Mario Tirelli
Negishi’s approach to Walrasian equilibria
Handout 2
2010
1. Introduction
Following Lange (1942) and Allais (1943), interior allocations of given resources ω are Walrasian equilibria
if and only if they maximize a utilitarian social welfare function (a weighted sum of utilities) on the set of
resource feasible allocations1 Welfare weights parameterizing Lange’s social welfare function capture the
relative importance of households’ welfare. This characterization has been applied to establish fundamental
properties of the equilibrium notion–existence, Negishi (1960) and Bewley (1969), determinacy with infinitely
lived households, Kehoe and Levine (1985), Kehoe, Levine and Romer (1990, 1992), and computability,
Mantel (1971), Ginsburgh and Van der Heyden (1988).
The rational of this approach (often called Negishi’s approach) is to characterize efficient allocations as
solutions of a central planner’s problem and then use welfare theorems to establish the equivalence of this
set of solutions with the set of equilibrium allocations. This approach is useful since, in many contexts,
solving a planner’s problem is, loosely speaking, much easier than directly establishing existence, or proving
uniqueness, or obtaining computations of competitive equilibria. This motivates our interest in the following
analysis and results.
2. The economy and its fundamentals
Consider a private ownership, Walrasian, economy with L goods and J technologies/firms and I < ∞
consumers. Each firm j is owned by consumers according to some fixed ownership share θj , Σi θi,j = 1.
From now on we fix the characteristics {(ui , Xi , θi,j )i , (Yj )j , ω} and identify an economy with an initial
distribution of endowments, (ωi )i , such that Σωi = ω.
Assumption 1.
(1) ui is continuous, strictly increasing, quasi-concave, Xi ⊂ RL
+ is closed, convex, for
all i;
(2) Yj := yj ∈ RL : Fj (yj ) ≤ 0 where Fj is a continous, strictly increasing, quasi-convex transformation function, with Fj (0) ≤ 0, for all j.
(3) ω 0
Under these assumptions, the set of resource feasible allocations,
A := {(x, y) ∈ Πi Xi × Πj Yj : Σi xi − ω − Σj yj ≤ 0} ,
is a nonempty, closed and bounded subset of RIL × RJL .
We latter focus on differentiable economies, namely on economies also satisfying the following assumption.
Assumption 2.
(1) ui is differentiable, for all i;
(2) Fj is differentiable, for all j.
3. Results
3.1. Pareto efficiency. First we characterize the set of Pareto efficient allocations as solutions to a programming problem for an economy satisfying assumption 1.
Proposition 3. Any allocation (x, y) solving problem (P):
max u1 (x1 )
s.t.
ui (xi ) ≥ ui , i = 2, .., I
Σi xi − ω − Σj yj ≤ 0
yj ∈ Yj , j = 1, .., J
1Lange characterizes Pareto optima in this way. So the above characterization follows from the two welfare theorems. (Lange
(1942) is aware of the first one, while Allais (1953) is among the first to rigourously prove the second one.)
2
at ū := {ūi }Ii=2 is Pareto efficient; denote the set of Pareto efficient allocations O, and the set of solutions
to (P), at ū, Xū .
Proof. By contradiction, suppose (x, y) ∈ Xu , but (x, y) ∈
/ O. Then, there exists a (x0 , y 0 ) ∈ A, that is
0
Pareto superior to (x, y) : ui (xi ) ≥ ui (xi ) with strict inequality for at least one i. Since (x, y) was feasible for
problem (P), and ui (x0i ) ≥ ui (xi ) ≥ ui for all i ≥ 2, (x0 , y 0 ) is also feasible for problem (P). Moreover, there
are two possibilities: either ui (x0i ) > ui (xi ) holds for i = 1 or for some i ≥ 2. In the first case we obtain a
contradiction to the assumption that (x, y) was solving (P). In the second case, suppose ui (x0i ) > ui (xi ), by
continuity of utilities and strict monotonicity of u1 , there exists an ε > 0 such that
ui (x0i − ε1x0i ) >
ui (xi )
u1 (x1 + ε1x0i ) >
u1 (x1 )
where 1x0i is an indicator function assuming values equal to one for all commodities l satisfying x0il > 0, and
zero otherwise. Clearly, this infinitesimal transfer from i to 1 is resource-feasible, hence it is feasible for the
constrained set of (P) and improves the value of the objective u1 . This delivers a contradiction.
Now we establish the reverse.
Proposition 4. Any Pareto efficient allocation (x, y) solves problem (P) for some choice of utility levels,
u := {ui }Ii=2 .
Proof. Suppose (x, y) ∈ O. Construct a profile of utility levels, ui := ui (x) for all i ≥ 2, and assume that
(x, y) ∈
/ Xu . The latter implies that there exists a (x0 , y 0 ) that is feasible for problem (P) and such that
0
u1 (x1 ) > u1 (x1 ). Analogously, (x0 , y 0 ) is such that ui (x0i ) ≥ ui (xi ), for all i, strictly for at least agent 1, and
such that the (x0 , y 0 ) ∈ A. But this contradicts the assumption that (x, y) was Pareto efficient.
The utility possibility set, U , is the set of utility levels which can be achieved through feasible allocations.
This can be constructed defining a mapping A → RI , such that for all x = (x1 , ..., xI ) ∈ A, (u1 , .., uI ) =
(u1 (xi ), .., uI (xI )). Indeed, the image of this mapping is U .
Proposition 5. If A is nonempty, closed and bounded, and preferences are represented by continuous utility
functions, U is closed and bounded above. If, in addition (to assumption 1), utilities are concave and
transformation functions convex, U is convex.
The proof is immediate by recalling that the image of a compact set by a continuous function is compact
(see proposition 16.AA.2 MWG). For the convexity part, do it as an exercise.
Proposition 6. The boundary of U ,
∂U := {u ∈ U : for no u0 ∈ U, u0i ≥ ui ∀ i, u0i > ui for some i} ,
is the Pareto frontier. An allocation (x, y) in A is Pareto optimal if and only if u(x) : (u1 (xi ), .., uI (xI ))
belongs to ∂U .
The proof is easy once you observe that a typical element of ∂U is a vector of utility levels u such that
any alternative vector u0 dominating u (i.e. u0 > u0 and u0 6= u) cannot possibly be achieved, i.e. is not in U
(see MWG Proposition 16.E.1). A boundary point of U may not be Pareto efficient when there are ”bads”
if one imposes the resource constraint in A holds with equality, without assuming free-disposal. Then, even
consuming the whole endowment of a bad only, does not lead the economy to a Pareto efficient allocation.
Instead, with free disposal, or with a weak feasibility constraint, the boundary of the utility possibility set
identifies utility payoffs of Pareto efficient allocations (see also MWG exercise 16.E.1, notice however that
the attainability in MWG is imposed with equality).
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3.2. Pareto efficiency and social welfare. Thus, any point on the Pareto frontier is efficient, but different
points correspond to different distributions of resources. Suppose to fix a distribution by giving weights λ
to the utility levels, and that the social welfare is represented by the levels of
Wλ (u) :=
X
λi ui .
i
If there are only two agents, fixing λ2 equal to zero implies that the social welfare is maximized by taking
an extreme point in the Pareto frontier, u = (u1 (ω̄), 0), which is one where all resources and welfare goes to
agent one; this is equivalent to fix ū2 = 0 and solve problem (P). We now establish the equivalence of fixing
distributions by fixing utility levels of I − 1 consumers and solve (P), and fixing welfare weights of I − 1
consumers and maximize Wλ (u) over U .
Proposition 7. Fix λ 0, if u := (u1 , .., uI ) maximize Wλ (u) over U , then u belongs to the Pareto frontier
∂U . Conversely, if U is convex, any vector of utility u in ∂U maximizes Wλ (u) for some, non-trivial, vector
of welfare weights λ > 0.
Proof. For the first part, observe if u ∈ U , but not in ∂U , there exists a u0 ∈ U with u0 > u and u0 6= u.
Since λ 0, the latter implies λ · u0 > λ · u, contradicting the fact that λ · u was maximizing Wλ on U .
For the second part one can use convexity of U and apply the Separating Hyperplane Theorem. The idea is
that U and the interior of a convex cone pointed on any vector u in ∂U , can be separated by an hyperplane
(the two sets have an empty intersection, are convex and nonempty). The hyperplane defines λ. λ ≥ 0
provided that U contains −RI+ , something that is equivalent to free-disposal (if λi < 0 for some i, λ · u
cannot be maximized on U ; at any u, it would always be feasible to lower ui indefinitely, and increase social
welfare).
Notice that this proposition says that any solution of the maximization of a social welfare function Wλ ,
at given welfare weights λ 0, is Pareto optimal and conversely, that any Pareto optimal allocation can be
represented as a maximum of a social welfare function for some distribution of welfare weights λ > 0.
Problem 8. Proposition 7 has been established without making use of differentiability. Next, consider a
differentiable economy, satisfying assumptions 1, 2. Differentiability allows for a simpler proof of this proposition exploiting first order conditions. Prove that an interior allocation satisfying the first order conditions
of (P), satisfies the first order conditions of problem (W), for some λ > 0; and, conversely, that an interior
allocation satisfying the first order conditions of problem (W), satisfies the first order conditions of problem
(P) for some ū.
(W)
max Wλ (x) such that (x, y) ∈ A
3.3. Negishi’s approach. Hereafter, we focus on differentiable economies, satisfying assumption 1 and 2.
The following theorem establishes the equivalence between the set of competitive equilibrium allocations and
the set of solutions of (W).
Before formally stating and proving this property, let us recall a few well know ones concerning interior
equilibria (i.e. equilibria in which commodities are all consumed and produced in strictly positive quantities
by all agents). First, since the economy under consideration is characterized by differentiable, concave,
utilities and transformation function, individual optimality is fully characterized by its respective first order
conditions; among these sufficient conditions of individual optimality, by restricting to interior allocations,
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(x∗ , y ∗ ) 0, the following hold:
Dui (x∗i )
− p∗
D1 ui (x∗i )
DFj (yj∗ )
p∗ −
D1 Fj (yj∗ )
(1)
=
0, for all i
=
0, for all j
Notice that these are the usual conditions, respectively, requiring that M RS = p∗ = M RT (a (L − 1)−vector
of conditions expressed using as the numéraire commodity l = 1). At no loss of generality, in the sequel we
normalize the price of commodity 1 to unity; so we take the equilibrium price vector associated to (x∗ , y ∗ )
to be p∗ := (1, p∗ ).
The main result is the following.
Theorem 9. (x∗ , y ∗ ) 0 is a competitive equilibrium allocation if and only if it maximizes Wλ (x) on A,
P
for some λ in Λ := λ ∈ RI++ : λi = 1 .
We prove this theorem by establishing the following two propositions.
Proposition 10. (x∗ , y ∗ ) 0 is a competitive equilibrium allocation only if it maximizes (W) at
(λ)
λi =
t
> 0, for all i
D1 ui (x∗i )
"
(t)
t :=
X
i
1
D1 ui (x∗i )
#−1
>0
Hence, λ belongs to Λ.
Proof. We brake the proof in a few steps.
1: (x∗ , y ∗ ) ∈ A. Indeed, satisfying market clearing, it also resource-feasible.
2: By Kuhn-Tucket theorem if (x∗ , y ∗ ) maximizes (W) at λ in Λ then there exist non-trivial vectors ρ
in RL and η in RJ such (x∗ , y ∗ ) maximizes
hX
X i X
yj −
ηj Fj (yj ).
Wλ (x) − ρ
xi − ω −
Given λ 0, the latest is a concave function, hence its maximum is fully characterized by first order
conditions, implying,
(2)
λi Dui (x∗i ) − ρ
=
0, for all i
ηDFj (yj∗ )
=
0, for all j
ρ−
3: We still have to recover the multipliers (ρ, η) which at (x∗ , y ∗ ) satisfy (2). We do so using the
equilibrium first order conditions for individual optimality, knowing that the equilibrium price is
p∗ = (1, p∗ ). First let ρ be a linear combination of equilibrium prices: ρ = tp∗ for some t > 0.
Indeed, plugging into equations (2)
λi D1 ui (x∗i ) − t =
λi Dl ui (x∗i )
−
tp∗l
=
0
0, for all l > 1
which by (1) hold at λi = D1 uti (x∗ ) . To recover the the last set of multipliers, you can guess
i
ηj = D1 Ftj (y∗ ) and verify that this implies that the second condition in (2) holds at yj∗ , ρ = tp∗ and
j
t > 0.
5
i−1
h
> 0; then, by the first equation in (2),
4: Finally, let t := Σi D1 u1i (x∗ )
i
#−1
"
X
t
1
1
λi =
=
>0
∗
∗
D1 ui (xi )
D1 ui (xi )
D1 ui (x∗i )
i
for all i. And λ is in Λ as claimed.
Remark 11. (Welfare weights) Welfare weights coincide with the reciprocal of the marginal utility of the
numéraire commodity (i.e. income). Since marginal utility of 1 is decreasing in x∗1 , agents with, say, higher
income are weighted relatively more in the social welfare function W . To see it more clearly, denote by µh
the reciprocal of the marginal utility of commodity 1 by consumer h (µh := [D1 uh (x∗h )]−1 ). Then,
µh
λh = P .
µi
Do you find this unusual? Draw yourself the utility possibility frontier in the usual space (u1 , u2 ) and the
tangent line of extreme points in which agents have very high or low u2 /u1 . For concave utilities, the slope
of the tangent line is −λ1 /λ2 , in absolute value, increasing in u1 and decreasing in u2 , i.e. increasing as
we move down on the frontier toward point in which u1 /u2 is the highest. So, this finding is not unusual;
think about it mathematically; if an allocation gives more to an agent i than another, h, of the economy total
resources, it is natural that the allocation maximizes a social welfare which weights relatively more i than
h; that is such a welfare function rationalizes the distributional choice implicit in the allocation considered,
regardless this is or is not “fair”.
How about t? t is substantially the analogue of the choice of a normalization of prices in any competitive
economy with locally-non-satiated consumers. Notice that when we guess ρ we take a positive linear combination of prices; this is clearly not unique; here we have the same nominal indeterminacy of equilibria (only
relative prices count to pin down allocations). The choice of t > 0 does only affect the normalization of λ.
For example if t := Σi D1 u1i (x∗ ) > 0 then λ, defined as above, lives in a subset of RI++ satisfying Σ λ1i = 1.
i
P
λi = 1 only if (x∗ , y ∗ )
Proposition 12. (x∗ , y ∗ ) 0 maximizes (W) at some λ in Λ := λ ∈ RI++ :
is a competitive equilibrium allocation.
Proof. We brake the proof in two steps.
1: Again, by Khun-Tucker theorem, there exist non-trivial multipliers (ρ, η) such that (x∗ , y ∗ ) maximize
hX
X i X
yj −
ηj Fj (yj ),
Wλ (x) − ρ
xi − ω −
which is a concave problem. First order conditions (2) hold with equality, under constrained qualification; implying
ρ = λi Dui (x∗i ), for all i.
Using λ ∈ Λ, and particularly Σλi = 1, we find
X −1
1
ρ1 =
>0
D1 ui (x∗i )
2: Now we are ready to decentralize (x∗ , y ∗ ) as a competitive equilibrium. Again, restricting to equilibria with the numerire being commodity 1, we let p∗ := ρ11 ρ. Moreover, provide every consumer i with
∗
a wealth endowment wi satisfying budget balance at (p∗ , x∗ ), i.e. such that wi := p·x∗i −Σj θi,j p·yi,j
.
2
∗ ∗
∗ ∗
It is easy to check that (x , y ) is an equilibrium allocation. Indeed, since (x , y ) are in A, markets
2In a pure exchange economy, one would fix w = p∗ · x∗ for all i, or equivalently ω = x∗ , and attain a no-trade equilibrium.
i
i
6
clear. Individual optimality follows by checking that x∗ satisfies (2) if and only if x∗ satisfies (1)
at p∗ . Similarly, one checks that production decisions are individually optimal observing that y ∗
satisfies (2) if and only if y ∗ satisfies (1), when evaluated at p∗ .
Remark 13 (From equilibrium allocations to equilibria). Observe that in neither of the propositions we
explicitly refer to an economy, i.e. to a particular distribution of initial endowments. The first proposition
clearly extends to any initial economy. The second, does also apply to any economy provided the equilibrium
allocation is attainable through market trade, at the (implicit) equilibrium prices (p∗ := M RSi (x∗ , y ∗ ), for
some i). How many of these economies (again, endowment distributions) are there? With fixed aggregate
resources ω in RL , their number is n := (I − 1)(L − 1). Let us explain why. Indeed, to be an equilibrium
the allocation must satisfy budget balance for (I − 1) consumers, which (given (x∗ , y ∗ )) imposes (I − 1)
restrictions to the choice of endowments; the budget constraint of the I th consumer is satisfied automatically
by Walras law. Moreover, endowments must sum to ω ∈ RL
++ , which we assumed given, implying L extra
constraints. Hence the economy is parameterized by a subset of endowments of dimension d = I − 1 + L; its
complement has dimension equal to the endowment space IL minus d, i.e. n. We can also interpret n as
the “dimension”of trade compatible with each equilibrium allocation. Notice that if aggregate resources are
not fixed in advance, d = I − 1, n = I(L − 1).
Remark 14 (Parameterizing equilibrium allocations). This last proposition says something about the parameterization needed to describe the set of equilibrium allocation. This is summarized by Λ. Given a λ in
Λ we find (hence establish existence of ) an equilibrium allocation (simply by Weierstrass theorem); unique
if individual utilities are strictly quasi-concave. This suggests that the set of equilibrium allocations is of
dimension (I − 1) the dimensionality of Λ (which is a subset of the interior of a (I − 1) simplex in RI ). In
other words, by the two welfare theorems the set of interior, Walrasian, equilibrium allocations is equivalent
to the set of interior Pareto optima which has dimension I − 1. Equivalence is rigourously established for
smooth economies3 using differential topology; namely by showing that they are both diffeomorphic (linked by
a bijective map, smooth in both directions) to the interior of a (I − 1)−simplex (cf. proof of 5.2.4 in Balasko
(1988)).
Interestingly, the parameterization used to characterize equilibrium allocations does only depend on the
types of consumers, it is (I − 1)−dimensional. Neither the degree of heterogeneity of technologies, nor
the number of commodities affects the parameterization. But how general is this result? The answer is it
is fully general provided one considers Walrasian economies so that all equilibrium allocations are Pareto
efficient. This occurs, at least generically, in competitive economies whose markets are complete; including
those of infinite number of commodities (such as infinite horizon economies) and with uncertainty. On the
contrary, if the economy is not Walrasian, the above analysis and parameterization will fail to capture all
the equilibria which are not Pareto efficient. This is a problem considering that in such economies most
equilibria are inefficient (often in the sense that, there is a generic –open and dense– subset of economies
which deliver Pareto inefficient equilibrium allocations). Examples of inefficiencies, or market failures, are
those in which agents are exposed to risk which they are unable to fully diversify, either because the number
of financial markets is limited (in the sense of Radner (1972)) or because trade is limited (e.g. there
are borrowing constraints). Some extensions to avoid this problem are provided for finite economies with
3For smooth economies we should strengthen our assumptions by letting utilities to be strictly quasi-concave, twice con-
tinuously differentiable, describing preferred sets which are contained in the closure of the commodity space Xi ⊂ RL
++ ; an
analogous differentiable structure has to be assumed for transformation functions. For a detailed analysis see the seminal work
by Debreu (1972, ’76) or more recent general treatment by Mas-Colell (1988, chp.2,3) or Balasko (1988).
7
incomplete financial markets (Siconolfi and Villanacci (1991), Tirelli (2008), Tirelli and Turner (2010)) and
for economies with infinite horizon, complete financial markets, but borrowing constraints (Bloise et al.
(2010)). Other examples are economies with externalities as in Kehoe, Levine and Romer (1992), or price
rigidities as in Ginsburgh and Van der Heyden (1985). In most of these extensions the idea is, first, to
weaken the efficiency notion from Pareto optimality to some second best efficiency, for which the two welfare
theorems apply.
References
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