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Arrow–Debreu model

In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.

The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954, and Lionel W. McKenzie independently in 1954, with later improvements in 1959.

The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria.

Arrow (1972) and Debreu (1983) were separately awarded the Nobel Prize in Economics for their development of the model. McKenzie, however, did not receive the award.

Formal statement
"The contents of both theorems [fundamental theorems of welfare economics] are old beliefs in economics. Arrow and Debreu have recently treated this question with techniques permitting proofs.""This statement is precisely correct; once there were beliefs, now there was knowledge.

But more was at stake. Great scholars change the way that we think about the world, and about what and who we are. The Arrow-Debreu model, as communicated in Theory of Value changed basic thinking, and it quickly became the standard model of price theory. It is the “benchmark” model in Finance, International Trade, Public Finance, Transportation, and even macroeconomics... In rather short order it was no longer “as it is” in Marshall, Hicks, and Samuelson; rather it became “as it is” in Theory of Value."

This section follows the presentation in, which is based on.

Intuitive description of the Arrow–Debreu model
The Arrow–Debreu model models an economy as a combination of three kinds of agents: the households, the producers, and the market. The households and producers transact with the market, but not with each other directly.

The households possess endowments (bundles of commodities they begin with), which one may think of as "inheritance". For the sake of mathematical clarity, all households are required to sell all their endowment to the market at the beginning. If they wish to retain some of the endowment, they would have to repurchase from the market later. The endowments may be working hours, use of land, tons of corn, etc.

The households possess proportional ownerships of producers, which can be thought of as joint-stock companies. The profit made by producer $$j$$ is divided among the households in proportion to how much stock each household holds for the producer $$j$$. Ownership is imposed at the beginning, and the households may not sell, buy, create, or discard them.

The households receive a budget, as the sum of income from selling endowments and dividend from producer profits.

The households possess preferences over bundles of commodities, which under the assumptions given, makes them utility maximizers. The households choose the consumption plan with the highest utility that they can afford using their budget.

The producers are capable of transforming bundles of commodities into other bundles of commodities. The producers have no separate utility functions. Instead, they are all purely profit maximizers.

The market is only capable of "choosing" a market price vector, which is a list of prices for each commodity, which every producer and household takes (there is no bargaining behavior—every producer and household is a price taker). The market has no utility or profit. Instead, the market aims to choose a market price vector such that, even though each household and producer is maximizing their own utility and profit, their consumption plans and production plans "harmonize". That is, "the market clears". In other words, the market is playing the role of a "Walrasian auctioneer".

Notation setup
In general, we write indices of agents as superscripts, and vector coordinate indices as subscripts.

useful notations for real vectors

 * $$x \succeq y$$ if $$\forall n, x_n \geq y_n$$
 * $$\R^N_+$$ is the set of $$x$$ such that $$x \succeq 0$$
 * $$\R_{++}^N$$ is the set of $$x$$ such that $$x \succ 0$$
 * $$\Delta_N = \left\{x\in \R^N: x_1, ..., x_N \geq 0, \sum_{n\in 1:N} x_n = 1\right\}$$ is the N-simplex. We often call it the price simplex since we will sometimes scale the price vector to lie on it.

market

 * The commodities are indexed as $$n\in 1:N$$. Here $$N$$ is the number of commodities that exists in the economy. It is a finite number.
 * The price vector $$p = (p_1, ..., p_N) \in \R_{++}^N$$ is a vector of length $$N$$, with each coordinate being the price of a commodity. The prices may be zero or positive.

households

 * The households are indexed as $$i\in I$$.
 * Each household begins with an endowment of commodities $$r^i\in \R^N_+$$.
 * Each household begins with a tuple of ownerships of the producers $$\alpha^{i,j} \geq 0$$. The ownerships satisfy $$\sum_{i\in I} \alpha^{i,j} = 1 \quad \forall j\in J $$.
 * The budget that the household receives is the sum of its income from selling endowments at the market price, plus profits from its ownership of producers:$$M^i(p) = \langle p, r^i\rangle + \sum_{j\in J}\alpha^{i,j}\Pi^j(p)$$($$M$$ stands for money)
 * Each household has a Consumption Possibility Set $$CPS^i\subset \R_+^N$$.
 * Each household has a preference relation $$\succeq^i$$ over $$CPS^i$$.
 * With assumptions on $$\succeq^i$$ (given in the next section), each preference relation is representable by a utility function $$u^i: CPS^i \to [0, 1]$$ by the Debreu theorems. Thus instead of maximizing preference, we can equivalently state that the household is maximizing its utility.
 * A consumption plan is a vector in $$CPS^i$$, written as $$x^i$$.
 * $$U_+^i(x^i)$$ is the set of consumption plans at least as preferable as $$x^i$$.
 * The budget set is the set of consumption plans that it can afford:$$B^i(p) = \{x^i \in CPS^i : \langle p, x^i \rangle \leq M^i(p)\}$$.
 * For each price vector $$p$$, the household has a demand vector for commodities, as $$D^i(p)\in \R_+^N$$. This function is defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.$$D^i(p) := \arg\max_{x^i \in B^i(p)} u^i(x^i)$$It may not be well-defined for all $$p \in \R^N_{++}$$. However, we will use enough assumptions such that that it is well-defined at equilibrium price vectors.

producers

 * The producers are indexed as $$j\in J$$.
 * Each producer has a Production Possibility Set $$PPS^j$$. Note that the supply vector may have both positive and negative coordinates. For example, $$(-1, 1, 0)$$ indicates a production plan that uses up 1 unit of commodity 1 to produce 1 unit of commodity 2.
 * A production plan is a vector in $$PPS^j$$, written as $$y^j$$.
 * For each price vector $$p$$, the producer has a supply vector for commodities, as $$S^j(p)\in \R^N$$. This function will be defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.$$S^j(p) := \arg\max_{y^j\in PPS^j} \langle p, y^j\rangle$$It may not be well-defined for all $$p \in \R^N_{++}$$. However, we will use enough assumptions such that that it is well-defined at equilibrium price vectors.
 * The profit is $$\Pi^j(p) := \langle p, S^j(p)\rangle = \max_{y^j\in PPS^j} \langle p, y^j\rangle$$

aggregates

 * aggregate consumption possibility set $$CPS = \sum_{i\in I}CPS^i$$.
 * aggregate production possibility set $$PPS = \sum_{j\in J}PPS^j$$.
 * aggregate endowment $$r = \sum_i r^i$$
 * aggregate demand $$D(p) := \sum_i D^i(p)$$
 * aggregate supply $$S(p) := \sum_j S^j(p)$$
 * excess demand $$Z(p) = D(p) - S(p) - r$$

the whole economy
\leq 0 \text{ if } p_n = 0 \\ = 0 \text{ if } p_n > 0 \end{cases}$$That is, if a commodity is not free, then supply exactly equals demand, and if a commodity is free, then supply is equal or greater than demand (we allow free commodity to be oversupplied).
 * An economy is a tuple $$(N, I, J, CPS^i, \succeq^i, PPS^j)$$. That is, it is a tuple specifying the commodities, the consumer preferences, consumption possibility sets, and the producers' production possibility sets.
 * An economy with initial distribution is an economy, along with an initial distribution tuple $$(r^i, \alpha^{i,j})_{i\in I, j\in J}$$ for the economy.
 * A state of the economy is a tuple of price, consumption plans, and production plans for each household and producer: $$((p_n)_{n\in 1:N}, (x^i)_{i\in I}, (y^j)_{j\in J})$$.
 * A state is feasible iff each $$x^i \in CPS^i$$, each $$y^j\in PPS^j$$, and $$\sum_{i\in I}x^i \preceq \sum_{j\in J}y^j + r$$.
 * The feasible production possibilities set, given endowment $$r$$, is $$PPS_r := \{y\in PPS: y+r \succeq 0\}$$.
 * Given an economy with distribution, the state corresponding to a price vector $$p$$ is $$(p, (D^i(p))_{i\in I}, (S^j(p))_{j\in J})$$.
 * Given an economy with distribution, a price vector $$p$$ is an equilibrium price vector for the economy with initial distribution, iff$$Z(p)_n \begin{cases}
 * A state is an equilibrium state iff it is the state corresponding to an equilibrium price vector.

Imposing an artificial restriction
The functions $$ D^i(p), S^j(p)$$ are not necessarily well-defined for all price vectors $$p$$. For example, if producer 1 is capable of transforming $$t$$ units of commodity 1 into $$\sqrt{(t+1)^2-1}$$ units of commodity 2, and we have $$p_1 / p_2 < 1$$, then the producer can create plans with infinite profit, thus $$\Pi^j(p) = +\infty$$, and $$S^j(p)$$ is undefined.

Consequently, we define "restricted market" to be the same market, except there is a universal upper bound $$C$$, such that every producer is required to use a production plan $$\|y^j\| \leq C$$, and each household is required to use a consumption plan $$\|x^i\| \leq C$$. Denote the corresponding quantities on the restricted market with a tilde. So for example, $$\tilde Z(p)$$ is the excess demand function on the restricted market.

$$C$$ is chosen to be "large enough" for the economy, so that the restriction is not in effect under equilibrium conditions (see next section). In detail, $$C$$ is chosen to be large enough such that:

 * For any consumption plan $$x$$ such that $$x \succeq 0, \|x\| = C$$, the plan is so "extravagant" that even if all the producers coordinate, they would still fall short of meeting the demand.
 * For any list of production plans for the economy $$(y^j\in PPS^j)_{j\in J}$$, if $$\sum_{j\in J} y^j + r \succeq 0$$, then $$\|y^j\| < C$$for each $$j\in J$$. In other words, for any attainable production plan under the given endowment $$r$$, each producer's individual production plan must lie strictly within the restriction.

Each requirement is satisfiable.

 * Define the set of attainable aggregate production plans to be $$PPS_r = \left\{\sum_{j\in J} y^j : y^j \in PPS^j \text{ for each } j\in J, \text{ and }\sum_{j\in J} y^j + r \succeq 0\right\}$$, then under the assumptions for the producers given above (especially the "no arbitrarily large free lunch" assumption), $$PPS_r$$ is bounded for any $$r \succeq 0$$ (proof omitted). Thus the first requirement is satisfiable.
 * Define the set of attainable individual production plans to be $$PPS_r^j := \{y^j \in PPS^j: y^j\text{ is a part of some attainable production plan under endowment }r\}$$then under the assumptions for the producers given above (especially the "no arbitrarily large transformations" assumption), $$PPS_r^j$$ is bounded for any $$j\in J, r \succeq 0$$ (proof omitted). Thus the second requirement is satisfiable.

The two requirements together imply that the restriction is not a real restriction when the production plans and consumption plans are "interior" to the restriction.

 * At any price vector $$p$$, if $$\|\tilde S^j(p)\| < C$$, then $$S^j(p) $$ exists and is equal to $$\tilde S^j(p) $$. In other words, if the production plan of a restricted producer is interior to the artificial restriction, then the unrestricted producer would choose the same production plan. This is proved by exploiting the second requirement on $$C$$.
 * If all $$S^j(p) = \tilde S^j(p)$$, then the restricted and unrestricted households have the same budget. Now, if we also have $$\|\tilde D^i(p)\| < C$$, then $$D^i(p) $$ exists and is equal to $$\tilde D^i(p) $$. In other words, if the consumption plan of a restricted household is interior to the artificial restriction, then the unrestricted household would choose the same consumption plan. This is proved by exploiting the first requirement on $$C$$.

These two propositions imply that equilibria for the restricted market are equilibria for the unrestricted market:

Existence of general equilibrium
As the last piece of the construction, we define Walras's law:

 * The unrestricted market satisfies Walras's law at $$p$$ iff all $$S^j(p), D^i(p)$$ are defined, and $$ \langle p, Z(p)\rangle = 0$$, that is,$$ \sum_{j\in J} \langle p,S^j(p)\rangle + \langle p, r\rangle = \sum_{i\in I} \langle p, D^i(p)\rangle$$
 * The restricted market satisfies Walras's law at $$p$$ iff $$ \langle p, \tilde Z(p)\rangle = 0$$.

Walras's law can be interpreted on both sides:

 * On the side of the households, it is saying that the aggregate household expenditure is equal to aggregate profit and aggregate income from selling endowments. In other words, every household spends its entire budget.
 * On the side of the producers, it is saying that the aggregate profit plus the aggregate cost equals the aggregate revenue.

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Note that the above proof does not give an iterative algorithm for finding any equilibrium, as there is no guarantee that the function $$f$$ is a contraction. This is unsurprising, as there is no guarantee (without further assumptions) that any market equilibrium is a stable equilibrium.

The role of convexity

In 1954, McKenzie and the pair Arrow and Debreu independently proved the existence of general equilibria by invoking the Kakutani fixed-point theorem on the fixed points of a continuous function from a compact, convex set into itself. In the Arrow–Debreu approach, convexity is essential, because such fixed-point theorems are inapplicable to non-convex sets. For example, the rotation of the unit circle by 90 degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non-convex. In contrast, the same rotation applied to the convex hull of the unit circle leaves the point (0,0) fixed. Notice that the Kakutani theorem does not assert that there exists exactly one fixed point. Reflecting the unit disk across the y-axis leaves a vertical segment fixed, so that this reflection has an infinite number of fixed points.

Non-convexity in large economies
The assumption of convexity precluded many applications, which were discussed in the Journal of Political Economy from 1959 to 1961 by Francis M. Bator, M. J. Farrell, Tjalling Koopmans, and Thomas J. Rothenberg. proved the existence of economic equilibria when some consumer preferences need not be convex. In his paper, Starr proved that a "convexified" economy has general equilibria that are closely approximated by "quasi-equilbria" of the original economy; Starr's proof used the Shapley–Folkman theorem.

Uzawa equivalence theorem
(Uzawa, 1962) showed that the existence of general equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras’s Law is equivalent to Brouwer fixed-Point theorem. Thus, the use of Brouwer's fixed-point theorem is essential for showing that the equilibrium exists in general.

Fundamental theorems of welfare economics
In welfare economics, one possible concern is finding a Pareto-optimal plan for the economy.

Intuitively, one can consider the problem of welfare economics to be the problem faced by a master planner for the whole economy: given starting endowment $$r$$ for the entire society, the planner must pick a feasible master plan of production and consumption plans $$((x^i)_{i\in I}, (y^j)_{j\in J})$$. The master planner has a wide freedom in choosing the master plan, but any reasonable planner should agree that, if someone's utility can be increased, while everyone else's is not decreased, then it is a better plan. That is, the Pareto ordering should be followed.

Define the Pareto ordering on the set of all plans $$((x^i)_{i\in I}, (y^j)_{j\in J})$$ by $$((x^i)_{i\in I}, (y^j)_{j\in J}) \succeq((x'^i)_{i\in I}, (y'^j)_{j\in J})$$ iff $$x^i \succeq^i x'^i$$ for all $$i\in I$$.

Then, we say that a plan is Pareto-efficient with respect to a starting endowment $$r$$, iff it is feasible, and there does not exist another feasible plan that is strictly better in Pareto ordering.

In general, there are a whole continuum of Pareto-efficient plans for each starting endowment $$r$$.

With the set up, we have two fundamental theorems of welfare economics:

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Proof idea: any Pareto-optimal consumption plan is separated by a hyperplane from the set of attainable consumption plans. The slope of the hyperplane would be the equilibrium prices. Verify that under such prices, each producer and household would find the given state optimal. Verify that Walras's law holds, and so the expenditures match income plus profit, and so it is possible to provide each household with exactly the necessary budget. $$

Convexity vs strict convexity
The assumptions of strict convexity can be relaxed to convexity. This modification changes supply and demand functions from point-valued functions into set-valued functions (or "correspondences"), and the application of Brouwer's fixed-point theorem into Kakutani's fixed-point theorem.

This modification is similar to the generalization of the minimax theorem to the existence of Nash equilibria.

The two fundamental theorems of welfare economics holds without modification.

Equilibrium vs "quasi-equilibrium"
The definition of market equilibrium assumes that every household performs utility maximization, subject to budget constraints. That is, $$\begin{cases} \max_{x^i} u^i(x^i) \\ \langle p, x^i\rangle \leq M^i(p) \end{cases}$$The dual problem would be cost minimization subject to utility constraints. That is,$$\begin{cases} u^i(x^i) \geq u^i_0\\ \min_{x^i} \langle p, x^i\rangle \end{cases}$$for some real number $$u^i_0$$. The duality gap between the two problems is nonnegative, and may be positive. Consequently, some authors study the dual problem and the properties of its "quasi-equilibrium" (or "compensated equilibrium" ). Every equilibrium is a quasi-equilibrium, but the converse is not necessarily true.

Accounting for strategic bargaining
In the model, all producers and households are "price takers", meaning that they simply transact with the market using the price vector $$p$$. In particular, behaviors such as cartel, monopoly, consumer coalition, etc are not modelled. Edgeworth's limit theorem shows that under certain stronger assumptions, the households can do no better than price-take at the limit of an infinitely large economy.

Setup
In detail, we continue with the economic model on the households and producers, but we consider a different method to design production and distribution of commodities than the market economy. It may be interpreted as a model of a "socialist" economy.

 * There is no money, market, or private ownership of producers.
 * Since we have abolished private ownership, money, and the profit motive, there is no point in distinguishing one producer from the next. Consequently, instead of each producer planning individually $$y^j \in PPS^j$$, it is as if the whole society has one great producer producing $$y\in PPS$$.
 * Households still have the same preferences and endowments, but they no longer have budgets.
 * Producers do not produce to maximize profit, since there is no profit. All households come together to make a state $$((x_i)_{i\in I}, y)$$—a production and consumption plan for the whole economy—with the following constraints:$$x^i \in CPS^i, y \in PPS, y\succeq \sum_i (x^i- r^i)$$
 * Any nonempty subset of households may eliminate all other households, while retaining control of the producers.

This economy is thus a cooperative game with each household being a player, and we have the following concepts from cooperative game theory:

Since we assumed that any nonempty subset of households may eliminate all other households, while retaining control of the producers, the only states that can be executed are the core states. A state that is not a core state would immediately be objected by a coalition of households.
 * A blocking coalition is a nonempty subset of households, such that there exists a strictly Pareto-better plan even if they eliminate all other households.
 * A state is a core state iff there are no blocking coalitions.
 * The core of an economy is the set of core states.

We need one more assumption on $$PPS$$, that it is a cone, that is, $$k \cdot PPS \subset PPS$$ for any $$k \geq 0$$. This assumption rules out two ways for the economy to become trivial.

 * The curse of free lunch: In this model, the whole $$PPS$$ is available to any nonempty coalition, even a coalition of one. Consequently, if nobody has any endowment, and yet $$PPS$$ contains some "free lunch" $$y\succ 0$$, then (assuming preferences are monotonic) every household would like to take all of $$y$$ for itself, and consequently there exists *no* core state. Intuitively, the picture of the world is a committee of selfish people, vetoing any plan that doesn't give the entire free lunch to itself.
 * The limit to growth: Consider a society with 2 commodities. One is "labor" and another is "food". Households have only labor as endowment, but they only consume food. The $$PPS$$ looks like a ramp with a flat top. So, putting in 0-1 thousand hours of labor produces 0-1 thousand kg of food, linearly, but any more labor produces no food. Now suppose each household is endowed with 1 thousand hours of labor. It's clear that every household would immediately block every other household, since it's always better for one to use the entire $$PPS$$ for itself.

Main results (Debreu and Scarf, 1963)
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In Debreu and Scarf's paper, they defined a particular way to approach an infinitely large economy, by "replicating households". That is, for any positive integer $$K$$, define an economy where there are $$K$$ households that have exactly the same consumption possibility set and preference as household $$i$$.

Let $$x^{i, k}$$ stand for the consumption plan of the $$k$$-th replicate of household $$i$$. Define a plan to be equitable iff $$x^{i, k} \sim^i x^{i, k'}$$ for any $$i\in I$$ and $$k, k'\in K$$.

In general, a state would be quite complex, treating each replicate differently. However, core states are significantly simpler: they are equitable, treating every replicate equally.

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Consequently, when studying core states, it is sufficient to consider one consumption plan for each type of households. Now, define $$C_K$$ to be the set of all core states for the economy with $$K$$ replicates per household. It is clear that $$C_1 \supset C_2 \supset \cdots$$, so we may define the limit set of core states $$C := \cap_{K=1}^\infty C_K$$.

We have seen that $$C$$ contains the set of market equilibria for the original economy. The converse is true under minor additional assumption:

The assumption that $$PPS$$ is a polygonal cone, or every $$CPS^i$$ has nonempty interior, is necessary to avoid the technical issue of "quasi-equilibrium". Without the assumption, we can only prove that $$C$$ is contained in the set of quasi-equilibria.

Accounting for nonconvexity
The assumption that production possibility sets are convex is a strong constraint, as it implies that there is no economy of scale. Similarly, we may consider nonconvex consumption possibility sets and nonconvex preferences. In such cases, the supply and demand functions $$S^j(p), D^i(p)$$ may be discontinuous with respect to price vector, thus a general equilibrium may not exist.

However, we may "convexify" the economy, find an equilibrium for it, then by the Shapley–Folkman–Starr theorem, it is an approximate equilibrium for the original economy.

In detail, given any economy satisfying all the assumptions given, except convexity of $$PPS^j, CPS^i$$ and $$\succeq^i$$, we define the "convexified economy" to be the same economy, except that

 * $$PPS'^j = \mathrm{Conv}(PPS^j)$$
 * $$CPS'^i = \mathrm{Conv}(CPS^i)$$
 * $$x \succeq'^i y$$ iff $$\forall z \in CPS^i, y \in \mathrm{Conv}(U_+^i(z)) \implies x \in \mathrm{Conv}(U_+^i(z)) $$.

where $$\mathrm{Conv}$$ denotes the convex hull.

With this, any general equilibrium for the convexified economy is also an approximate equilibrium for the original economy. That is, if $$p^*$$ is an equilibrium price vector for the convexified economy, then $$\begin{align} d(D'(p^*) - S'(p^*), D(p^*) - S(p^*)) &\leq N\sqrt{L} \\ d(r, D(p^*) - S(p^*)) &\leq N\sqrt{L} \end{align}$$where $$d(\cdot, \cdot)$$ is the Euclidean distance, and $$L$$ is any upper bound on the inner radii of all $$PPS^j, CPS^i$$ (see page on Shapley–Folkman–Starr theorem for the definition of inner radii).

The convexified economy may not satisfy the assumptions. For example, the set $$\{(x, 0): x \geq 0\}\cup \{(x,y): xy = 1, x > 0\}$$ is closed, but its convex hull is not closed. Imposing the further assumption that the convexified economy also satisfies the assumptions, we find that the original economy always has an approximate equilibrium.

Accounting for time, space, and uncertainty
The commodities in the Arrow–Debreu model are entirely abstract. Thus, although it is typically represented as a static market, it can be used to model time, space, and uncertainty by splitting one commodity into several, each contingent on a certain time, place, and state of the world. For example, "apples" can be split into "apples in New York in September if oranges are available" and "apples in Chicago in June if oranges are not available".

Given some base commodities, the Arrow–Debreu complete market is a market where there is a separate commodity for every future time, for every place of delivery, for every state of the world under consideration, for every base commodity.

In financial economics the term "Arrow–Debreu" most commonly refers to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (the price of such a security being a so-called "state price"). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities.

Since the work of Breeden and Lizenberger in 1978, a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.

Accounting for the existence of money
"No theory of money is offered here, and it is assumed that the economy works without the help of a good serving as medium of exchange.""To the pure theorist, at the present juncture the most interesting and challenging aspect of money is that it can find no place in an Arrow–Debreu economy. This circumstance should also be of considerable significance to macroeconomists, but it rarely is."Typically, economists consider the functions of money to be as a unit of account, store of value, medium of exchange, and standard of deferred payment. This is however incompatible with the Arrow–Debreu complete market described above. In the complete market, there is only a one-time transaction at the market "at the beginning of time". After that, households and producers merely execute their planned productions, consumptions, and deliveries of commodities until the end of time. Consequently, there is no use for storage of value or medium of exchange. This applies not just to the Arrow–Debreu complete market, but also to models (such as those with markets of contingent commodities and Arrow insurance contracts) that differ in form, but are mathematically equivalent to it.

Computing general equilibria
Scarf (1967) was the first algorithm that computes the general equilibrium. See Scarf (2018) and Kubler (2012) for reviews.

Number of equilibria
Certain economies at certain endowment vectors may have infinitely equilibrium price vectors. However, "generically", an economy has only finitely many equilibrium price vectors. Here, "generically" means "on all points, except a closed set of Lebesgue measure zero", as in Sard's theorem.

There are many such genericity theorems. One example is the following:

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title: Arrow–Debreu model


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United Ireland

United Ireland (Éire Aontaithe), also referred to as Irish reunification or a New Ireland,     is the proposition that all of the island of Ireland should be a single sovereign state. At present, the island is divided politically: the sovereign state of Ireland (legally described also as the Republic of Ireland) has jurisdiction over the majority of Ireland, while Northern Ireland, which lies entirely within (but consists of only 6 of 9 counties of) the Irish province of Ulster, is part of the United Kingdom. Achieving a united Ireland is a central tenet of Irish nationalism and Republicanism, particularly of both mainstream and dissident republican political and paramilitary organisations. Unionists support Northern Ireland remaining part of the United Kingdom and oppose Irish unification.

Ireland has been partitioned since May 1921, when the implementation of the Government of Ireland Act 1920 created the states of Southern Ireland and Northern Ireland within the United Kingdom, with the former becoming independent, and the other petitioning to remain a part of the UK. The Anglo-Irish Treaty, which led to the establishment in December 1922 of a dominion called the Irish Free State, recognised partition, but this was opposed by anti-Treaty republicans. When the anti-Treaty Fianna Fáil party came to power in the 1930s, it adopted a new constitution which claimed sovereignty over the entire island. The Irish Republican Army (IRA) had a united Ireland as its goal during the conflict with British security forces and loyalist paramilitaries from the 1960s to the 1990s known as The Troubles. The Good Friday Agreement signed in 1998, which ended the conflict, acknowledged the legitimacy of the desire for a united Ireland, while declaring that it could be achieved only with the consent of a majority of the people of both jurisdictions on the island, and providing a mechanism for ascertaining this in certain circumstances.

In 2016, Sinn Féin called for a referendum on a united Ireland following Brexit, the decision by the United Kingdom to leave the European Union (EU). The decision had increased the perceived likelihood of a united Ireland, in order to avoid the possible requirement for a hard border between Northern Ireland and the Republic of Ireland, though the imposition of a hard border has not, as yet, eventuated. Fine Gael Taoiseach Enda Kenny successfully negotiated that in the event of reunification, Northern Ireland will become part of the EU, just as East Germany was permitted to join the EU's predecessor institutions by reuniting with the rest of Germany after the fall of the Berlin Wall.

The majority of Ulster Protestants, almost half the population of Northern Ireland, favour continued union with Great Britain, and have done so historically. Four of the six counties have Irish Catholic majorities, and majorities voting for Irish nationalist parties, and Catholics have become the plurality in Northern Ireland as of 2021. The religious denominations of the citizens of Northern Ireland are only a guide to likely political preferences, as there are both Protestants who favour a united Ireland, and Catholics who support the union. Two surveys in 2011 identified a significant number of Catholics who favoured the continuation of the union without identifying themselves as unionists or British.

Legal basis for future change
Article 3.1 of the Constitution of Ireland "recognises that a united Ireland shall be brought about only by peaceful means with the consent of a majority of the people, democratically expressed, in both jurisdictions in the island". This provision was introduced in 1999 after implementation of the Good Friday Agreement, as part of replacing the old Articles 2 and 3, which had laid a direct claim to the whole island as the national territory.

The Northern Ireland Act 1998, a statute of the Parliament of the United Kingdom, provides that Northern Ireland will remain within the United Kingdom unless a majority of the people of Northern Ireland vote to form part of a united Ireland. It specifies that the Secretary of State for Northern Ireland "shall exercise the power [to hold a referendum] if at any time it appears likely to him that a majority of those voting would express a wish that Northern Ireland should cease to be part of the United Kingdom and form part of a united Ireland". Such referendums may not take place within seven years of each other.

The Northern Ireland Act 1998 supersedes previous similar legislative provisions. The Northern Ireland Constitution Act 1973 also provided that Northern Ireland remained part of the United Kingdom unless a majority voted otherwise in a referendum, while under the Ireland Act 1949 the consent of the Parliament of Northern Ireland was needed for a united Ireland. In 1985, the Anglo-Irish Agreement affirmed, while providing for devolved government in Northern Ireland, and an advisory role for the Republic of Ireland government, that any change in the status of Northern Ireland would only come about with the consent of a majority of the people of Northern Ireland.

Home Rule, resistance and the Easter Rising
The Kingdom of Ireland as a whole had become part of the United Kingdom of Great Britain and Ireland under the Acts of Union 1800. From the 1870s, support for some form of an elected parliament in Dublin grew. In 1870, Isaac Butt, who was a Protestant, formed the Home Government Association, which became the Home Rule League. Charles Stewart Parnell, also a Protestant, became leader in 1880, and the organisation became the Irish National League in 1882. Despite the religion of its early leaders, its support was strongly associated with Irish Catholics. In 1886, Parnell formed a parliamentary alliance with Liberal Party Prime Minister William Ewart Gladstone and secured the introduction of the First Home Rule Bill. This was opposed by the Conservative Party and led to a split in the Liberal Party, the Liberal Unionist Party. Opposition in Ireland was concentrated in the heavily Protestant counties in Ulster. The difference in religious background was a legacy of the Ulster Plantation in the early seventeenth century. In 1893, the Second Home Rule Bill passed in the House of Commons, but was defeated in the House of Lords, where the Conservatives dominated. A Third Home Rule Bill was introduced in 1912, and in September 1912, just under half a million men and women signed the Ulster Covenant to swear they would resist its application in Ulster. The Ulster Volunteer Force were formed in 1913 as a militia to resist Home Rule.

The Government of Ireland Act 1914 (previously known as the Third Home Rule Bill) provided for a unitary devolved Irish Parliament, a culmination of several decades of work from the Irish Parliamentary Party. It was signed into law in September 1914 in the midst of the Home Rule Crisis and at the outbreak of the First World War. On the same day, the Suspensory Act 1914 suspended its actual operation.In 1916, a group of revolutionaries led by the Irish Republican Brotherhood launched the Easter Rising, during which they issued a Proclamation of the Irish Republic. The rebellion was not successful and sixteen of the leaders were executed. The small separatist party Sinn Féin became associated with the Rising in its aftermath as several of those involved in it were party members.

The Irish Convention held between 1917 and 1918 sought to reach agreement on manner in which Home Rule would be implemented after the war. All Irish parties were invited, but Sinn Féin boycotted the proceedings. By the end of the First World War, a number of moderate unionists came to support Home Rule, believing that it was the only way to keep a united Ireland in the United Kingdom. The Irish Dominion League opposed partition of Ireland into separate southern and northern jurisdictions, while arguing that the whole of Ireland should be granted dominion status with the British Empire. At the 1918 election Sinn Féin won 73 of the 105 seats; however, there was a strong regional divide, with the Ulster Unionist Party (UUP) winning 23 of the 38 seats in Ulster. Sinn Féin had run on a manifesto of abstaining from the United Kingdom House of Commons, and from 1919 met in Dublin as Dáil Éireann. At its first meeting, the Dáil adopted the Declaration of Independence of the Irish Republic, a claim which it made in respect of the entire island. Supporters of this Declaration fought in the Irish War of Independence.

Two jurisdictions
During this period, the Government of Ireland Act 1920 repealed the previous 1914 Act, and provided for two separate devolved parliaments in Ireland. It defined Northern Ireland as "the parliamentary counties of Antrim, Armagh, Down, Fermanagh, Londonderry and Tyrone, and the parliamentary boroughs of Belfast and Londonderry" and Southern Ireland "so much of Ireland as is not comprised within the said parliamentary counties and boroughs". Section 3 of this Act provided that the parliaments may be united by identical acts of parliament:

Sinn Féin did not recognise this act, treating elections to the respective parliaments as a single election to the Second Dáil. While the Parliament of Northern Ireland sat from 1921 to 1972, the Parliament of Southern Ireland was suspended after its first meeting was boycotted by the Sinn Féin members, who comprised 124 of its 128 MPs. A truce in the War of Independence was called in July 1921, followed by negotiations in London between the government of the United Kingdom and a Sinn Féin delegation. On 6 December 1921, they signed the Anglo-Irish Treaty, which led to the establishment of the Irish Free State the following year, a dominion within the British Empire.

With respect to Northern Ireland, Articles 11 and 12 of the Treaty made special provision for it including as follows:

The Prime Minister of Northern Ireland, Sir James Craig, speaking in the House of Commons of Northern Ireland in October 1922 said that "when 6 December [1922] is passed the month begins in which we will have to make the choice either to vote out or remain within the Free State". He said it was important that that choice be made as soon as possible after 6 December 1922 "in order that it may not go forth to the world that we had the slightest hesitation". On 7 December 1922, the day after the establishment of the Irish Free State, the Houses of the Parliament of Northern Ireland resolved to make the following address to the King so as to exercise the rights conferred on Northern Ireland under Article 12 of the Treaty:

The King received it the following day. These steps cemented Northern Ireland's legal separation from the Irish Free State.

In Irish republican legitimist theory, the Treaty was illegitimate and could not be approved. According to this theory, the Second Dáil did not dissolve and members of the Republican Government remained as the legitimate government of the Irish Republic declared in 1919. Adherents to this theory rejected the legitimacy of both the Irish Free State and Northern Ireland.

The report of Boundary Commission in 1925 established under the Treaty did not lead to any alteration in the border.

Within Northern Ireland, the Nationalist Party was an organisational successor to the Home Rule Movement, and advocated the end of partition. It had a continuous presence in the Northern Ireland Parliament from 1921 to 1972, but was in permanent opposition to the UUP government.

A new Constitution of Ireland was proposed by Éamon de Valera in 1937 and approved by the voters of the Irish Free State (thereafter simply Ireland). Articles 2 and 3 of this Constitution claimed the whole island of Ireland as the national territory, while claiming legal jurisdiction only over the previous territory of the Irish Free State.

Article 15.2 allowed for the "creation or recognition of subordinate legislatures and for the powers and functions of these legislatures", which would have allowed for the continuation of the Parliament of Northern Ireland within a unitary Irish state.

In 1946, former Prime Minister Winston Churchill told the Irish High Commissioner to the United Kingdom, "I said a few words in Parliament the other day about your country because I still hope for a United Ireland. You must get those fellows in the north in, though; you can't do it by force. There is not, and never was, any bitterness in my heart towards your country." He later said, "You know I have had many invitations to visit Ulster but I have refused them all. I don't want to go there at all, I would much rather go to southern Ireland. Maybe I'll buy another horse with an entry in the Irish Derby."

Under the Republic of Ireland Act 1948, Ireland declared that the country may officially be described as the Republic of Ireland and that the President of Ireland had the executive authority of the state in its external relations. This was treated by the British Commonwealth as ending Irish membership. In response, the United Kingdom passed the Ireland Act 1949. Section 1(2) of this act affirmed the provision in the Treaty that the position of Ireland remained a matter for the Parliament of Northern Ireland:

Between 1956 and 1962, the IRA engaged in a border campaign against British Army and Royal Ulster Constabulary outposts with the aim of ending British rule in Northern Ireland. This coincided with brief electoral success of Sinn Féin, which won four seats at the 1957 Irish general election. This was its first electoral success since 1927, and it did not win seats in the Republic of Ireland again until 1997. The border campaign was entirely unsuccessful in its aims. In 1957, Prime Minister Harold Macmillan wrote that "I do not think that a United Ireland - with de Valera as a kind of Irish Nehru would do us much good. Let us stand by our friends."

Calls for unification, start of the Troubles
The Northern Ireland civil rights movement emerged in 1967 to campaign for civil rights for Catholics in Northern Ireland. Tensions between republican and loyalist groups in the north erupted into outright violence in the late 1960s. In 1968 the Irish Taoiseach, Jack Lynch, raised the issue of partition in London: "It has been the aim of my government and its predecessors to promote the reunification of Ireland by fostering a spirit of brotherhood among all sections of the Irish people. The clashes in the streets of Derry are an expression of the evils which partition has brought in its train." He later stated to the press that the ending of partition would be "a just and inevitable solution to the problems of Northern Ireland."

Lynch renewed his call to end partition in August 1969 when he proposed negotiations with Britain with the hope of merging the Irish Republic and Northern Ireland into a federal type state. Lynch proposed that the two parliaments continue to function with a Council of Ireland having authority over the entire country. The Prime Minister of Northern Ireland James Chichester-Clark rejected the proposal. In August 1971 Lynch proposed that the Government of Northern Ireland (Stormont) be replaced with an administration that would share power with Catholics. The next day the Northern Prime Minister Brian Faulkner rejected Lynch's statement and stated that "no further attempt by us to deal constructively with the present Dublin government is possible." Later in 1971 British Labour Party leader (and future Prime Minister) Harold Wilson proposed a plan that would lead to a united Ireland after a 15-year transitional period. He called for the establishment of a commission that would examine the possibility of creating a united Ireland which would be agreed upon by all three parliaments. The northern Prime Minister rejected the proposal and reiterated the desire that Northern Ireland remain an integral part of the United Kingdom. The Irish Taoiseach indicated the possibility of amending the Irish constitution to accommodate the Protestants of Northern Ireland and urged the British government to "declare its interest in encouraging the unity of Ireland".

In 1969 the British government deployed troops in what would become the longest continuous deployment in British military history Operation Banner. The Provisional Irish Republican Army (IRA) had begun a thirty-year campaign against British security forces with the aim of winning a united Ireland.

In 1970, the Social Democratic and Labour Party (SDLP) was established to campaign for civil rights and a united Ireland by peaceful, constitutional means. The party rose to be the dominant party representing the nationalist community until the early twenty-first century.

In 1972, the parliament of Northern Ireland was suspended, and under the Northern Ireland Constitution Act 1973, it was formally abolished. Section 1 of the 1973 Act stated,

A border poll was held in Northern Ireland in 1973. The SDLP and Sinn Féin called for a boycott of the poll. 98.9% of votes cast supported remaining part of the United Kingdom. The poll was overwhelmingly boycotted by nationalists, and the turnout was therefore 58.7%. The pro-UK vote did however represent 57.5% of the entire electorate, notwithstanding the boycott.

In 1983, the Irish government led by Taoiseach Garret FitzGerald established the New Ireland Forum as a consultation on a new Ireland. Though all parties in Ireland were invited, the only ones to attend were Fine Gael, Fianna Fáil, the Labour Party and the SDLP. Its report considered three options: a unitary state, i.e., a united Ireland; a federal/confederal state; and joint sovereignty. These options were rejected by Prime Minister Margaret Thatcher. In 1985, the governments of Ireland and of the United Kingdom signed the Anglo-Irish Agreement; the British government accepted an advisory role for the Irish government in the future of Northern Ireland. Article 1 of the Agreement stated that the future constitutional position of Northern Ireland would be a matter for the people of Northern Ireland:

In the Downing Street Declaration, Taoiseach Albert Reynolds and Prime Minister John Major issued a joint statement, in which Major, "reiterated on behalf of the British Government, that they have no selfish strategic or economic interest in Northern Ireland".

Good Friday Agreement
The Good Friday Agreement in 1998 was a culmination of the peace process. The agreement acknowledged nationalism and unionism as "equally legitimate, political aspirations". In the Northern Ireland Assembly, all members would designate as Unionist, Nationalist, or Other, and certain measures would require cross-community support. The agreement was signed by the governments of Ireland and of the United Kingdom. In Northern Ireland, it was supported by all parties who were in the Northern Ireland Forum with the exception of the Democratic Unionist Party and the UK Unionist Party, and it was supported by all parties in the Oireachtas. It was also opposed by dissident republicans, including Republican Sinn Féin and the 32 County Sovereignty Movement. It was approved in referendums in Northern Ireland and in the Republic of Ireland.

Included in the Agreement were provisions which became part of the Northern Ireland Act 1998 on the form of a future referendum on a united Ireland. In essence the Belfast/Good Friday Agreement provided the opportunity for self determination and mutual respect. Those born in Northern Ireland could identify as Irish, British or both and would be entitled to hold a passport from both, or either country. The freedom of movement, allowed citizens of either jurisdiction to live in which ever part of the island they wanted, therby enabling them to choose which state they paid taxes to or claimed benefits from. The 'Two state' solution advocated for conflict resolution in other jurisdictions therefore applied. Provision within the Agreement allows for a simple majority to vote in favour of Irish Unification, but does nothing to explain how the dissolution of the two state solution, leads to a peaceful and prosperous new country when potentially 13% of the 'new' country are forced into it against their will and have no allegiance to it nor incentive for it to succeed. A fear of political, civil and economic turmoil and a lack of protection for minority rights, as experienced by the Catholic community in Northern Ireland and the Protestant community in the Republic of Ireland historically, is a key driver towards the desire for the maintenance of the status quo on both sides of the border.

On the establishment of the institutions in 1999, Articles 2 and 3 of the Constitution of Ireland were amended to read:

Brexit, the Northern Ireland Protocol and elections
In a referendum in June 2016, England and Wales voted to leave the European Union. The majority of those voting in Northern Ireland and in Scotland, however, voted for the UK to remain. Of the parties in the Assembly, only the Democratic Unionist Party (DUP), the Traditional Unionist Voice (TUV) and People Before Profit (PBP) had campaigned for a Leave vote. Irish politicians began the discussion regarding possible changes to the border between Northern Ireland and the Republic of Ireland. The status and treatment of Northern Ireland and Gibraltar, the only parts of the United Kingdom which would have new land borders with the EU following the UK withdrawal, became important to the negotiations, along with access to the regional development assistance scheme (and new funding thereof) from the European Union.

Sinn Féin cited these concerns as the basis for new discussion on a united Ireland. These calls were rejected by the British government and Unionist politicians, with Theresa Villiers arguing that there was no evidence that opinion in Northern Ireland had shifted towards being in favour of a united Ireland.

2017 Assembly election
In the 2017 Assembly election, the DUP lost ten seats and came just one seat ahead of Sinn Féin. Sinn Féin used this opportunity to call for a Northern Ireland referendum on a united Ireland.

Theoretical return to EU confirmed in a United Ireland
The Brexit Secretary, David Davis, confirmed to Mark Durkan, the SDLP MP for Foyle, that in the event of Northern Ireland becoming part of a united Ireland, "Northern Ireland would be in a position of becoming part of an existing EU member state, rather than seeking to join the EU as a new independent state." Enda Kenny pointed to the provisions that allowed East Germany to join the West and the EEC during the reunification of Germany as a precedent. In April 2017 the European Council acknowledged that, in the event of Irish unification, "the entire territory of such a united Ireland would [...] be part of the European Union." The SDLP manifesto for the 2017 UK general election called for a referendum on a united Ireland after the UK withdraws from the EU. However the Secretary of State for Northern Ireland at the time, James Brokenshire, said the conditions for a vote are "not remotely satisfied".

2017 general election
After the 2017 election, the UK government was reliant on confidence and supply from the Democratic Unionist Party. The deal supported the Conservative led government through the Brexit negotiation process. The 2020 Brexit withdrawal agreement included the Northern Ireland Protocol, which established different trade rules for the territory than Great Britain. While Northern Ireland would de jure leave the single market, it would still enforce all EU customs rules, while Britain would diverge. This would result in a regulatory "border in the Irish Sea" rather than a border between Northern Ireland and the Republic of Ireland, and caused fears from unionist politicians about Brexit causing a weakening of the UK.

Brexit negotiations continue
The new UK prime minister Boris Johnson continued to claim no trade border would take form as late as August 2020, despite having negotiated its creation. Dominic Cummings later claimed that Johnson did not understand the deal at the time it was signed, while Ian Paisley Jr claimed that Johnson had privately promised to "tear up" the deal after it was agreed. In September, Johnson sought to unilaterally dis-apply parts of the Northern Ireland protocol, despite acknowledging that this broke international law. The bill was rejected by the House of Lords, resulting in several provisions being withdrawn before it passed in December 2020- shortly before the protocol was due to come into effect.

The implementation of the protocol, and the new regulatory hurdles had a negative effect on east–west trade, and drew strong condemnation from unionist figures, including DUP members such as First Minister Arlene Foster. Staff making the required checks were threatened, resulting in a temporary suspension of checks at Larne and Belfast ports. In February 2021, several unionist parties began a legal challenge, alleging that the protocol violated the Act of Union 1800, the bill which had originally merged Ireland with the United Kingdom, as well as the Good Friday Agreement. The challenge was dismissed in June, with the court deciding that the protocol- and other legislation in the intervening 200 years- had effectively repealed parts of the Act of Union. On 4 March the Loyalist Communities Council withdrew its support for the peace agreement- while indicating that opposition to it should not be in the form of violence. Riots erupted in loyalist areas at the end of the month, continuing until 9 April. The protocol's implementation, and opposition within the DUP, resulted in the announcement of Foster's resignation on 28 April. The Irish Times interviewed loyalist Shankill Road residents that month and found significant anger at the DUP, and accusations that the community had been "sold short" on the protocol. Foster was replaced by Paul Givan later that year, though he too resigned in February 2022 over the continued existence of the protocol.

The UK government sought to re-negotiate the protocol, a prospect poorly received by EU leaders such as Emmanuel Macron. When discussing the effects of the protocol in June 2021, Leo Varadkar outlined a vision for a united Irish state with devolved representation in the North. He added "It should be part of our mission as a party to work towards it." Talks aimed at amending the customs checks required by the protocol began in October; though Maroš Šefčovič indicated that the protocol itself will not be re-negotiated. In December, the UK's chief negotiator Lord Frost resigned his post over "concerns about the current direction of travel".

2022 Assembly election
After the results of the 2022 Northern Ireland Assembly election, Sinn Féin were set to become the largest party in the assembly for the first time in history, with the DUP coming in second place. Sinn Féin won 27 seats, compared to the DUP's 25. Sinn Féin said that it will be at least a decade-long plan for Irish unity, which would only happen after an island-wide conversation.

2023-24 and return to Stormont
In February 2023, UK Prime Minister Rishi Sunak and President of the European Commission Ursula von der Leyen announced a new agreement called the Windsor Framework including a green lane for trade between Britain and Northern Ireland and a red lane for Republic of Ireland and EU trade. Sinn Fein called for a restoration of devolved governance following the deal whilst the DUP continued their boycott.

On 31 January 2024, a deal between the DUP and the UK government led to the abolition of "routine" checks on goods from Britain sent to Northern Ireland with the intention of staying there. A new body, Intertrade UK will be formed to promote trade within the UK, modelled on the all Ireland body, InterTradeIreland. The deal also includes UK government ministers being compelled to inform Parliament if a Bill they are introducing will have "significant adverse implications for Northern Ireland's place in the UK internal market". On the basis of the deal, the DUP decided to return to devolved governance in Stormont.

On 3 February, Michelle O'Neill made history by becoming the first ever Irish nationalist First Minister. After taking office as First Minister, O'Neill stated that she expected a referendum on the reunificiation to be held within the next decade, which would be in accordance with the 1998 Good Friday Agreement signed by the UK and Ireland.

Shared Island
In 2021, the Irish Government launched the "Shared Island" initiative, to fund projects enhancing cross-border cooperation. In February 2024, it was announced that a total of €1 billion of funding from the Irish government was commited for:

 * A5 road (€600m committed)
 * Construction of Narrow Water Bridge between the Mourne Mountains and the Cooley Peninsula
 * New hourly rail service between Belfast and Dublin
 * €50m towards Casement Park
 * €10m towards a visitor experience at the Battle of the Boyne site
 * Cross- border women’s entrepreneurship

The Narrow Water bridge, linking Omeath to Warrenpoint, began construction in June 2024. During an Ireland's Future event in the same month, Taoiseach Leo Varadkar proposed that the Irish government set up a State fund using current budget surpluses that could then be used in the event of a United Ireland. Also at the event, Michelle O'Neill promised that Casement Park would be built "on my watch".

Timescale proposals
In 2020, Ireland Taoiseach Micheál Martin said that a referendum on Irish unity should not be held for 5 years, adding, "once Brexit happened, it should not be a catalyst for something like a border poll. I thought that would be too divisive and would only exacerbate the tensions there because of Brexit itself".

Sinn Féin president Mary Lou McDonald suggested in 2020 that an "All-Ireland Forum on Unity" be created to plan for all aspects of reunification, including a referendum by 2025, claiming that "Irish Unity is the best idea for the future of the country". In 2021, Gerry Adams told the Irish Government that it should begin planning for a border poll and that one could happen within three years.

In April 2022, DUP leader Jeffrey Donaldson has said that Northern Ireland does not need a border poll at all and that it would be divisive.

In May 2022, Mary Lou McDonald said that a referendum would be possible within 5 years. First minister elect, Michelle O’Neill launched Sinn Fein's manifesto at the Canal Court Hotel in Newry in March 2023 which was a 16-page document including a commitment to set a date on a unity referendum. In February 2024, Mary Lou McDonald said that she expected the referendums to occur by 2030. Political scientist Brendan O'Leary also suggested the year 2030 as a potential tipping point.

In October 2023, Taoiseach Leo Varadkar said that it was not currently the right time to hold a referendum because the evidence indicated the referendum would not be won. If a referendum was held, work would need to be done to convince unionists that this was the right path and that a United Ireland would be a "warm home" for them.

The UK government-DUP deal published on 31 January 2024 said there was "no realistic prospect of a border poll". On 4 February, the day after becoming First Minister, Michelle O'Neill contested this, outlining her vision of a "decade of opportunity"; in other words, a border poll could be held in 10 years.

Criteria
In May 2022, Tánaiste Leo Varadkar said the criteria for a referendum had not yet been met and also called for clarity on the mechanism for a border poll. Michelle O'Neill also called for clarity on the criteria for a referendum. In September 2022, Northern Ireland shadow secretary Peter Kyle (of the Labour party) said that he would set out border poll criteria.

In the May 2023 Northern Ireland local elections, the three largest unionist parties received over 38 per cent of the vote and the parties in favour of a united Ireland received 41 per cent of the vote. Jeffrey Donaldson suggested that the criteria for a border poll had not been met because unionists had more seats. This however is only true when excluding pro-irish unity independents and People Before Profit. When asked about the criteria, Northern Ireland secretary Chris Heaton-Harris said that those interested “need to read the Belfast/Good Friday Agreement, it’ll give you a good clue”. In October 2023, Heaton-Harries added that there was no basis to suggest a majority in Northern Ireland currently supported a United Ireland and that the UK government supported all aspects of the Good Friday Agreement, including the use of a simple majority of >50% in the event of a referendum.

In June 2023, UUP leader Doug Beattie also suggested that the criteria has not been met for a border poll. He added that restoration of the executive would halt a swing in support of Sinn Féin and a United Ireland. Ian Paisley Jr. suggested that support to unify Ireland is not adequate and that should a vote come about, a supermajority and a turnout quota should be required for a border poll. John Major has also called upon the UK government to "spell out" the criteria for a border poll.

Parties of Ireland
Within the Northern Ireland Assembly, MLAs designate as Unionist, Nationalist or Other. The DUP (25 seats), the UUP (9 seats), the TUV (1 seat) and the Independent MLAs Claire Sugden and Alex Easton are designated as Unionist; Sinn Féin (which won 27 seats in the 2022 Northern Ireland Assembly election) and the SDLP (8 seats) are designated as Nationalist; the Alliance Party (17 seats) and PBP (1 seat) are designated as Other. However People Before Profit are in favour of Irish Unity.

Within the Oireachtas, there has traditionally been broad support for a united Ireland, with differences over the twentieth century on how it would be achieved. This includes Sinn Féin, which has had seats in the Dáil since 1997. The initial party constitution of Fianna Fáil in 1926 under Éamon de Valera included as the first of its aims, "To secure the Unity and Independence of Ireland as a Republic". In 1937, de Valera proposed the Constitution of Ireland which laid claim to the whole island of Ireland. In the 1980s, led by Charles Haughey, the party opposed the consideration of options other than a unitary state in the New Ireland Forum Report and opposed the Anglo-Irish Agreement; this stance led in part to the Des O'Malley and Mary Harney leaving Fianna Fáil and establishing the Progressive Democrats, a party that lasted from 1985 to 2008. Fianna Fáil leaders Albert Reynolds and Bertie Ahern led Irish governments in favour of the Downing Street Declaration and the Good Friday Agreement respectively.

When formed in 1933, Fine Gael initially used the subtitle United Ireland. Fine Gael leader Garret FitzGerald convened the New Ireland Forum in 1983 and negotiated the Anglo-Irish Agreement. In the aftermath of the vote on Brexit, Enda Kenny sought assurances on the position of Northern Ireland in the case of a united Ireland. The Irish Labour Party has adopted a similar approach to Fine Gael in government to a united Ireland.

Minor parties
In a survey of TDs conducted by TheJournal.ie on support for a border poll and a united Ireland conducted in December 2016, only TDs from the Anti-Austerity Alliance (now Solidarity) stated they were opposed to a united Ireland at the present moment.

There are a number of minor nationalist parties, including the Irish Republican Socialist Party, which supports a united socialist Irish state and is affiliated with the Irish National Liberation Army. Another such party, Republican Sinn Féin, linked to the Continuity IRA, maintain the Irish republican legitimist theory that neither state in Ireland is legitimate. Its Éire Nua (in English, New Ireland) policy advocates a unified federal state with regional governments for the four provinces and the national capital in Athlone. None of these parties has significant electoral support.

British parties
Of the British parties, the Conservative Party is explicitly unionist; it has formally been called the Conservative and Unionist Party since a merger with the Liberal Unionist Party in 1912. The UUP was affiliated with the National Union of Conservative and Unionist Associations until 1985. The Northern Ireland Conservatives are a minor unionist party in Northern Ireland.

Historically, there has been support for a united Ireland within the left of the British Labour Party, and in the 1980s it became official policy to support a united Ireland by consent. The policy of "unity by consent" continued into the 1990s, eventually being replaced by a policy of neutrality in line with the Downing Street Declaration. The former Labour leader Jeremy Corbyn supports a united Ireland, although he has said that it is "up for the Irish people to decide" whether to remain part of the UK. They do not organise electorally in Northern Ireland, respecting the SDLP as their sister party within the Party of European Socialists. Similarly, the Liberal Democrats co-operate with the Alliance Party and share their support of the Good Friday Agreement while expressing reservations about what they perceive as 'institutionalised sectarianism' in the agreement. Former Alliance leader Lord Alderdice is a member of the Liberal Democrats in the House of Lords. One supporter of a United Ireland in the Liberal Democrats was Michael Meadowcroft, MP for Leeds West between 1983 and 1987.

Economic
Neale Richmond of Fine Gael says that unifying Ireland as a unitary state within the EU would benefit the economy across the island. He also argues it would allow the simplification of the systems of healthcare, public transport, education and more.

Sinn Féin says that economic powers held in London contribute to Northern Ireland being the slowest growing economy of the British Isles and also negatively impacts the economy of the Republic of Ireland. They cite lower paid and less secure jobs in the North and say that a United Ireland would allow for a more coordinated economic strategy and increasing investment, productivity and improving infrastructure, particularly in the border region. Sinn Féin also suggests that unity will improve public revenue returns, overall output, and higher-skilled employment.

Cultural
The Irish rugby team has been cited has a unifying force across Ireland and a "gesture of unity". The team has been described as "showing the best what island has to offer" and has sometimes been described as a United Ireland team.

Reconciliation
Jim O'Callaghan of Fianna Fáil suggests that a United Ireland would bring people on the island closer together, reconcile old conflicts and increase the number of opportunities for young people.

Identity
Many Unionist Protestants in Northern Ireland argue they have a distinct identity that would be overwhelmed in a united Ireland. They cite the decline of the small Protestant population of the Republic of Ireland since independence from the United Kingdom, the economic cost of unification, their place in a key international player within the UK and their mainly non-Irish ancestry. Unionist people in Northern Ireland primarily find their cultural and ethnic identity from the Scottish and English planters (colonists), whose descendants can be found in the three counties of Ulster which are governed by the Republic of Ireland. Such individuals celebrate their Scots heritage each year like their counterparts in the other six counties. While Catholics in general consider themselves to be Irish, Protestants generally see themselves as British, as shown by several studies and surveys performed between 1971 and 2006.

Many Protestants do not consider themselves as primarily Irish, as many Irish nationalists do, but rather within the context of an Ulster or British identity. A 1999 survey showed that a little over half of Protestants felt "Not at all Irish", while the rest "felt Irish" in varying degrees.

Constitutional options
A report by University College London found four constitutional options for a United Ireland;

 * A single central legislature in e.g Dublin. This model is considered the historic choice for many Irish republicans but may not be seen as favourable by some unionists.
 * Maintain the devolved institutions in the North but with sovereignty transferred from London to Dublin.
 * A two unit federal state.
 * A confederation of two independent states. UCL suggested this would not meet the terms of unity as clearly (1998 Belfast Good Friday Agreement).

Ireland

 * Demographics of Northern Ireland
 * Politics of Northern Ireland
 * Irish nationalism
 * Protestant Irish nationalists
 * Ulster nationalism

Other

 * Scottish independence
 * Welsh independence
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title: United Ireland


text:
Civil unions in Quebec

Civil unions in Quebec are available in Quebec to both opposite-sex and same-sex couples, which attempts to create the same rights for the partners as a traditional marriage.

As a result of a range of activism and to the M. v. H. decision, the National Assembly of Quebec voted unanimously in 2002 to amend the Civil Code of Quebec to create a status of civil union in Quebec, available to both opposite-sex and same-sex couples and largely having the same rights as marriage. The law was enacted on June 24, 2002.

A civil union is contracted into by same-sex or opposite-sex partners 18 years of age and older, who are not otherwise married, not in another civil union, or who are not closely related, following prescribed formalities similar to the regime of marriage. The civil union carries obligations and benefits equivalent to that of marriage including the obligation of support (CCQ art. 585) and the establishment of a family patrimony and family residence (CCQ art. 521.6) and may otherwise be modified by a contract (CCQ 521.8) similar to a pre-nuptial agreement and may agree to a particular property regime similar to available matrimonial regimes. It creates a family connection between the spouses and their relatives (CCQ art. 521.7).

Judicial conciliation is merited "when the spouses cannot agree on their rights and performance of their duties" (CCQ art. 521.9). Such union may be annulled within three years if irregularly contracted (CCQ arts. 521.10-521.11). A civil union ends at death of one of the partners or may be dissolved by judicial dissolution or by a 'transaction agreement' and 'joint declaration' before a civil law notary and recorded en minute if both partners consent and they settle all the consequences of the dissolution (CCQ arts. 521.13-521.16).

Judicial dissolution is merited when "the interests of the common children of the spouses are at stake" or where the parties cannot otherwise agree (CCQ art. 521.17 para. 1). Provisional orders of support, custody and access may be entered during the pendency of the dissolution action (CCQ art. 521.71 para. 2) and the court may, upon or after pronouncing dissolution decide maintenance, custody and education issues in the best interests of and with due regard to the rights of the children (CCQ art. 521.17, para. 3).

The act establishing the regime of civil union also modified rules creating filiation for biological children of one of the partners, and for adopted children as well as the recognition of parental authority and child support obligations so they will apply to civil union couples as well as married couples.

In March 2004, Quebec same-sex couples won the right to marry. Today, couples (both opposite- and same-sex) can choose between civil marriage and civil unions.

title: Civil unions in Quebec