Download Lecture 7: General Equilibrium - Welfare

Lecture 7: General Equilibrium - Welfare
HS 12
Overview
1
Setting the Stage
2
First Welfare Theorem
3
Second Welfare Theorem
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
2/11
Setting the Stage
Again, we consider an exchange economy.
In this case
P
P
feasibility requires xi ∈ Rn+ and i∈I xi = i∈I ei .
an allocation x∗ is a Walrasian equilibrium allocation (WEA)
if there exists a Walrasian equilibrium price vector p∗ such
that
x∗ := (x1 (p∗ , p∗ · e1 ), x2 (p∗ , p∗ · e2 ), . . . , xI (p∗ , p∗ · eI )).
As a there might be several equilibria p∗ for given
endowments e, it is useful to define the following set:
For an economy with initial endowments e, we denote the
set of Walrasian equilibrium allocations by W (e).
Similarly, let F (e) denote the set of feasible allocations.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
3/11
Setting the Stage: Feasibility of WEAs
Every Walrasian equilibrium allocation is feasible:
W (e) ⊂ F (e).
The proof is trivial (Try writing it down!)
The same result holds in a economy with production.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
4/11
First Welfare Theorem: Statement
Every Walrasian equilibrium allocation is Pareto-efficient.
The only assumption needed to prove this result is that all
utility functions are strictly increasing.
The textbook proves a stronger result (Theorem 5.6) – we
will not discuss the core in this lecture.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
5/11
First Welfare Theorem: Proof
Consider x∗ ∈ W (e) and let p∗ be the associated vector of
equilibrium prices.
Consider any allocation x̃ (feasible or not) such that
u i (x̃) ≥ u i (x∗ ) holds with at least one strict inequality.
Then p∗ · x̃ ≥ p∗ · ei holds with at least one strict inequality.
(Why?)
Adding up the inequalities from the previous step yields:
"
#
X
X
∗
i
i
p ·
x −
e > 0.
i∈I
i∈I
This implies that x̃ is not feasible. (Why?)
Hence, x∗ is Pareto-efficient. (Why?)
The same logic works in an economy with production.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
6/11
First Welfare Theorem: Discussion
The First Welfare Theorem is a remarkable result,
highlighting the role of the market in coordinating individual
actions.
All individuals strive only to achieve the best for themselves
. . . nevertheless, the social outcome resulting from a
competitive market system is Pareto-efficient.
However, we may care about more than efficiency, such as
“justness” or “fairness.”
Indeed, many efficient allocations might not be desirable,
because they violate appealing conditions of distributional
justice.
So we may wonder: Is there a fundamental conflict
between what can be achieved through competitive
markets and some notion of distributional justice?
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
7/11
Second Welfare Theorem: Statement
Assume that all utility functions u i are continuous, strongly
increasing, and
quasiconcave on Rn+ . Assume in
P strictly
addition that i∈I ei 0.
For every Pareto-efficient allocation x̄ there exists a
redistribution of the initial endowments ē ∈ F (e) such that
starting from these endowment x̄ is the unique Walrasian
equilibrium allocation of the exchange economy:
{x̄} = W (ē).
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
8/11
Second Welfare Theorem: Proof
1
Set ē = x̄.
2
The assumptions ensure the existence of a WEA x̂ for the
exchange economy with endowments x̄. It remains to
show that x̂ = x̄ must hold.
3
x̂ is feasible in the original economy and satisfies
u i (x̂i ) ≥ u i (x̄i ) for all i ∈ I. (Why?)
4
5
Because x̄ is Pareto-efficient, it follows that u i (x̂i ) = u i (x̄i )
holds for all i ∈ I.
Strict quasiconcavity of u i (and the fact that the consumer
can afford both x̂i and x̄i under the Walrasian equilibrium
prices) implies that x̂ = x̄ holds.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
9/11
Second Welfare Theorem with Production
A counterpart to the Second Welfare Theorem holds in
economies with production.
The formulation and proof (See Theorem 5.15) is
somewhat different because
appropriate assumptions on the production sets Y j have to
be imposed.
agents are not only endowed with goods but also with profit
shares.
the profit maximizing behavior of firm has to be taken into
account.
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
10/11
Second Welfare Theorem: Discussion
The Second Welfare Theorem tells us that we can reach
any Pareto-efficient allocation in a decentralized way.
Thus if we have a distributional objective (some notion of
“fairness”, “justice”, or “sustainability”), we can attain this
objective in a competitive market system as long as it also
insists on efficiency.
So a market economy is not only a tool for achieving some
efficient allocation, it grants us the choice between all
possible efficient allocations.
Note, however, that this is not a piece of practical advice: if
we had enough information to “use” the Second Welfare
Theorem, we might as well forget about markets and
impose that allocation directly . . ..
Advanced Economic Theory
Lecture 7: General Equilibrium - Welfare
11/11