Existence of an equilibrium in arrowian markets for consumption externalities ✩

We study the existence of quasi-equilibria and equilibria for pure exchange economies with consumption externalities and Arrowian markets with personalized Lindahl prices. We provide examples showing ﬁrst that quasi-equilibria and equilibria of the externality economy fail to exist under assumptions guaranteeing existence for economies without externalities. We show that the externality economy has identical equilibrium allocations of an appropriately deﬁned constant returns to scale production economy without externalities. We exploit this equivalence to map sufﬁcient conditions for the existence of quasi-equilibria and equilibria of the production economy into sufﬁcient conditions of the pure exchange economy with externalities, thereby unveiling suitable irreducibility conditions and survival conditions. ©


Introduction
In the presence of consumption externalities, restoring the two Fundamental Theorems of Welfare Economics requires setting up markets where individuals face personalized Lindahl prices and choose allocations, that is, profiles of consumption bundles one for each of the individuals in the economy.This was observed a long time ago by Samuelson (1954), Arrow (1969), and Laffont (1976).
Surprisingly, the literature on the existence of competitive equilibria in economies with markets for externalities (hereafter Arrow-Lindahl equilibria) is very limited.Foley (1970) proves the existence of Arrow-Lindahl equilibria in economies with private and public goods, without consumption externalities and with monotonic preferences.Bergstrom (1976b) proves the existence of an equilibrium for so-called communal commodities and shows that it encompasses the case of markets for external effects through a technical construction. 1All existence results in the literature are based on specific fixed-points like arguments.
We study economies with consumption externalities in preferences, and find general conditions for the existence of quasi-equilibria and equilibria.This is not a trivial task.Even when preferences are locally non-satiated and individual endowments are strictly positive, quasi-equilibria and equilibria may fail to exist.We provide two examples that make this point.The existence failures of markets for externalities were hidden in the literature by technical assumptions playing the role of the survival and irreducibility assumptions. 2 We do not develop a direct argument, but rather follow the intuition of Arrow that in externality economies, "individual i's consumption is regarded as the production of joint outputs, one for each individual whose utility is affected by individual i's consumption" (Arrow, 1969, see pages 9-10).We show that a pure exchange economy with consumption externalities where trade takes place in Arrowian markets for externalities yields the same equilibrium allocations of an appropriately defined production economy with constant returns to scale and no externalities.Once this equivalence is established, existence of equilibrium, quasi-equilibrium, irreducibility and survival for economies with consumption externalities follow from well known and understood results for standard production economies.
The heavy lifting in our argument is done by the construction of the production economy and in particular of its technology.The technology transforms inputs of physical commodities into a collection of as many identical allocations as individuals in the economy.The total amount of physical commodities in an allocation cannot exceed the total amount of inputs used in the production process.The technology displays constant returns to scale, thereby generating zero equilibrium profits and making it irrelevant to specify individual property rights over production activities.Further, the chosen form of the technology restricts equilibrium prices to satisfy the classical Lindahl compatibility conditions.The latter establishes equivalence in equilibrium prices, and thus equilibrium allocations of the two economies.
1 There are other more recent papers on the subject.Crès (1996) focuses on the symmetry breaking properties of equilibria with consumption externalities.Existence is generic in the space of preferences under strong conditions of smoothness and irreducibility.Conley and Smith (2005) proves existence of quasi-equilibria and equilibria treating externality markets directly instead of indirectly through Arrowian commodities.Thus, there is no relation between the prices of the ordinary commodities and the prices of the externality rights.del Mercato and Florenzano (2009), anddel Mercato (2010) prove the existence of Arrow-Lindalh quasi-equilibria in economies with consumption externalities. 2 See, for example, Assumptions (S), (E), and (F) in Bergstrom (1976b).
We exploit the equivalence of feasible allocations to map classical irreducibility and survival conditions of the production economy into equivalent such conditions of the economy with externalities.In doing this, we understand the reasons for the failure of existence of quasi-equilibria and equilibria of the externality economy.
Without free disposal in the externality economy, the equivalent production economy does not satisfy the standard weak survival assumption, explaining the lack of quasi-equilibria in our first example.In the second example there is free disposal, and thus quasi-equilibria exist, but equilibria do not because the equivalent production economy is not irreducible.Indeed, irreducibility for externality economies is more demanding.In economies without externalities, irreducibility means that at any feasible allocation, any group of individuals can be made better off (in the Pareto sense) by adding to their consumption bundles something of the privately owned resources of the complementary group.For economies with externalities, any group must choose an allocation that specifies not only their consumption bundles, but also those of the individuals in the complementary group.This makes it harder to find an allocation (Pareto) dominating the target feasible allocation for all the members of the chosen group.
The paper is organized as follows.Section 2 describes the fundamentals, the basic assumptions and the structure of the markets.Section 3 provides examples of nonexistence of quasiequilibrium and equilibrium in economies satisfying the assumptions of Section 2. Section 4 is the key part of the paper describing the equivalent production economy without externalities.Section 5 shows arguments for the existence of a quasi-equilibrium and equilibrium and it defines notions of irreducibility and survival for the economies with externalities.

The basic model and its assumptions
We consider a pure exchange economy with a finite number of physical commodities labeled by the superscript ∈ L = {1, . . ., L}, and a finite set of individuals labeled by the subscript i ∈ I = {1, . . ., I }.Each individual i is characterized by a consumption set X i ⊂ R L , an endowment of commodities e i ∈ R L , and preferences represented by a utility function allowing for possible consumption external effects: Individuals' utility functions, consumption sets and endowments satisfy classical assumptions listed below.

Assumption A.
(1) X i is closed, convex, bounded from below, and 0 ∈ X i .
(2) u i is continuous, quasi-concave, and locally non-satiated on X. 3(3) e i ∈ R L + . 4e also assume the existence of a free disposal technology of physical commodities, i.e., an allocation x = (x i ) i∈I ∈ X is a feasible allocation of the economy E if i∈I x i ≤ i∈I e i .
While in economies without externalities Assumption A is standard, Starrett (1972) pointed out that negative external effects may create nonconvexities in preferences.As is well known, without convexity competitive equilibria may fail to exist.Since this paper is concerned with existence of equilibrium in a finite economy, we must assume that utilities are quasi-concave.5

Arrowian markets for consumption externalities
Market structure and equilibrium follow the classical definitions in Arrow (1969) and Laffont (1976).
Thus, an extended consumption vector of individual i is where x ii is the effective consumption of individual i, while x ih is the external effect of individual h's consumption as perceived by individual i.Equivalently, x ih is the demand of i for the consumption of h.A price system is where p ∈ R L is the price of the L physical commodities, while p ih ∈ R L is the price paid by individual i to h for the consumption externality created by individual h on i.Individual i faces a personalized price for her own consumption, i.e., the price p ii = p − h =i p hi .By definition of p ii , p satisfies the classical compatibility condition on prices à la Lindahl, that is: With this notation, individual i's budget constraint is defined below.6 At equilibrium, the extended consumption vectors are coherent, that is is the value transfer (positive or negative) from individual i to h.Definition 1 (Equilibrium with markets for externalities).An equilibrium of E is a pair of an allocation x * = x * i i∈I ∈ X and a price system p * = 0 with p * ≥ 0, satisfying the two following properties. 71) For all i, x * ∈ argmax {u i (x) : x ∈ B i ( p * )}, and (2) x * is feasible, i.e., i∈I x * i ≤ i∈I e i .
By local non-satiation, at equilibrium, all budget constraints are satisfied with equality, and therefore p * • i∈I x * i − e i = 0.For a quasi-equilibrium, the cost of x * is equal to the value of e i , and any consumption bundle that is weakly preferred to x * cannot cost less.That is, (1) in the definition above is replaced by Under local non-satiation, an equilibrium is a quasi-equilibrium, but not vice versa.

Examples of non-existence of (quasi-)equilibrium
This section consists of two examples.The first shows that quasi-equilibria (hence equilibria) may fail to exist in the absence of a free disposal technology.The second example suggests that the notion of irreducibility is far more demanding for economies with externalities than for standard economies.Indeed, strictly positive endowments and local non-satiation are not sufficient for irreducibility.

Non-existence of a quasi-equilibrium without free disposal
For standard pure exchange economies (without external effects) satisfying Assumption A and e i 0, all i ∈ I, quasi-equilibria exist even when the presence of free disposal is dispensed with, that is, when feasibility reads i∈I x i = i∈I e i . 8The example shows that this is not the case for economies with consumption externalities.We come back on this issue in Subsection 5.1.
Consider an economy with one physical commodity, three individuals, , and utility functions: We show that this economy does not have quasi-equilibria.
We break the argument in three simple steps.Suppose by contradiction that a quasiequilibrium (x * , p * ) exists.Through out the argument we use the fact that by definition p * ii = p * − h =i p * hi , for all i.
The market clearing (or feasibility) condition i∈I x * i = i∈I e i implies that aggregate consumption is strictly positive, i.e., i∈I x * i > 0. Therefore, since Assume without loss of generality that u 2 (x * ) < 0. Recall that by Step 1, p * ≥ 0. Then we prove the following step.
Step 2: In both cases, the interior of the budget set B 2 ( p) is nonempty implying that x * (solution to the cost minimization problem) is a solution to individual 2's utility maximization problem.9Further, since p * e 2 ≥ 0, in both cases 0 is in the budget set.But, then u 2 (x * ) must be greater than u 2 (0) = 0 contradicting u 2 (x * ) < 0.
Step However, by definition 0 = p * = 3 h=1 p * hi for all i, thereby implying that Proof.Since u 3 is strictly increasing in x 3 , the optimality conditions of individual 3 for cost minimization imply that p * 33 ≥ 0. By Steps 1 and 2, p * 13 ≥ 0, p * 23 ≥ 0, and p * = 0. Therefore, Then, if it were either p * 31 < 0 or p * 32 < 0, since p * 33 = 0, there could not be solution to the cost minimization problem of individual 3. Therefore, p * 31 = p * 32 = 0. Then since all prices are non negative and 0 = p * = h p * hi , all i, it must be p * = 0.
Step 3 shows that a quasi-equilibrium cannot exist as the latter requires p * = 0. Therefore, without free disposal, quasi-equilibria may fail to exist.
With free disposal, the allocation x * = 0 and the price p * defined by p * = p * 12 = p * 13 = p * 21 = p * 31 = 0, and and it is easily verified that at p * the allocation x * solves the cost minimization problems of every individual.

Non-existence of an equilibrium
The economy satisfies Assumption A with strictly positive endowments and individuals' utility functions strictly increasing in their own consumption.Under such conditions, all quasiequilibria of economies without externalities are equilibria.However, the example shows that for economies with externalities equilibria may fail to exist.
There is one physical commodity and three individuals, X i = R + , e i ∈ R ++ for i ∈ I = {1, 2, 3}.Utility functions are: Let (x * , p * ) be an equilibrium.Recall that, by definition, p * ii = p * − h =i p * hi for all i.The existence of a solution to the individual maximization problems has the following immediate implications.
(1) Since u i is strictly increasing in x i , p * ii > 0 for every i.
Claim: There cannot be a competitive equilibrium.
We conclude with two further observations.This economy has a continuum of quasiequilibria.They are described by p * = p * 12 = p * 13 = p * 21 = p * 31 = 0, p * 23 = p * 32 = −1, and r] and r = e 1 + e 2 + e 3 .In the economy with individuals 2 and 3 only, equilibria exist with p * = 0, p * 23 = p * 32 = −1, and x * 2 = x * 3 ≤ 1 2 (e 2 + e 3 ).Hence, the presence of individual 1, who is not affected by any external effects and whose consumption does not generate external effects on any other individual prevents the existence of an equilibrium.We further discuss this issue in Subsection 5.2.

The production economy behind the consumption externalities
In this section, we construct a standard production economy without externalities, E Y .We show that there is a simple one-to-one mapping between the equilibria of the two economies E Y and E. Thus existence of (quasi-)equilibria of E follows from the existence of equilibria of E Y .Similarly, survival and irreducibility conditions for E are derived from survival and irreducibility conditions for E Y .The main idea behind the construction of E Y is in the definition of the production set Y that transforms inputs of L physical commodities into I identical allocations, one for each individual.
The production economy E Y has L + (LI )I commodities.A generic bundle of R L × R (LI )I is denoted as (a, a 1 , . . ., a i , . . ., a I ) where a ∈ R L and a i = (a ih ) h∈I ∈ R LI for every i ∈ I.The consumption set of individual i is now defined as: The utility function of individual i is The next lemma describes the relation between the feasible allocations of the economies E and E Y .

Proof.
(1) By definition of Y , x 1 = x i = y i for all i ∈ I. Then setting x = x 1 and recalling that y i = (y ih ) h∈I , for every i ∈ I, the definition of Y implies that i∈I x i = i∈I y ii ≤ i∈I e i .Therefore x = x 1 is a feasible allocation of E.
Since the production set Y exhibits constant returns to scale, at equilibrium optimal profit must be zero.Thus profit maximization restricts the search of an equilibrium price of E Y to the price domain A price π is in * if and only if it satisfies restrictions identical to those generated by the Lindahl conjecture in the price domain of E, the economy with externalities.This is spelled out in the next key lemma.
Lemma 3. The price system π is in * if and only if 0 ≤ h∈I p hi ≤ p and h∈I p hi does not depend on i.
Proof.By definition, y ∈ Y implies that z = i∈I y ii + b for some b ∈ R L + .Thus, profits at (π, y) are Let A be a matrix of dimension 2L × (LI + L) defined as Thus π ∈ * iff the system of linear inequalities v • c > 0, Ac ≥ 0 does not have a solution c ∈ R LI +L .By the Theorem of the Alternative, this is true iff there exists θ Remark 4. Let y = −z, ( y i ) i∈I ∈ Y be a profit maximization bundle at π ∈ * .Lemma 3 implies that h∈I p hi = p whenever z > 0.Then, since feasibility in E Y entails z = i∈I e i , at an equilibrium of the production economy, the Lindahl compatibility condition h∈I p hi = p holds true for all commodities .
As E Y is a standard production economy (without externalities), we omit the definitions of equilibrium and quasi-equilibrium.
The next proposition exploits Lemmas 2 and 3 in order to show the existence of a simple one-to-one mapping between the equilibria of the economies E and E Y .Importantly such a mapping yields equivalent allocation-price pairs in the sense specified by the statement of the next proposition.
Proof.First, by Lemma 2, the mappings between (ξ, y) ∈ × Y and x * ∈ X preserve feasibility.Next we pick π, y as defined in (1) and we show that π ∈ * and that y maximizes profits at π .Since by construction h∈I p hi = p * ≥ 0, for all i ∈ I, Lemma 3 implies π ∈ * .Then, implies that y maximizes profits at π .Therefore, in order to conclude the argument it suffices to verify the optimality of individuals' choices in (1) and (2) at respectively π and p * .This is implied by the equalities π • η i = p * • e i and π • ξ i = p * i • x i , for ξ i = 0, . . ., 0, x i , 0, . . ., 0 , all x i ∈ X, and i ∈ I.It is verified that the later holds true since, by Lemma 3, h∈ I p h1 = h∈I p hi , for all i ∈ I.

Example (a).
We describe the production economy E Y and some of its properties by means of a simple example.Consider a 2 × 2 economy with the following characteristics.Endowments are e 1 0 and e 2 0, the consumption sets are X 1 = X 2 = R + × R ++ , and the quasi-linear utilities are: where α = 0 is a parameter measuring the intensity of the external effect of the second consumer on the first one, which could be either negative, α > 0, or positive, α < 0. We assume that e 1 1 > 1 and e 1 2 > 1 + α > 0. 10 The production set of E Y is: The consumption sets of E Y are: The utility functions are ũ1 (ξ 1 ) = u 1 (x 11 , x 12 ) and ũ2 (ξ 2 ) = u 2 (x 21 , x 22 ), and the initial endowments η i = (e i , 0 4 , 0 4 ), all i = 1, 2. The price vector is:

Equilibrium prices satisfy:
(1) p * ii 0, all i = 1, 2, because u i is strongly monotone in x ii .
The profit maximization problem is then The three properties above imply that q * 0 and therefore that the technological constraint is binding, i.e., y 11 + y 22 = z.Further, by Lemma 2-(2), at equilibrium, x 12 = x 22 = x 2 , x 21 = x 11 = x 1 , y ii = x ii = x i , all i = 1, 2, and z = e 1 + e 2 .Then, at equilibrium, the market clearing conditions must hold true in the market of physical commodities, i.e., Further, since (e 1 + e 2 ) 0, by Remark 4, it must be that q * = p * .We determine equilibrium allocations of the economy E. We choose the price normalization p * 2 = 1 and limit attention to interior equilibrium allocations x * 1 , x * 2 0. Equilibrium price and allocations of E are: , where r 2 = e 2 1 + e 2 2 .By Proposition 5, the corresponding competitive equilibrium of the production economy with the pro-10 As is well known, with quasi-linear preferences, some lower bounds on the endowments are required to get an 2+α , 0 , (0, 0) , (1+α)r 2 2+α , 1 .Since utilities are separable, competitive equilibrium allocations of the economy without externalities (α = 0) are competitive equilibrium allocations of the economy with externalities (α = 0) without markets for externalities.Such equilibrium allocations are never Pareto optimal when α = 0.The introduction of markets for externalities restores the Pareto optimality in the economy E. Indeed, the economy E Y is a standard production economy and as such its equilibrium allocations satisfy the First Fundamental Theorem of Welfare Economics.Then, by Proposition 5, the same properties hold true for the equilibrium allocations of economy with externalities E.

Existence of an equilibrium with markets for externalities
We prove that quasi-equilibria exist and we provide sufficient conditions under which quasiequilibria are equilibria, namely an irreducibility condition combined with a survival condition.Contrary to what happens in exchange economies without externalities, the argument requires an irreducibility condition even with strictly positive endowments and locally non-satiated preferences.

Existence of a quasi-equilibrium with markets for externalities
The following theorem is a consequence of Lemma 3, Proposition 5, and a general result, Proposition 2.2.2 in Florenzano (2003), for the existence of a quasi-equilibrium in production economies without externalities.
Theorem 6 (Existence of a quasi-equilibrium of the economy E).Under Assumption A, there exists a quasi-equilibrium (x * , p * ) of the economy E with p * ≥ 0.
Proof.Consider the production economy E Y defined in Section 4. By Assumption A, consumption sets are closed, convex and bounded from below, the sets P i (ξ i ) = ξ i ∈ i : ũi ξ i > ũi (ξ i ) are nonempty and convex, while the sets Further, by construction, the production set Y is closed, convex, and satisfies the possibility of inaction and the impossibility of free production.Therefore, the attainable consumption sets are nonempty and compact.Further, the weak survival assumption η i ∈ i − Y for all i ∈ I is satisfied since 0 ∈ i and −η i ∈ Y .Thus the assumptions of Proposition 2.2.2 in Florenzano (2003) are satisfied: E Y has a quasi-equilibrium with π = 0 that by Lemma 3 and Proposition 5 implies the existence of a quasi-equilibrium of E with p * ≥ 0.
Example in Subsection 3.1 shows that without free disposal the economy with externalities E may fail to have a quasi-equilibrium.This is because, in the absence of the free disposal condition i∈I y ii ≤ z, the production economy E Y does not satisfy the weak survival condition Florenzano (2003).

Irreducibility conditions
We derive equivalent irreducibility conditions of the economy with externalities from known conditions of irreducibility for production economies without externalities.We adapt the classical irreducibility conditions of McKenzie (1959McKenzie ( , 1961McKenzie ( , 1981) ) and Bergstrom (1976a) to the production economy E Y .

Assumption I.
The economy E Y is McKenzie-Bergstrom irreducible if for every feasible allocation (ξ i ) i∈I , y ∈ × Y and for every pair of non-empty sets I 1 and I 2 with I 1 ∪ I 2 = I and I 1 ∩ I 2 = ∅, there exist θ h > 0, h ∈ I 2 , ξ i i∈I ∈ and y ∈ Y satisfying the two following conditions: (1) It is verified that: (1) if Y = {0} and I 2 is a singleton, Assumption I coincides with the irreducibility condition of Bergstrom (1976a) for a classical exchange economy without free disposal, and (2) if the economy satisfies the irreducibility conditions of McKenzie (1959McKenzie ( , 1961McKenzie ( , 1981)), then it satisfies Assumption I.
We now translate Assumption I into an equivalent irreducibility condition for the economy with consumption externalities E.An externality economy is irreducible when any subset of individuals can pick a Pareto improving allocation whose total resources are less or equal than the sum of the endowments of the first group plus a weighted sum of endowments of the residual.

Assumption E (Irreducibility in E ). The economy E is irreducible if for every feasible allocation
x ∈ X and for every pair of non-empty sets I 1 and I 2 with I 1 ∪ I 2 = I and I 1 ∩ I 2 = ∅, there exist θ h > 0, h ∈ I 2 , and an allocation x = ( xi ) i∈I ∈ X satisfying the two following conditions: (1) The next proposition exploits the definition of the production technology Y to show that Assumptions I and E are equivalent.Proof.Let us assume that Assumption I is satisfied.Let x be a feasible allocation of E and I 1 and I 2 as in Assumption E.Then, by definition of Y and feasibility, y = − i∈I e i , x, . . ., x, . . ., x and ξ i = (0 L , 0 LI , . . ., 0 LI , x, 0 LI , . . ., 0 LI ) for all i ∈ I is a feasible allocation of E Y .Let y ∈ Y and ξ i i∈I as given by Assumption I. Then y = −z , y i i∈I with z ∈ R L + and y i = x for all i ∈ I for some For all i, ξ i = 0 L , 0 LI , . . ., 0 LI , x i , 0 LI , . . ., 0 LI for some x i ∈ X.Since η i = (e i , 0 LI , . . ., 0 LI , . . ., 0 LI ) for all i ∈ I, Condition 1 in Assumption I reads: Conversely, assume now that the economy E satisfies Assumption E. Let (ξ i ) i∈I , y be a feasible allocation of E Y and I 1 and I 2 as in Assumption I.Then, x such that ξ 1 = (0, x, 0, . . ., 0) is a feasible allocation of E .Let x as given by Assumption E.
Assumption E is satisfied in an economy without externalities if e i 0, all i ∈ I, and the utility functions are locally non-satiated.It is also satisfied with externalities if there are only two consumers, e i > 0, all i = 1, 2, and preferences are strongly monotonic in own consumption.In such a situation, without loss of generality, set I 1 = {1} and I 2 = {2} and choose x2 = x 2 and x1 ≤ x 1 + e 2 with u 1 ( x1 , x2 ) > u 1 (x 1 , x2 ).However, in the economy of the example in Subsection 3.2 with three individuals, although e i > 0 and preferences are strongly monotonic in own consumption, an equilibrium does not exist.Hence such an economy does not satisfy Assumption E with I 1 = {2, 3}, I 2 = {1} at the feasible allocation (0, r/2, r/2) with r = e 1 + e 2 + e 3 .
In general, whether or not an economy is irreducible depends on the sign and strength of the externalities.12If some commodities do not display negative external effects, under standard monotonicity in own consumption, the economy is irreducible.The next definition and proposition make this idea precise.Definition 8. Good * is socially desirable if for every pair of allocations x and x with x i = x i , = * , and x * i ≥ x * i , all i, with at least one strict inequality, it holds true that If there is at least one socially desirable commodity and individuals are positively endowed with it, the economy is irreducible.This is stated in the next proposition whose simple proof is left to the reader.Proposition 9.Under Assumption A, suppose that there exists at least one socially desirable commodity * ∈ L, and that e * i > 0 for all i ∈ I.Then, Assumption E is satisfied.
Thus, if the economy is not irreducible, all commodities necessarily display negative externalities.The example below suggests that the converse is not always true.In one commodity economies with more than two individuals, irreducibility fails if there are at least two individuals whose consumption has negative and sizable effects on the other.Therefore, even when all commodities display negative external effects, irreducibility holds whenever these effects are not too strong.

Example (b).
Consider a slightly different version of the economy given in Subsection 3.2, where u 1 (x 1 , x 2 , x 3 ) = x 1 and the utility functions of individuals 2 and 3 are now u 2 (x 1 , x 2 , x 3 ) = x 2 − αx 3 and u 3 (x 1 , x 2 , x 3 ) = x 3 − δx 2 , with parameters α ≥ 0 and δ ≥ 0 controlling the size of the negative externality.Irreducibility is satisfied if and only if αδ < 1.Hence, irreducibility fails if both α > 0 and δ > 0 and the combined effect (measure by αδ) is large.To show this, notice that since e i > 0, all i, and θ h > 0, h ∈ I 2 , Assumption E holds true if and only if for each feasible allocation x there is an allocation x satisfying condition (2) in Assumption E. If I 1 = {i}, the allocation xi > x i and xh = x h , for all h ∈ I 2 , satisfies condition (2).Similarly, if I 1 = {1, i} with i = 1, the allocation x1 > x 1 , xi = x i and xh = x h satisfies condition (2).Thus, the economy is irreducible if and only if condition (2) is satisfied for I 1 = {2, 3}.Condition (2) holds if and only if ( x2 − x 2 ) − α ( x3 − x 3 ) ≥ 0 and ( x3 − x 3 ) − δ ( x2 − x 2 ) ≥ 0, with at least one strict inequality.By Farkas's Lemma those inequalities are simultaneously satisfied if and only if αδ < 1.

Survival conditions
The survival assumption is a sufficient condition guaranteeing that at a quasi-equilibrium at least one individual's consumption is not a minimal expenditure bundle.When combined with the irreducibility conditions, it implies that at a quasi-equilibrium no individual's consumption is a minimal expenditure bundle, that is, that individuals consumptions are utility maximizers.Hence a quasi-equilibrium is an equilibrium.We posit the following assumption.

Assumption S (Survival in E)
. For all i ∈ I, Int X i = ∅ and i∈I e i ∈ R L ++ ∩ Int i∈I X i .
Assumption S is clearly satisfied under standard survival assumptions for exchange economies without externalities, i.e., X i = R L + for all i ∈ I and i∈I e i 0. The following proposition shows that if the economy E satisfies Assumption S, then the production economy E Y satisfies the survival assumption of Florenzano (2003), under which at a quasi-equilibrium at least one individual's consumption is a utility maximizer.
The production technology is described by the production setY = y = (−z, y 1 , . . ., y i , . . ., y I ) ∈ R L − × R (LI )I : y i = y h , ∀i, h ∈ I and i∈I y ii ≤ zIt is verified that Y is a closed convex cone that satisfies the possibility of inaction and the impossibility of free production, i.e., Y ∩ R L + × R (LI )I + = {0}.The condition i∈I y ii ≤ z allows for free disposal of physical commodities.
Proposition 7. The economy E Y is McKenzie-Bergstrom irreducible if and only if the economy E satisfies Assumption E. 11