Editing Arrow–Debreu model (section)

== Extensions ==

=== Accounting for strategic bargaining ===
In the model, all producers and households are "[[Market power|price takers]]", meaning that they transact with the market using the price vector <math>p</math>. 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 <math>y^j \in PPS^j</math>, it is as if the whole society has one great producer producing <math>y\in PPS</math>.
* 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''' <math>((x_i)_{i\in I}, y)</math>—a production and consumption plan for the whole economy—with the following constraints:<math display="block">x^i \in CPS^i, y \in PPS, y\succeq \sum_i (x^i- r^i)</math>
* Any nonempty subset of households may eliminate all other households, while retaining control of the producers.

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

* 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.
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.

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

* The curse of free lunch: In this model, the whole <math>PPS</math> is available to any nonempty coalition, even a coalition of one. Consequently, if nobody has any endowment, and yet <math>PPS</math> contains some "free lunch" <math>y\succ 0</math>, then (assuming preferences are monotonic) every household would like to take all of <math>y</math> 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 <math>PPS</math> 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 <math>PPS</math> for itself.

==== Main results (Debreu and Scarf, 1963) ====
{{Math theorem|name=Proposition|math_statement= Market equilibria are core states.
}}

{{Math proof|title=Proof|proof=

Define the price hyperplane <math>\langle p, q \rangle = \langle p, \sum_j y^j\rangle</math>. Since it's a supporting hyperplane of <math>PPS</math>, and <math>PPS</math> is a convex cone, the price hyperplane passes the origin. Thus <math>\langle p, \sum_j y^j\rangle =  \langle p, \sum_i x^i - r^i\rangle = 0</math>.

Since <math>\sum_j \langle p, y^j\rangle</math> is the total profit, and every producer can at least make zero profit (that is, <math>0 \in PPS^j</math> ), this means that the profit is exactly zero for every producer. Consequently, every household's budget is exactly from selling endowment.

<math display="block">\langle p, x^i \rangle = \langle p, r^i\rangle</math>

By utility maximization, every household is already doing as much as it could. Consequently, we have <math>\langle p, U^i_{++}(x^i)\rangle > \langle p, r^i\rangle</math>.

In particular, for any coalition <math>I' \subset I</math>, and any production plan <math>x'^i</math> that is Pareto-better, we have

<math display="block"> \sum_{i\in I'} \langle p, x'^i \rangle >\sum_{i\in I'} \langle p, r^i \rangle</math>
and consequently, the point <math>\sum_{i\in I'} x'^i - r^i</math> lies above the price hyperplane, making it unattainable.
}}

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 <math>K</math>, define an economy where there are <math>K</math> households that have exactly the same consumption possibility set and preference as household <math>i</math>.

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

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.

{{Math theorem|name=Proposition|math_statement= Any core state is equitable.}}

{{Math proof|title=Proof|proof=  We use the "underdog coalition".

Consider a core state <math>x^{i, k}</math>. Define average distributions <math>\bar x^{i} := \frac 1K \sum_{k\in K} x^{i,k}</math>.

It is attainable, so we have <math>K \sum_{i} (\bar x^i - r^i) \in PPS</math>

Suppose there exist any inequality, that is, some <math>x^{i, k} \succ^i x^{i, k'}</math>, then by convexity of preferences, we have <math>\bar x^i \succ^i x^{i, k'}</math>, where <math>k'</math> is the worst-treated household of type <math>i</math>.

Now define the "underdog coalition" consisting of the worst-treated household of each type, and they propose to distribute according to <math>\bar x^i</math>. This is Pareto-better for the coalition, and since <math>PP</math> is conic, we also have <math>\sum_i(\bar x^i - r^i) \in PPS</math>, so the plan is attainable. Contradiction.
}}

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

We have seen that <math>C</math> contains the set of market equilibria for the original economy. The converse is true under minor additional assumption:<ref>(Starr 2011) Theorem 22.2</ref>

{{Math theorem
| name = (Debreu and Scarf, 1963)
| note = 
| math_statement = If <math>PPS</math> is a polygonal cone, or if every <math>CPS^i</math> has nonempty interior with respect to <math>\R^N</math>, then <math>C</math> is the set of market equilibria for the original economy.
}}

The assumption that <math>PPS</math> is a polygonal cone, or every <math>CPS^i</math> has nonempty interior, is necessary to avoid the technical issue of "quasi-equilibrium". Without the assumption, we can only prove that <math>C</math> 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 <math>S^j(p), D^i(p)</math> 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 lemma#Shapley–Folkman–Starr theorem|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 <math>PPS^j, CPS^i</math> and <math>\succeq^i</math>, we define the "convexified economy" to be the same economy, except that

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

where <math>\mathrm{Conv}</math> 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 <math>p^*</math> is an equilibrium price vector for the convexified economy, then<ref>(Starr 2011), Theorem 25.1</ref><math display="block">\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}</math>where <math>d(\cdot, \cdot)</math> is the Euclidean distance, and <math>L</math> is any upper bound on the inner radii of all <math>PPS^j, CPS^i</math> (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 <math>\{(x, 0): x \geq 0\}\cup \{(x,y): xy = 1, x > 0\}</math> 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 ===
{{see|Financial economics#State prices|State prices#Application to financial assets|Contingent claim analysis}}
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 divided 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 prices|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 [[Douglas Breeden|Breeden]] and [[Robert Litzenberger|Lizenberger]] in 1978,<ref>{{cite journal |title=Prices of State-Contingent Claims Implicit in Option Prices |first1=Douglas T. |last1=Breeden |first2=Robert H. |last2=Litzenberger |journal=[[Journal of Business]] |volume=51 |issue=4 |year=1978 |pages=621–651 |jstor=2352653 |doi=10.1086/296025|s2cid=153841737 }}</ref> a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in [[financial economics]].<ref>{{cite journal |last1=Almeida |first1=Caio |last2=Vicente |first2=José |year=2008 |title=Are interest rate options important for the assessment of interest risk? |journal=Working Papers Series N. 179, Central Bank of Brazil |url=http://www.bcb.gov.br/pec/wps/ingl/wps179.pdf }}</ref>

=== Accounting for the existence of money ===
{{blockquote|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.|Gérard Debreu|Theory of value: An axiomatic analysis of economic equilibrium (1959)|source=}}{{blockquote|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.|[[Frank Hahn]]|The foundations of monetary theory (1987)|source=}}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.<ref>(Starr 2011) Exercise 20.15</ref>

=== Computing general equilibria ===
{{Main|Computable general equilibrium}}
Scarf (1967)<ref>{{Cite journal |last=Scarf |first=Herbert |date=September 1967 |title=The Approximation of Fixed Points of a Continuous Mapping |url=http://dx.doi.org/10.1137/0115116 |journal=SIAM Journal on Applied Mathematics |volume=15 |issue=5 |pages=1328–1343 |doi=10.1137/0115116 |issn=0036-1399}}</ref> was the first algorithm that computes the general equilibrium. See Scarf (2018)<ref>{{Citation |last=Scarf |first=Herbert E. |title=Computation of General Equilibria |date=2018 |url=http://dx.doi.org/10.1057/978-1-349-95189-5_451 |work=The New Palgrave Dictionary of Economics |pages=1973–1984 |place=London |publisher=Palgrave Macmillan UK |doi=10.1057/978-1-349-95189-5_451 |isbn=978-1-349-95188-8 |access-date=2023-01-06}}</ref> and Kubler (2012)<ref>{{Citation |last=Kubler |first=Felix |title=Computation of General Equilibria (New Developments) |url=http://dx.doi.org/10.1057/9781137336583.0296 |work=The New Palgrave Dictionary of Economics, 2012 Version |year=2012 |place=Basingstoke |publisher=Palgrave Macmillan |doi=10.1057/9781137336583.0296 |isbn=9781137336583 |access-date=2023-01-06}}</ref> for reviews.

=== Number of equilibria ===
{{See also|General equilibrium theory#Uniqueness}}
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]].<ref>{{Citation |last=Debreu |first=Gérard |title=Stephen Smale and the Economic Theory of General Equilibrium |date=June 2000 |url=http://dx.doi.org/10.1142/9789812792815_0025 |work=The Collected Papers of Stephen Smale |pages=243–258 |publisher=World Scientific Publishing Company |doi=10.1142/9789812792815_0025 |isbn=978-981-02-4991-5 |access-date=2023-01-06}}</ref><ref>{{Citation |last=Smale |first=Steve |title=Chapter 8 Global analysis and economics |date=1981-01-01 |url=https://www.sciencedirect.com/science/article/pii/S1573438281010126 |series=Handbook of Mathematical Economics |volume=1 |pages=331–370 |publisher=Elsevier |doi=10.1016/S1573-4382(81)01012-6 |isbn=978-0-444-86126-9 |language=en |access-date=2023-01-06}}</ref>

There are many such genericity theorems. One example is the following:<ref>{{Cite journal |last=Debreu |first=Gérard |date=December 1984 |title=Economic Theory in the Mathematical Mode |url=http://dx.doi.org/10.2307/3439651 |journal=The Scandinavian Journal of Economics |volume=86 |issue=4 |pages=393–410 |doi=10.2307/3439651 |jstor=3439651 |issn=0347-0520}}</ref><ref>(Starr 2011) Section 26.3</ref>

{{Math theorem
| name = Genericity
| math_statement = For any strictly positive endowment distribution <math>r^1, ..., r^I \in \R_{++}^N</math>, and any strictly positive price vector <math>p\in \R_{++}^N</math>, define the excess demand <math>Z(p, r^1, ..., r^I)</math> as before.

If on all <math>p, r^1, ..., r^I \in \R_{++}^N</math>,
* <math>Z(p, r^1, ..., r^I)</math> is well-defined,
* <math>Z</math> is differentiable,
* <math>\nabla_p Z</math> has <math>(N-1)</math>,

then for generically any endowment distribution <math>r^1, ..., r^I \in \R_{++}^N</math>, there are only finitely many equilibria <math>p^* \in \R_{++}^N</math>.
}}

{{Math proof|title=Proof (sketch)|proof=

Define the "equilibrium manifold" as the set of solutions to <math>Z=0</math>. By Walras's law, one of the constraints is redundant. By assumptions that <math>\nabla_p Z</math> has rank <math>(N-1)</math>, no more constraints are redundant. Thus the equilibrium manifold has dimension <math>N \times I</math>, which is equal to the space of all distributions of strictly positive endowments <math>\R_{++}^{N \times I}</math>.

By continuity of <math>Z</math>, the projection is closed. Thus by Sard's theorem, the projection from the equilibrium manifold to <math>\R_{++}^{N \times I}</math> is critical on only a set of measure 0. It remains to check that the preimage of the projection is generically not just discrete, but also finite.
}}