Arrow–Debreu model

In mathematical economics, the Arrow–Debreu model suggests 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 is 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; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists. With weaker conditions, uniqueness fails maximally, according to the Sonnenschein–Mantel–Debreu theorem.

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

Preliminary concepts

Convex sets and fixed points

A quarter turn of the convex unit disk leaves the point (0,0) fixed but moves every point on the non–convex unit circle.

Main article: Kakutani fixed-point theorem

See also: Convex set, Compact set, Continuous function, Fixed-point theorem, and Brouwer fixed-point theorem

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

See also: Shapley–Folkman lemma and Market failure

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. Ross M. Starr (1969) 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.

Formal statement

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".

How an Arrow-Debreu model moves from beginning to end.

households	producers
receive endowment and ownership of producers
sell all endowment to the market
	plan production to maximize profit
	enter purchase agreements between the market and each other
	perform production plan
	sell everything to the market
	send all profits to households in proportion to ownership
plan consumption to maximize utility under budget constraint
buy the planned consumption from the market

Notation setup

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

useful notations for real vectors

• $x\succeq y$ iff $\forall n,x_{n}\geq y_{n}$

• $\mathbb {R} _{+}^{N}$ is the set of $x$ such that $x\succeq 0$

• $\mathbb {R} _{++}^{N}$ is the set of $x$ such that $x\succ 0$
• $\Delta _{N}=\left\{x\in \mathbb {R} ^{N}:x_{1},...,x_{N}\geq 0,\sum _{n\in 1:N}x_{n}=1\right\}$ 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 $p=(p_{1},...,p_{N})\in \mathbb {R} _{++}^{N}$ 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 $r^{i}\in \mathbb {R} _{+}^{N}$.
• 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 $CPS^{i}\subset \mathbb {R} _{+}^{N}$.
• 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 $D^{i}(p)\in \mathbb {R} _{+}^{N}$. 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 $p\in \mathbb {R} _{++}^{N}$. 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 $S^{j}(p)\in \mathbb {R} ^{N}$. 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 $p\in \mathbb {R} _{++}^{N}$. 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

• 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$$Z(p)_{n}{\begin{cases}\leq 0{\text{ if }}p_{n}=0\\=0{\text{ if }}p_{n}>0\end{cases}}$$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).
• A state is an equilibrium state iff it is the state corresponding to an equilibrium price vector.

Assumptions

on the households

assumption	explanation	can we relax it?
$CPS^{i}$ is closed	Technical assumption necessary for proofs to work.	No. It is necessary for the existence of demand functions.
local nonsatiation: $\forall x\in CPS^{i},\epsilon >0,$ $\exists x'\in CPS^{i},x'\succ ^{i}x,\|x'-x\|<\epsilon$	Households always want to consume a little more.	No. It is necessary for Walras's law to hold.
$CPS^{i}$ is strictly convex	strictly diminishing marginal utility	Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section.
$CPS^{i}$ is convex	diminishing marginal utility	Yes, to nonconvexity, with Shapley–Folkman lemma.
continuity: $U_{+}^{i}(x^{i})$ is closed.	Technical assumption necessary for the existence of utility functions by the Debreu theorems.	No. If the preference is not continuous, then the excess demand function may not be continuous.
$U_{+}^{i}(x^{i})$ is strictly convex.	For two consumption bundles, any bundle strictly between them is strictly better than the lesser.	Yes.
$U_{+}^{i}(x^{i})$ is convex.	For two consumption bundles, any bundle between them is no worse than the lesser.	Yes.
The household always has at least one feasible consumption plan.	no bankruptcy	No. It is necessary for the existence of demand functions.

on the producers

assumption	explanation	can we relax it?
$PPS^{j}$ is strictly convex	diseconomies of scale	Yes.
$PPS^{j}$ is convex	no economies of scale	Yes.
$PPS^{j}$ contains 0.	Producers can close down for free.
$PPS^{j}$ is a closed set	Technical assumption necessary for proofs to work.	No. It is necessary for the existence of supply functions.
$PPS\cap \mathbb {R} _{+}^{N}$ is bounded	There is no arbitrarily large "free lunch".	No. Economy needs scarcity.
$PPS\cap (-PPS)$ is bounded	The economy cannot reverse arbitrarily large transformations.

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:

Theorem—If $p$ is an equilibrium price vector for the restricted market, then it is also an equilibrium price vector for the unrestricted market. Furthermore, we have ${\tilde {D}}^{i}(p)=D^{i}(p),{\tilde {S}}^{j}(p)=S^{j}(p)$.

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.

Theorem—${\tilde {Z}}$ satisfies weak Walras's law: For all $p\in \mathbb {R} _{++}^{N}$, $$\langle p,{\tilde {Z}}(p)\rangle \leq 0$$ and if $\langle p,{\tilde {Z}}(p)\rangle <0$, then ${\tilde {Z}}(p)_{n}>0$ for some $n$.

Proof sketch

If total excess demand value is exactly zero, then every household has spent all their budget. Else, some household is restricted to spend only part of their budget. Therefore, that household's consumption bundle is on the boundary of the restriction, that is, $\|{\tilde {D}}^{i}(p)\|=C$. We have chosen (in the previous section) $C$ to be so large that even if all the producers coordinate, they would still fall short of meeting the demand. Consequently there exists some commodity $n$ such that ${\tilde {D}}^{i}(p)_{n}>{\tilde {S}}(p)_{n}+r_{n}$

Theorem—An equilibrium price vector exists for the restricted market, at which point the restricted market satisfies Walras's law.

Proof sketch

By definition of equilibrium, if $p$ is an equilibrium price vector for the restricted market, then at that point, the restricted market satisfies Walras's law.

${\tilde {Z}}$ is continuous since all ${\tilde {S}}^{j},{\tilde {D}}^{i}$ are continuous.

Define a function $$f(p)={\frac {\max(0,p+\gamma {\tilde {Z}}(p))}{\sum _{n}\max(0,p+\gamma {\tilde {Z}}(p)_{n})}}$$on the price simplex, where $\gamma$ is a fixed positive constant.

By the weak Walras law, this function is well-defined. By Brouwer's fixed-point theorem, it has a fixed point. By the weak Walras law, this fixed point is a market equilibrium.

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.

Corollary—An equilibrium price vector exists for the unrestricted market, at which point the unrestricted market satisfies Walras's law.

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 $r$.

First fundamental theorem of welfare economics—Any market equilibrium state is Pareto-efficient.

Proof sketch

The price hyperplane separates the attainable productions and the Pareto-better consumptions. That is, the hyperplane $\langle p^{*},q\rangle =\langle p^{*},D(p^{*})\rangle$ separates $r+PPS_{r}$ and $U_{++}$, where $U_{++}$ is the set of all $\sum _{i\in I}x'^{i}$, such that $\forall i\in I,x'^{i}\in CPS^{i},x'^{i}\succeq ^{i}x^{i}$, and $\exists i\in I,x'^{i}\succ ^{i}x^{i}$. That is, it is the set of aggregates of all possible consumption plans that are strictly Pareto-better.

The attainable productions are on the lower side of the price hyperplane, while the Pareto-better consumptions are strictly on the upper side of the price hyperplane. Thus any Pareto-better plan is not attainable.

• Any Pareto-better consumption plan must cost at least as much for every household, and cost more for at least one household.
• Any attainable production plan must profit at most as much for every producer.

Second fundamental theorem of welfare economics—For any total endowment $r$, and any Pareto-efficient state achievable using that endowment, there exists a distribution of endowments $\{r^{i}\}_{i\in I}$ and private ownerships $\{\alpha ^{i,j}\}_{i\in I,j\in J}$ of the producers, such that the given state is a market equilibrium state for some price vector $p\in \mathbb {R} _{++}^{N}$.

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.

Proof

Since the state is attainable, we have $\sum _{i\in I}x^{i}\preceq \sum _{j\in J}y^{j}+r$. The equality does not necessarily hold, so we define the set of attainable aggregate consumptions $V:=\{r+y-z:y\in PPS,z\succeq 0\}$. Any aggregate consumption bundle in $V$ is attainable, and any outside is not.

Find the market price $p$.

Define $U_{++}$ to be the set of all $\sum _{i\in I}x'^{i}$, such that $\forall i\in I,x'^{i}\in CPS^{i},x'^{i}\succeq ^{i}x^{i}$, and $\exists i\in I,x'^{i}\succ ^{i}x^{i}$. That is, it is the set of aggregates of all possible consumption plans that are strictly Pareto-better. Since each $CPS^{i}$ is convex, and each preference is convex, the set $U_{++}$ is also convex.

Now, since the state is Pareto-optimal, the set $U_{++}$ must be unattainable with the given endowment. That is, $U_{++}$ is disjoint from $V$. Since both sets are convex, there exists a separating hyperplane between them.

Let the hyperplane be defined by $\langle p,q\rangle =c$, where $p\in \mathbb {R} ^{N},p\neq 0$, and $c=\sum _{i\in I}\langle p,x^{i}\rangle$. The sign is chosen such that $\langle p,U_{++}\rangle \geq c$ and $\langle p,r+PPS\rangle \leq c$.

Claim: $p\succ 0$.

Suppose not, then there exists some $n\in 1:N$ such that $p_{n}<0$. Then $\langle p,r+0-ke_{n}\rangle >c$ if $k$ is large enough, but we also have $r+0-ke_{n}\in V$, contradiction.

We have by construction $\langle p,\sum _{i\in I}x^{i}\rangle =c$, and $\langle p,V\rangle \leq c$. Now we claim: $\langle p,U_{++}\rangle >c$.

For each household $i$, let $U_{+}^{i}(x^{i})$ be the set of consumption plans for $i$ that are at least as good as $x^{i}$, and $U_{++}^{i}(x^{i})$ be the set of consumption plans for $i$ that are strictly better than $x^{i}$.

By local nonsatiation of $\succeq ^{i}$, the closed half-space $\langle p,q\rangle \geq \langle p,x^{i}\rangle$ contains $U_{+}^{i}(x^{i})$.

By continuity of $\succeq ^{i}$, the open half-space $\langle p,q\rangle >\langle p,x^{i}\rangle$ contains $U_{++}^{i}(x^{i})$.

Adding them up, we find that the open half-space $\langle p,q\rangle >c$ contains $U_{++}$.

Claim (Walras's law): $\langle p,r+\sum _{j}y^{j}\rangle =c=\langle p,\sum _{i}x^{i}\rangle$

Since the production is attainable, we have $r+\sum _{j}y^{j}\succeq \sum _{i}x^{i}$, and since $p\succ 0$, we have $\langle p,r+\sum _{j}y^{j}\rangle \geq \langle p,\sum _{i}x^{i}\rangle$.

By construction of the separating hyperplane, we also have $\langle p,r+\sum _{j}y^{j}\rangle \leq c=\langle p,\sum _{i}x^{i}\rangle$, thus we have an equality.

Claim: at price $p$, each producer $j$ maximizes profit at $y^{j}$,

If there exists some production plan $y'^{j}$ such that one producer can reach higher profit $\langle p,y'^{j}\rangle >\langle p,y^{j}\rangle$, then

$$\langle p,r\rangle +\sum _{j\in J}\langle p,y'^{j}\rangle >\langle p,r\rangle +\sum _{j\in J}\langle p,y^{j}\rangle =c$$

but then we would have a point in $r+PPS$ on the other side of the separating hyperplane, violating our construction.

Claim: at price $p$ and budget $\langle p,x^{i}\rangle$, household $i$ maximizes utility at $x^{i}$.

Otherwise, there exists some $x'^{i}$ such that $x'^{i}\succ ^{i}x^{i}$ and $\langle p,x'^{i}\rangle \leq \langle p,x^{i}\rangle$. Then, consider aggregate consumption bundle $q':=\sum _{i'\in I,i'\neq i}x^{i}+x'^{i}$. It is in $U_{++}$, but also satisfies $\langle p,q'\rangle \leq \sum \langle p,x^{i}\rangle =c$. But this contradicts previous claim that $\langle p,U_{++}\rangle >c$.

By Walras's law, the aggregate endowment income and profit exactly equals aggregate expenditure. It remains to distribute them such that each household $i$ obtains exactly $\langle p,x^{i}\rangle$ as its budget. This is trivial.

Here is a greedy algorithm to do it: first distribute all endowment of commodity 1 to household 1. If household 1 can reach its budget before distributing all of it, then move on to household 2. Otherwise, start distributing all endowment of commodity 2, etc. Similarly for ownerships of producers.

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.

The two fundamental theorems of welfare economics holds without modification.

converting from strict convexity to convexity

strictly convex case	convex case
$PPS^{j}$ is strictly convex	$PPS^{j}$ is convex
$CPS^{i}$ is strictly convex	$CPS^{i}$ is convex
$\succeq ^{i}$ is strictly convex	$\succeq ^{i}$ is convex
${\tilde {S}}^{j}(p)$ is point-valued	${\tilde {S}}^{j}(p)$ is set-valued
${\tilde {S}}^{j}(p)$ is continuous	${\tilde {S}}^{j}(p)$ has closed graph
$\langle p,{\tilde {Z}}(p)\rangle \leq 0$	$\langle p,z\rangle \leq 0$ for any $z\in {\tilde {Z}}(p)$
...	...
equilibrium exists by Brouwer's fixed-point theorem	equilibrium exists by Kakutani's fixed-point theorem

Extensions

Accounting for time, space, and uncertainty

Further information: Financial economics § State prices, State prices § Application to financial assets, and 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 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.

— Gérard Debreu, Theory of value: An axiomatic analysis of economic equilibrium (1959)

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)

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.

See also

• Model (economics)
• Incomplete markets
• Fisher market - a simpler market model, in which the total quantity of each product is given, and each buyer comes only with a monetary budget.
• List of asset pricing articles
• Financial economics § Underlying economics

Further reading

• Athreya, Kartik B. (2013). "The Modern Macroeconomic Approach and the Arrow–Debreu–McKenzie Model". Big Ideas in Macroeconomics: A Nontechnical View. Cambridge: MIT Press. pp. 11–46. ISBN 978-0-262-01973-6.
• Geanakoplos, John (1987). "Arrow–Debreu model of general equilibrium". The New Palgrave: A Dictionary of Economics. Vol. 1. pp. 116–124.
• Takayama, Akira (1985). Mathematical Economics (2nd ed.). London: Cambridge University Press. pp. 255–284. ISBN 978-0-521-31498-5.
• Düppe, Till (2012). "Arrow and Debreu de-homogenized". Journal of the History of Economic Thought. 34 (4): 491–514. CiteSeerX 10.1.1.416.2120. doi:10.1017/s1053837212000491. S2CID 15771197.

External links

• Notes on the Arrow–Debreu–McKenzie Model of an Economy, Prof. Kim C. Border California Institute of Technology
• "The Fundamental Theorem" of Finance; part II. Prof. Mark Rubinstein, Haas School of Business