Efficiency and markets
Massimo D’Antoni
Dept of Economics and Statistics, University of Siena
Course of Public Economics
Academic Year 2023-2024
Last update — February 27, 2024
Efficiency and welfare
The basis of normative analysis
▶ Welfare = utility = individual preferences
▶ respect individual preferences: no paternalism!
▶ preferences are given (what if they are endogenous?)
▶ Economize on value judgements (or ban them completely, see L. Robbins,
“Interpersonal Comparisons of Utility: A Comment”, The Economic
Journal, Vol. 48, 1938)
▶ do not allow for comparison of "utility" levels or variations among individuals
(at the core of the "New" welfare economics of the 1930s)
▶ look only at consequences (outcomes) of actions for the individuals
Is it reasonable? Does it allow us to justify social choice?
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Pareto efficiency
(strong) Pareto dominance x dominates x′ if U h (x) ⩾ U h (x′ ) for all h and
U h (x) > U h (x′ ) per some h.
(weak) Pareto dominance x dominates x′ if U h (x) > U h (x′ ) for all h
Pareto efficiency x is efficient in X if no x′ ∈ X dominates x
Remarks:
▶ An outcome x is efficient when, given utility levels uk for each k ≠ h, it
solves
max
U h (x′ ) s.t. U k (x′ ) = uk for k ≠ h
′
x ∈X
▶ Pareto efficiency does not require any interpersonal comparison of utility: it
is consistent with the idea that utility only reflects subjective preferences
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
An economy with private goods
Efficient allocations of private goods (pure exchange)
▶ Assume that x is an allocation of n goods (whose total quantities are given)
among H individuals
▶ x = (x1 , . . . , xH ), where
▶ xh = (xh , . . . , xnh ), where X is given
1
Í
no production: X = h 𝜔h with 𝜔h h’s vector of endowments
▶ The allocation x is Pareto efficient (optimal) if it is feasible and it is not
dominated in X
Í
h
▶ H
h=1 x ⩽ X (feasible);Í
h
h h
h h
▶ there is no x̄ such that H
h=1 x̄ ⩽ X and such that u (x̄ ) > u (x ) for all h.
▶ In the 2 × 2 case efficient allocation can be represented in the Edgeworth
Box
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Competitive equilibrium: the first fundamental theorem
An allocation x and a price vector p = (p1 , . . . , pn ) represent a competitive
equilibrium if:
Í
h
1. H
h=1 x = X, the allocation is feasible (we may also describe this condition
as saying that demand equals supply for each commodity);
2. for each individual h, there is no vector x̄h such that uh (x̄h ) > uh (xh ) and at
Í
Í
the same time ni=1 pi x̄ih ⩽ ni=1 pi xih (i.e.~we assume that xh is the preferred
vector for h among those of equal or lower cost at given prices).
The competitive equilibrium is defined by the compatibility of optimizing
choices of price-taking individuals.
Theorem (First fundamental theorem of welfare economics)
If (x, p) is a competitive equilibrium, the allocation x is Pareto optimal.
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Proof of the First fundamental theorem of welfare economics
We deny the theorem by assuming there is x̄ feasible which Pareto dominates
x. In other words,
▶ we assume there is x̄h such that uh (x̄h ) > uh (xh ) for every individual in the
economy;
▶ if x is a competitive equilibrium, then from property 2 follows that
n
n
∑︁
∑︁
pi x̄ih >
pi xih
for all h.
i=1
i=1
▶ Summing the inequalities for all hs and using property 1, we have
n
H
n
H
n
∑︁
∑︁
∑︁
∑︁
∑︁
h
h
pi
x̄i >
pi
xi =
pi Xi
i=1
h=1
i=1
h=1
i=1
▶ but this is not compatible with the assumption that x̄ is feasible, because
with prices pi all positive, feasibility requires that
Ín
ÍH h Ín
i=1 pi h=1 x̄i ⩽
i=1 pi Xi .
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The first fundamental theorem: some remarks
▶ The theorem has been proved using the weak notion of Pareto optimality.
Could you extend the proof to the case of a strong Pareto optimum?
▶ The theorem does not say that a competitive equilibrium exists, only that if
it exist, it must be Pareto optimal. In order to make sure the equilibrium
exists we need some additional conditions on preferences and technology.
▶ Assumptions of the theorem:
▶ complete set of markets (all markets exist, or all interaction through markets)
▶ markets are competitive
Í
▶ goods are private ( h xh = X and xh affects only h’s utility)
whenever one of these assumptions is not verified, market is not able any
more to secure efficiency
▶ Efficiency is not welfare maximization
▶ equity matters
▶ can all efficient allocations be reached as competitive equilibria?
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The second fundamental theorem of welfare economics
Theorem (Second fundamental theorem of welfare economics)
When preferences are convex, for each efficient allocation x there is a vector
of prices p such that (x, p) is a competitive equilibrium.
Convexity matters:
2
Y
E
F
1
Massimo D’Antoni
X
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The second fundamental theorem: remarks
▶ In abstract, the theorem says that there is no conflict between market and
equity
▶ It states the possibility to separate equity and efficiency, where the former is
secured by a distribution of endowments, while the latter is secured by the
market mechanism
▶ This implies that the pursue of efficiency is always justified
▶ However, this requires that we are able to redistribute resources
(endowments) using non distortionary instruments such as a system of
personalized lump sum taxes and transfers — this is unrealistic!
▶ In practice, given the unavailability of such instruments, redistribution
implies waste (the leaky bucket metaphor)
▶ it follows that separation is not possible: efficiency and equity should be
jointly considered in policy evaluation
▶ and there might be circumstances where efficiency can be sacrified to
pursue equity
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Extensions of the general equilibrium model
We considered the simplified case of pure exchange. The model can be easily
extended:
▶ production (we require convexity of production sets and competitive
markets)
▶ time, location, etc. (we redefine what a commodity is)
▶ uncertainty: we consider contingent commodities defined across all
possible "states of the world"
The fundamental theorems carry on to these cases. However, the assumption
of a complete set of markets becomes more and more demanding:
▶ we require that for each commodity a price is defined on a competitive
market
▶ Arrow securities are a way to reduce the number of markets. However, the
assumption of complete markets remains very strong.
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Market failures
The First Theorem allows us to identify reasons why the market might not
secure efficiency:
▶ when each individual’s utility depends, positively or negatively, on
someone else’s consumption (externality), so that some effect of
individuals’ decisions are not internalized (they are not reflected in prices)
Í
▶ when there is joint consumption of some goods (the condition h xh = X in
the theorem is not satisfied), as in the case of public goods, expecially when
exclusion is not possible
▶ when individuals are not price takes (markets are not competitive)—this
may be the case when there are economies of scale (nonconvexity of
production)
▶ when some markets do not exist, due to information asymmetries (moral
hazard or adverse selection)
▶ more in general, transaction costs may make the working of some markets
prohibitively costly
In each of these cases, efficiency can be improved in principle by using the
coercive power Course
of the
state.
of Public Economics, Academic Year
Massimo D’Antoni
2023-2024
Dept of Economics and Statistics, University of Siena
Ideal taxes and transfers, first and second best
Lump sum taxation
▶ According to the 2nd Fundamental Theorem of welfare economics, any
Pareto optimal allocation can be implemented by competitive markets and a
suitable distribution of endowments
▶ hence, provided that we are able to redistribute individuals’ endowments
(taxes/transfers depend only on endowments), equity can be reached
without giving up efficiency
▶ Lump sum taxes: taxes whose amount is not affected by taxpayers’
behavior (only by characteristics they take as given)
▶ such taxes only have an income effect, not a substitution effect
▶ of course, in order to redistribute, they should not be uniform, but related to
variables relevant from the point of view of equity (earning innate abilities etc.)
▶ are there examples of such taxes? unfortunately, personalized lump sum taxes
implementing the optimal allocation are not generally feasible
▶ we must rely on taxes on quantities indirectly related to the innate ability
(income, consumption, wealth, etc.)
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Equity and efficiency in the first best
▶ With efficient redistributive taxes available, we need just one instrument
(individualized lump sum taxes) to attain equity
In this first best situation:
▶ lump sum taxes always dominate other redistributive instruments: they are
the only instrument to be used to reach equity;
▶ equity can never justify departures from the efficiency conditions;
▶ efficiency never conflicts with equity.
None of these above claims is warranted once we rule out lump sum taxes and
we are forced to rely on distortionary redistributive instruments.
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The Second Best Theorem
▶ A general result due to Lipsey and Lancaster ("The General Theory of
Second Best", Rev.Econ.Studies, 1956):
if there is introduced into a general equilibrium system a constraint which
prevents the attainment of one of the Paretian conditions, the other Paretian
conditions, although still attainable, are, in general, no longer desireable.
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The Second Best Theorem /2
▶ To illustrate, consider a problem maxx f (x), which gives the n first order
conditions for optimality:
𝜕f
=0
i = 1, . . . , n
𝜕xi
▶ however, assume that one of this condition cannot be satisfied, or
𝜕f /𝜕x1 = k > 0. The new optimization problem becomes
𝜕f
max f (x) s.t.
=k
x
𝜕x1
so that we have now e new set of (completely different) optimality
conditions:
𝜕f
𝜕2f
−𝜆
=0
i = 1, . . . , n
𝜕xi
𝜕x1 𝜕xi
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The second best with regard to efficiency and equity
In a second best world:
▶ efficiency constrains the achievement of equity (we trade-off equity and
efficiency);
▶ equity can justify departures from efficiency;
▶ in general, there is no single redistributive instrument dominating the
others; on the contrary, it may well be that the distortionary effect rises
rapidly with the use of one instrument, and this is a case for diversification
of redistributive devices;
▶ in general, the social optimum is not on the efficiency frontier; in the social
optimum, some (or even all) of the conditions which characterize efficiency
are not satisfied (second-best theorem).
. . . and things turn out to be much more complicated.
[See Blackorby (1990)]
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The second best with regard to efficiency and equity /2
We take into account the fact that we can only redistribute using imperfect
instruments, which instroduce distortions
u2
u1 = u2
E
F
A
▶ point A is the market outcome
without redistribution
▶ point E (first best) is the preferred
point on the frontier constrained
only by resources and technology
▶ point F (second best) is the preferred
point on the frontier constrainted by
the available redistributive
instruments
u1
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Potential Pareto improvements and the
compensation test
Potential Pareto improvements and the compensation test
▶ Pareto dominance does not require utility comparisons, however this cames
at a cost: it is not possible to compare situations where there are winners
and losers (the order induced by the Pareto criterion is partial)
▶ The notion of efficiency implicitly used in most economic analysis is
potential efficiency, or equivalent wealth maximization
▶ the implicit idea is that whenever you increase total wealth, even if at the
expense of equity, there is the possibility to redistribute such additional
wealth so that a Pareto improvement can be attained
▶ this idea was introduced by Kaldor and Hicks with their compensation test
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The compensation test
▶ Consider a change from B to A
▶ if this is not a Pareto improvements, there are winners and losers
▶ (Kaldor’s version) consider a transfer which compensates losers
▶ if winners, should they be asked to compensate loser, still prefer A over B,
then A must be socially preferred to B
▶ (Hicks’ version) consider a monetary transfer which is equivalent to the
change for winners
▶ if losers prefer the change to A rather than "bribing" winners to give up the
change, A must be socially preferred to B
▶ The compensation tests identify potential Pareto improvements
i.e. it may be turned to a Pareto improvement by a suitable redistribution of
resources among the individuals
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The compensation test
Kaldor
Hicks
U2
U2
A0
B
B
A
A
U1
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
U1
Dept of Economics and Statistics, University of Siena
The compensation test
▶ Let U h (i, Rh ) h’s utility in state i (i = A, B) where Rh is income (we assume
it does not change)
▶ Let 1 be the winner and 2 the loser, so that
U 1 (A, R1 ) > U 1 (B, R1 )
U 2 (A, R2 ) < U 2 (B, R2 )
▶ Let y1 and y2 the compensating variations for the change from B to A,
defined by
U h (A, Rh − yh ) = U h (B, Rh )
h
y is positive if h is a winner, it is negative if h is a loser
▶ Kaldor test requires that y1 + y2 > 0, or |y1 | > |y2 |.
▶ Similarly, the equivalent variations y1 and y2 are defined
U h (A, Rh ) = U h (B, Rh + yh )
▶ using the equivalent variations we have Hicks’ test
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
The compensation test: comment
▶ It does not take into account different value given to resources/income by
individuals; the implicit assumption is that the marginal utility of income is
the same across individuals, but this is not true in general
▶ It may be contradictory (A better than B and B better than A)
▶ note that although A ≻K B rules out B ≻H A, it does not rule out B ≻K A
▶ to avoid this contradiction, we can make the criterion stricter by requiring that
both A ≻K B and A ≻H B before concluding that A is better than B
▶ Using monetary equivalent/compensating variations at given prices can
give rise to the Boadway’s paradox
▶ Boadway’s paradox: although the change from B to A involve only a
redistribution of resources among individuals, A results to be better than B. This
is because the redistribution of resources itself can affect prices
▶ Although the test is usually not mentioned in textbooks, the comparison of
compensating/equivalent variation is implicit when we assess welfare
change using aggregate demand (e.g. consumer’s surplus)
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Equivalent and compensating variation (reminder)
x2
U0
U1
CV = compensating variation
0
e(q1 , U )
EV = equivalent variation
CV
y
EV
e( p1 , U 1 )
E1
E0
x1
The compensating and equivalent variation of a price change p → q are
CV = e(q, U 0 ) − y = e(q, U 0 ) − e(q, U 1 )
EV = y − e(p, U 1 ) = e(p, U 0 ) − e(p, U 1 )
Massimo D’Antoni0
Course of Public Economics, Academic Year
2023-2024
Dept of
and Statistics,
where U (U 1 ) is the initial (final) utility level and
yEconomics
is income
(note University
that of Siena
Equivalent and compensating variation (reminder)
Since 𝜕e(p, U)/𝜕pi = hi (p, U), where hi (p, U) is the compensated (Hicksian)
demand for good i, we have
∫ q
∫ q
de( p̃, U 0 )
0
0
h( p̃, U 0 )dp̃
CV = e(q, U ) − e(p, U ) =
dp̃ =
d
p̃
p
p
∫ q
∫ q
1
de( p̃, U )
EV = e(q, U 1 ) − e(p, U 1 ) =
dp̃ =
h( p̃, U 1 )dp̃
d
p̃
p
p
pi
qi = pi + ti
pi
xi ( p, y)
hi ( p, U 0 )
hi ( p, U 1 )
Massimo D’Antoni
Course of Public Economics,
′ Academic Year
2023-2024 i
x
xi
xi
Dept of Economics and Statistics, University of Siena
Change in consumers’ surplus
▶ Change in consumers’ surplus (for a change in price from p to p′ of good x)
is defined as:
∫ ′
p
x( p̃, R)dp̃
p
where x(p, R) is the demand function (it depends on income R)
▶ When the utility function is quasi-linear (no income effect), the consumers’
surplus is equal to the equivalent and compensating variation:
Prove as an exercise using the previous formulas
▶ In general "There is no well-defined economic question to which the change
in consumers’ is the answer" (Auerbach, 2002)
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Consumers’ surplus when more than one price changes
▶ With more than one price changing, it is not path independent: the measure
of excess burden is affected by the order in which the price changes are
considered.
The problem of "integrability"
Consider a change from (p1 , p2 ) to (q1 , q2 ). The expression for the consume surplus
is a line integral. If we start from the first price we have:
∫ q1
∫ q2
ΔS1 = −
x1 ( p̃1 , p2 )dp̃1 −
x2 (q1 , p̃2 )dp̃2
p1
p2
while starting from the second price we have:
∫ q2
∫
ΔS2 = −
x1 (p1 , p̃2 )dp̃2 −
p2
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
q1
x2 ( p̃1 , q2 )dp̃1
p1
Dept of Economics and Statistics, University of Siena
Equivalent/compensating variation and consumer surplus
CV
∆CS
EV
▶ There is a clear order among the three measures of the effect of a price
increase: CV ⩾ ΔSurplus ⩾ EV (the reverse is true when price decreases).
▶ They are closer the less relevant is the income effect of the price change
▶ When the income effect is null, the change in consumer surplus is equal to
the CV and the EV
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Exercises (compensation test paradoxes)
Paradox 1
Consider two alternative states A and B, and let the utilities U h (x, R) of
individuals h = 1 and 2 as a function of state x and wealth R satisfy the
following:
U 1 (A, 1000) = U 1 (B, 1200)
U 1 (A, 700) = U 1 (B, 1000)
U 1 (B, 1000) = U 1 (A, 1250)
U 2 (B, 750) = U 2 (A, 1000)
Consider the case that both individuals have initial wealth 1000.
1. Does a change from A (status quo) to B satisfies the Kaldor principle?
2. Imagine that we have changed to B (and no compensation has taken place).
Does a change from B to A staisfy the Kaldor principle?
N.B. The preferred state decreases the monetary value of the change
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
Dept of Economics and Statistics, University of Siena
Paradox 2 (Boadway’s paradox)
Kaldor’s criterion amounts to comparing losers’ and winners’ compensating
variations.
y
a
▶ Consider the change from
equilibrium B to A
▶ identify the compensating variation
measured in units of y
▶ does the change satisfy the Kaldor
criterion?
▶ Does it make sense?
O2
b
c
A
C
B
D
O1
Massimo D’Antoni
Course of Public Economics, Academic Year
2023-2024
x
Dept of Economics and Statistics, University of Siena