Download General equilibrium, efficiency and Public Good Notes

General equilibrium, efficiency and Public Good
Notes
Simona Montagnana
1. Explain carefully how the concept of Pareto efficiency differs from
that of social welfare. Discuss the meaning of the statement The market
mechanism is efficient, and explain whether this is a normative or a positive statement.
Pareto efficiency is a state of allocation of resources in which it is impossible to
make any one individual better off without making at least one individual worse
off. The First Fundamental Theorem of Welfare (FFTW) says that every Walrasian
equilibrium is a Pareto optimum. The implication of FFTW is that the market
mechanism is neutral in terms of distribution of welfare across agents (consumers).
Every Pareto-efficient competitive outcome is based on an initial endowment of resources distributed among the economic agents and is achieved through the market.
Therefore, the market mechanism is efficient represents a positive statement.
Social welfare is a normative concept, which depends on different social criteria
(value judgement, distributive justice, ethics, politics, etc). The Second Fundamental Theorem of Welfare (SFTW) says that every Pareto optimum allocation is a
Walrasian equilibrium, under certain conditions. The implication of SFTW is that
efficiency and equity are two distinct issues. The distributional issues should be
resolved by changing the initial endowments, and not by changing the competitive
pricing mechanism. The SFTW states that any Pareto efficient allocation might be
decentralised by a suitable choice of lump-sum transfers.
Provide a model to explain the meaning of these statements (...).
2. Explain the price normalization concept, and why it is important.
Recall that what matters in the Walrasian model are relative prices, because "relative prices" are used to make comparisons of different goods. For this reason we have
to normalize prices using one price as numeraire. The price normalization consists
in giving to a commodity the role played by money for expressing the prices of the
other commodities in quantities of that commodity (such a commodity is known as
the numeraire).
More generally, in an exchange economy general equilibrium model with n markets,
we only need to find a set of prices where n − 1 of the markets are in equilibrium,
and Walras’ Law says that the nth market must also have demand and supply equate
there too.
The last statement essentially says that given an economy with n goods all we need
to find are n − 1 equilibrium prices. However, on further thought, how can it be possible that we could solve for n unknowns with n − 1 equations. Essentially, the point
is that we have only n − 1 independent prices. This means that we can choose one of
the prices as the normalizing price so that all other prices are measured relative to it.
3. Explain the concept of Competitive (Walrasian) equilibrium
Competitive (Walrasian) equilibrium for an pure exchange economy model is:
a pair (p∗ , x∗ ), s.t:
P
P
xi (p∗ , p∗ wi ) ≤
wi
i
i
where p∗ is a competitive equilibrium, if there is no good for which there is a positive
excess demand.
More simply a Walrasian equilibrium is a set of prices such that each consumer
is choosing his or her most-preferred affordable bundle, and all consumers’ choices
are compatible in the sense that demand equals supply in every market. We know
that if each agent is choosing the best bundle that he can afford, then his marginal
rate of substitution between the two goods must be equal to the ratio of the prices
(not always holds).
Competitive (Walrasian) equilibrium in the 2 × 2 pure exchange model
∗
∗
A∗
B∗
is a price vector p∗ = (p∗1 , p∗2 ) and consumption bundles xA = (xA
=
1 , x2 ), x
B∗
B∗
(x1 , x2 ), s.t.:
• The bundle consumed by each agent is the best affordable bundle given price
∗
∗
vector p∗. That is xA solves the consumer choice problem (1) and xB solves
the symmetric consumer choice problem for agent B given prices (p1 , p2 ) =
(p∗1 , p∗2 )
• Markets clear (feasibility).
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4. Using an example with one public good and one private good (or
wealth), derive a necessary condition for achieving an efficient outcome
when the level of the public good varies continuously and provide an
intuitive interpretation of the condition.
We have to write the utility functions of two consumers 1 and 2 in terms of private
and public goods, where: x1 and x2 measure the private consumption (as wealth)
of each individual, g1 and g2 measure their contribution to the public good. Let
say that the quality of public good is measured by G (as quality that represent a
continuous variable, with value ) and c(G) is its cost function.
Let say that the quality of public good is measured by G (as quality that represent a continuous variable, with value G ≤ 0) and c(G) is its cost function. The
allocation (x1 , x2 , G) and the utility functions for the two consumers are, as follows:
u1 (x1 , G); u2 (x2 , G).
The two consumers face a constraint that depends on x1 , x2 , c(G), and their endowment w1 , w2 :
x1 + x2 + c(G) = w1 + w2
Which is the necessary condition for achieving an efficient outcome?
A Pareto optimum allocation is one that maximises consumer 1?s utility given a
utility level of consumer 2, that we can fix at u
The optimization problem is then:
Let solve using a lagrangian method:
L = u1 (x1 , G) − λ[u2 (x2 , G) − u] − u[x1 + x2 + c(G) − w1 − w2 ]
and differentiate with respect to x1 , x2 , and G:
∂u1 (x1 ,G)
∂x1
L
∂x1
=
L
∂x2
(x2 ,G)
= −λ ∂u2∂x
−µ=0
2
L
∂G
=
∂u1 (x1 ,G)
∂G
−µ=0
(x2 ,G)
− λ ∂u2∂G
− µ ∂c(G)
=0
∂G
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The first equation gives µ:
µ=
∂u1 (x1 , G)
∂x1
(1)
From the second equation we get :
µ
∂u2 (x2 , G)
=−
λ
∂x2
(2)
Then, we multiply the third equation by µ1 :
1 ∂u1 (x1 , G) 1 ∂u2 (x2 , G) 1 ∂c(G)
− λ
− µ
=0
µ
∂G
µ
G
µ ∂G
(3)
We substitute (1) and (2) into the latter, obtaining:
∂u1 (x1 ,G)
∂G
∂u1 (x1 ,G)
∂x1
+
∂u2 (x2 ,G)
∂G
∂u2 (x2 ,G)
∂x2
=
∂c(G)
∂G
M UG
M UG
+
= M C(G)
M U x1 M U x2
M RS1 + M RS2 = M C(G)
which is the necessary condition for this problem.
To check whether this condition is verified try with a numerical example.
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