) ≥
u^{l}(x^{*l})
But x^{*} is a Pareto efficient allocation.
There cannot be any other feasible bundle that makes everyone strictly
better off. So the above inequality must hold as an equality.
u^{l}(x^{~l}) =
u^{l}(x^{*l})
Now, this equation implies that if x^{~l}
solves
max u^{l}(x^{l})
s.t. p^{~} x^{l} ≤
p^{~} x^{*l}
then x^{*l} must also be a solution to the same
problem. In other words, x^{*l} maximizes l's
utility subject to the budget constraint at prices
p^{~}. Furthermore,
x^{*} ≤ w by definition of feasibility. So
(p^{~}, x^{*}) must be a
competitive equilibrium.
It should be noted that the endowments need not be redistributed to
x^{*} to make the second welfare theorem work.
Any new endowment vector x' that satisfies p^{~}
x'^{l} = p^{~}
x^{*l} for all l will equally does the trick.
ECONOMY WITH PRODUCTION
We modify the description of the pure exchange economy by introducing
the following:

Let y
= { y^{1}, ..., y^{K} }
represent the net output supply of K firms in the
economy. (Input demands are treated as negative output. Again, each element
of the list is a vector of n goods.)
 Let T^{lk} represent
the share of firm k owned by consumer l.
A competitive equilibrium is
represented by a list (x^{*},
y^{*}, p^{*}) such that
 x^{*l} maximizes l's utility subject to
the budget constraint
p^{*} x^{l} ≤
p^{*} w^{l} +
∑_{k}
T^{lk} p^{*}
y^{k}
 p^{*} y^{*k}(k)
≥
p^{*} y^{k} for all
y^{k} that is feasible given firm k's
production technology. (That is, all firms maximize their profits.)
 Demand does not exceed supply for each good. That is,
∑_{l}
( x^{*l}  w^{l} ) ≤
∑_{k} y^{*k}
It can be proved that a competitive equilibrium exists if
 preferences are convex (utility functions are quasiconcave)
 production sets are convex (production functions are concave)
 a number of technical assumptions a satisfied
The convexity assumptions are sufficient to guarantee that the demand and
supply functions are continuous in prices.
Note that the assumption of convex production sets is not innocuous. For
example, the usual Ushaped cost curves used in
partial equilibrium analysis
do not satisfy convexity.
Note also that competitive equilibrium need not be unique.
It can also be shown that the first and second welfare theorems
continue to hold.
CHARACTERIZATION OF WELFARE THEOREMS USING MARGINAL CONDITIONS
Consider a simple model with two persons (A and B), two final
goods (X and Y), and
two inputs (K and L). Pareto efficiency is modeled by:
maximize
U^{A}(X^{A},
Y^{A})
subject to
U^{B}(X^{B},
Y^{B})
= u
X^{A} + X^{B} =
F^{X}(K^{X},
L^{X})
Y^{A} + Y^{B} =
F^{Y}(K^{Y},
L^{Y})
K^{X} + K^{Y} = K
L^{X} + L^{Y} = L
Form the Lagrangian:
L = U^{A}(X^{A},
Y^{A}) + λ (U^{B}
(X^{B},Y^{B})  u) +
μ^{X} (F^{X}
(K^{X},L^{X}) 
X^{A}  X^{B})
+ μ^{Y} (F^{Y}
(K^{Y},L^{Y}) 
Y^{A}  Y^{B}) +
ν^{K}
(K  K^{X}  K^{Y}) +
ν^{L}
(L  L^{X}  L^{Y})
The FOCs are:
(1) U^{A}_{X} 
μ^{X} = 0
(2) U^{A}_{Y} 
μ^{Y} = 0
(3) λ U^{B}_{X} 
μ^{X} = 0
(4) λ U^{B}_{Y} 
μ^{Y} = 0
(5) μ^{X}
F^{X}_{K} 
ν^{K}
= 0
(6) μ^{X}
F^{X}_{L} 
ν^{L}
= 0
(7) μ^{Y}
F^{Y}_{K} 
ν^{K}
= 0
(8) μ^{Y}
F^{Y}_{L} 
ν^{L}
= 0
The first two equations imply MRS^{A}(X,Y) =
μ^{X}/μ^{Y}.
The third and fourth
equations imply MRS^{B}(X,Y) =
μ^{X}/μ^{Y}.
Therefore, we must have
MRS^{A}(X,Y) = MRS^{B}(X,Y)
The fifth and sixth equations imply MRTS^{X}(K,L)
= ν^{K}/ν^{L}.
The last two
equations imply MRTS^{Y}(K,L) =
ν^{K}/ν^{L}.
Therefore, we must have
MRTS^{X}(K,L) = MRTS^{Y}(K,L)
Equations (5) and (7) imply
F^{Y}_{K}/
F^{X}_{K} =
μ^{X}/μ^{Y}.
The left hand side of this
equality is called the marginal rate of transformation between X and Y.
That is, it tells us how much additional Y can be produced if the output of X
is reduced by one unit. Note that MRT can also be defined as
F^{Y}_{L}/
F^{X}_{L},
because equations (6) and (8) imply that this ratio is also equal to
F^{Y}_{K}/
F^{X}_{K}. Since
μ^{X}/μ^{Y}
equals MRS(X,Y), Pareto efficiency
requires
MRT(X,Y) = MRS(X,Y)
In summary, Pareto efficient requires the following three sets of marginal
conditions to be satisfied:
 MRS between any two goods must be the same across any pair of
consumersachieved by consumers maximizing utility
 MRTS between any two goods (factors) must be the same across any pair of
producersachieved by producers minimizing costs
 MRS between any two goods must be the same as their MRTachieved
by producers maximizing profits

Lecture Notes

Competitive Markets
Coase theorem