Mario Tirelli
On the sign of equilibrium prices
Handout 1
2010
1. Preliminary definitions
Consider a standard, pure exchange, Walrasian, economy with L commodities which are
available in strictly positive amount, ω À 0,
Eω := (ºi , Xi , ωi )Ii=1 .
Every economy satisfies the following properties.
Assumption 1. For each consumer i
a preferences (ºi ) are locally-non-satiated (LNS), continuous, reflexive and transitive
ordering;
b initial endowments ωi are nonnegative;
c Xi ⊂ RL+ is nonempty, closed and convex.
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Fix initial endowments (ωi )i such that
i ωi = ω. Then,
Definition 1. A competitive equilibrium is a price vector p∗ in RL satisfying
Z(p∗ ) = 0
2. “Goods” and “bads”
The economy we are considering may both contain commodities which are “goods” or
“bads”. Since these categories depend on individual testes, or preferences, we distinguish
between commodities which are “bads” from those which are a “bad for some consumer”.
Definition 2. A commodity l is a bad for consumer i if for every xi in Xi
xi − λ1l Âi xi , for all λ > 0
where 1l is an L−vector with all elements equal to zero except for the the lth that is equal to
one.
Definition 3. A commodity l is a bad if it is a bad for every consumer i.
Definition 4. A commodity l is a good if it is not a bad, and there exists a consumer i such
that for every xi in Xi ,
xi + λ1l Âi xi , for all λ > 0.
So we decided to label as good a commodity which is not a bad for everyone, and for
which at least one consumer has strongly monotonic preferences in that commodity. This
still allows for commodities to be neither goods nor bads, something we disregard because
irrelevant for our analysis.
3. The sign of prices at equilibrium
We now characterize the sign of prices at equilibrium.
Proposition 5. In an economy in which all commodities are goods, p∗ is a competitive
equilibrium price only if p∗ À 0.
Proof. By contradiction, suppose p∗l ≤ 0. Then, by definition 4, and labeling with x∗ the
equilibrium allocation, there exists a consumer i such that x∗i + λ1l Âi x∗i for all λ > 0;
moreover, this new bundle would be budget feasible (for all λ > 0). This contradicts the
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individual optimality of x∗i , and ultimately the fact that p∗ is an equilibrium.
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An immediate, but useful, corollary follows.
Corollary 6. In an economy in which some individual has strongly monotone preferences,
p∗ is a competitive equilibrium price only if p∗ À 0.
Proposition 7. If commodity l is a bad and p∗ is a competitive equilibrium, p∗l < 0.
Proof. By contradiction, suppose p∗l ≥ 0. Then, by definition 3, for all consumers i, there
exists a λ > 0 such that
xi − λ1l Âi xi , for all λ > 0, and all xi ∈ Xi .
Clearly, the latter holds also at ωi . Moreover, if ωi,l > 0, xi (λ) := ωi − λ1l is budget feasible
at p∗ ,
p∗ · xi (λ) ≤ p∗ · ωi ,
for all λ > 0 such that x(λ) ≥ 0. If instead ωi,l = 0, xi,l = 0 is still budget feasible. Therefore, it will be individually optimal for i to supply all her endowment of good l: xil (p∗ ) = 0,
or zil (p∗ ) = −ωil . This holds for all consumers i. Hence, summing over i, and using the
assumption that ω l > 0,
Zl (p∗ ) = −ω l < 0
contradicting the fact that p∗ is a competitive equilibrium.
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Remark 8 (Externalities and “bads”). Are negative externalities “bads”? There are two
features which differentiate externalities from bads. By definition, an externality is a consumption (or production) activity of some agent which has a negative economic impact (measurable as an economic loss) on another agent, for which the latest receives no payment. In
other words, first, externalities are non-tradable; second, agents who suffer from a negative
externality has no direct control on it. In our Walrasian economy, instead, bads are tradable
like all other commodities, and it is feasible for agents who own bads to decide not to consume
them. The closest bads can get to externality is if we assume that agents can control the level
of their own exposure to externalities by trading property rights (or emission permits). We
shall come back to externalities latter in the course.
4. The sign of prices at equilibrium: the case of free-disposal
When there are bads, it is important to explore whether it is possible for the individuals
to dispose or waste these commodities. We say that there is free-disposal (hereafter FD) if
individuals are allowed to through away commodities at no cost. FD has essentially the effect
of separating trading activities from consumption decisions: one can buy a bad and through
it away at no cost; whether or not the latest is optimal essentially depends on market prices,
as you will soon understand.
Let us provide a formal definition of FD.1
Definition 9. A consumer i has free-disposal if at (p, ωi ) she chooses xi , such that xi ºi x
ei
for all x
ei in
¾
½
p · vi ≤ p · ω i
Bi (p, ωi ) = xi ∈ Xi :
xi ≤ vi , vi ∈ RL+
1I
borrowed this definition from the class notes of Paolo Siconolfi.
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Observe that the control variables of consumer i are both xi and vi . So, under FD, i has
to increase vil to consume more of commodity l. On the contrary, she can keep constant, or
even reduce, xil while increasing the level of trade in the commodity, vil − ωil , through vil .
Obviously, this implies that she will end up not consuming some of the good l, which will
be thrown away. Whether this is, or is not, an optimal behavior for the individual depends
on prices; it will be so if throw away the quantity of good that is not consumed is effectively
not costly, at market prices, namely if pl < 0. This leads to the following.
Lemma 10. Under FD, the individual demand is well defined if p > 0.
Proof. Prove it as an exercise [Hint. do it by contradiction, supposing pl < 0 for some l and
noticing that this allows the consumer to financially support any level of expenditure]
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Since FD may, in principle, imply that agents optimally dispose of some commodity (the
bads), we should modify our definition of equilibrium as follows.
Definition 11. A competitive equilibrium of a pure exchange economy with FD is a price
vector p∗ in RL satisfying
Z(p∗ ) ≤ 0
We can now use lemma 10 to establish the following.
Proposition 12. Under FD, every equilibrium p∗ is such that p∗ > 0
Solve the following problem to observe that equilibria with excess supply on some markets
may exist.
Problem 13. Assume there is FD and that commodity L is a bad. Prove that every equilibrium has an excess supply of L, and that the equilibrium price p∗ satisfies p∗L = 0.
Notice that since, under FD, equilibrium prices are nonnegative, we could concentrate all
our analysis on nonnegative prices. If we do so, we are basically back to our original economy
without FD; namely for given preferences, endowments, and nonnegative prices, a consumer
with FD will behave exactly as one without FD.
Lemma 14. If xi is an optimal allocation under FD, at p > 0, with xi ≤ vi , it is also
optimal at vi0 = xi .
Proof. Let xi be individually optimal at p > 0 and vi , satisfying xi ≤ vi and p · vi ≤ p · ωi .
Since p > 0, p · xi ≤ p · vi ≤ p · ωi , implying p · xi ≤ p · ωi . Since preferences do only depend
on vi , xi is also optimal at vi0 = xi .
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Hence, if we restrict our analysis to a nonnegative price domain, we do not need to keep
track of free-disposal, since the individual budget sets may, at no loss of generality, be reduced
to the standard one, without FD.
This last lemma and LNS imply the following.
Proposition 15. In an economy with FD, every competitive equilibrium p∗ satisfies Walras
law,
p∗ · Z(p∗ ) = 0.
Proof. First, FD implies that p∗ > 0, by proposition 12. By lemma 14, we know that the
corresponding equilibrium allocation (x∗i )i is such that x∗i is individually optimal for all i as
without FD (i.e. with respect to the standard budget set). Reasoning by contradiction, if
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p∗ · Z(p∗ ) < 0 it must be that p · x∗i − p · ωi < 0 for some i. But LNS says that for all ² > 0
there exists a x0i (²), kx∗i − x0i (²)k < ², satisfying x0i (²) Âi x∗i . By continuity of the vector
product, for ² small enough, p · x0i (²) − p · ωi ≤ 0, x0i (²) Â x∗i , contradicting the fact that (x∗i )i
is an equilibrium allocation.
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