Arrow–Debreu model

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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.^[1]

The model is central to the theory of general (economic) equilibrium, and it is used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954,^[1] and Lionel W. McKenzie independently in 1954,^[2] with later improvements in 1959.^[3][4]

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.^[5]

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. The Arrow-Debreu model, as communicated in the Theory of Value, changed basic thinking and 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,^[6] which is based on.^[7]

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), one may think of as "inheritance." For mathematical clarity, all households must sell all their endowment to the market at the beginning. If they wish to retain some of the endowments, they would have to repurchase them from the market later. The endowments may be working hours, land use, 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 ${\displaystyle j}$ is divided among the households in proportion to how much stock each household holds for the producer ${\displaystyle j}$. Ownership is imposed initially, and the households may not sell, buy, create, or discard them.

The households receive a budget, 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 they can afford using their budget.

The producers can transform 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 utility and profit, their consumption and production plans "harmonize." That is, "the market clears". In other words, the market is playing the role of a "Walrasian auctioneer."

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

• ${\displaystyle x\succeq y}$ if ${\displaystyle \forall n,x_{n}\geq y_{n}}$
• ${\displaystyle \mathbb {R} _{+}^{N}}$ is the set of ${\displaystyle x}$ such that ${\displaystyle x\succeq 0}$
• ${\displaystyle \mathbb {R} _{++}^{N}}$ is the set of ${\displaystyle x}$ such that ${\displaystyle x\succ 0}$
• ${\displaystyle \Delta _{N}=\left\{x\in \mathbb {R} ^{N}:x_{1},...,x_{N}\geq 0,\sum _{n\in 1:N}x_{n}=1\right\}}$ is the N-simplex. We often call it the price simplex since we sometimes scale the price vector to lie on it.

• The commodities are indexed as ${\displaystyle n\in 1:N}$. Here ${\displaystyle N}$ is the number of commodities in the economy. It is a finite number.
• The price vector ${\displaystyle p=(p_{1},...,p_{N})\in \mathbb {R} _{++}^{N}}$ is a vector of length ${\displaystyle N}$, with each coordinate being the price of a commodity. The prices may be zero or positive.

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

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

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

• An economy is a tuple ${\displaystyle (N,I,J,CPS^{i},\succeq ^{i},PPS^{j})}$. It is a tuple specifying the commodities, consumer preferences, consumption possibility sets, and producers' production possibility sets.
• An economy with initial distribution is an economy, along with an initial distribution tuple ${\displaystyle (r^{i},\alpha ^{i,j})_{i\in I,j\in J}}$ for the economy.
• A state of the economy is a tuple of price, consumption plans, and production plans for each household and producer: ${\displaystyle ((p_{n})_{n\in 1:N},(x^{i})_{i\in I},(y^{j})_{j\in J})}$.
• A state is feasible iff each ${\displaystyle x^{i}\in CPS^{i}}$, each ${\displaystyle y^{j}\in PPS^{j}}$, and ${\displaystyle \sum _{i\in I}x^{i}\preceq \sum _{j\in J}y^{j}+r}$.
• The feasible production possibilities set, given endowment ${\displaystyle r}$, is ${\displaystyle PPS_{r}:=\{y\in PPS:y+r\succeq 0\}}$.
• Given an economy with distribution, the state corresponding to a price vector ${\displaystyle p}$ is ${\displaystyle (p,(D^{i}(p))_{i\in I},(S^{j}(p))_{j\in J})}$.
• Given an economy with distribution, a price vector ${\displaystyle p}$ is an equilibrium price vector for the economy with initial distribution, iff$${\displaystyle Z(p)_{n}{\begin{cases}\leq 0{\text{ if }}p_{n}=0\\=0{\text{ if }}p_{n}>0\end{cases}}}$$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.

on the households

assumption	explanation	can we relax it?
${\displaystyle CPS^{i}}$ is closed	Technical assumption necessary for proofs to work.	No. It is necessary for the existence of demand functions.
local nonsatiation: ${\displaystyle \forall x\in CPS^{i},\epsilon >0,}$ ${\displaystyle \exists x'\in CPS^{i},x'\succ ^{i}x,\|x'-x\|<\epsilon }$	Households always want to consume a little more.	No. It is necessary for Walras's law to hold.
${\displaystyle CPS^{i}}$ is strictly convex	strictly diminishing marginal utility	Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section.
${\displaystyle CPS^{i}}$ is convex	diminishing marginal utility	Yes, to nonconvexity, with Shapley–Folkman lemma.
continuity: ${\displaystyle U_{+}^{i}(x^{i})}$ 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.
${\displaystyle U_{+}^{i}(x^{i})}$ is strictly convex.	For two consumption bundles, any bundle between them is better than the lesser.	Yes, to mere convexity, with Kakutani's fixed-point theorem. See the next section.
${\displaystyle U_{+}^{i}(x^{i})}$ is convex.	For two consumption bundles, any bundle between them is no worse than the lesser.	Yes, to nonconvexity, with Shapley–Folkman lemma.
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?
${\displaystyle PPS^{j}}$ is strictly convex	diseconomies of scale	Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section.
${\displaystyle PPS^{j}}$ is convex	no economies of scale	Yes, to nonconvexity, with Shapley–Folkman lemma.
${\displaystyle PPS^{j}}$ contains 0.	Producers can close down for free.
${\displaystyle PPS^{j}}$ is a closed set	Technical assumption necessary for proofs to work.	No. It is necessary for the existence of supply functions.
${\displaystyle PPS\cap \mathbb {R} _{+}^{N}}$ is bounded	There is no arbitrarily large "free lunch".	No. Economy needs scarcity.
${\displaystyle PPS\cap (-PPS)}$ is bounded	The economy cannot reverse arbitrarily large transformations.

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

Consequently, we define "restricted market" to be the same market, except there is a universal upper bound ${\displaystyle C}$, such that every producer is required to use a production plan ${\displaystyle \|y^{j}\|\leq C}$. Each household is required to use a consumption plan ${\displaystyle \|x^{i}\|\leq C}$. Denote the corresponding quantities on the restricted market with a tilde. So, for example, ${\displaystyle {\tilde {Z}}(p)}$ is the excess demand function on the restricted market.^[8]

${\displaystyle C}$ 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, ${\displaystyle C}$ is chosen to be large enough such that:

• For any consumption plan ${\displaystyle x}$ such that ${\displaystyle x\succeq 0,\|x\|=C}$, 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 ${\displaystyle (y^{j}\in PPS^{j})_{j\in J}}$, if ${\displaystyle \sum _{j\in J}y^{j}+r\succeq 0}$, then ${\displaystyle \|y^{j}\|<C}$for each ${\displaystyle j\in J}$. In other words, for any attainable production plan under the given endowment ${\displaystyle r}$, 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 ${\displaystyle PPS_{r}=\left\{\sum _{j\in J}y^{j}:y^{j}\in PPS^{j}{\text{ for each }}j\in J,{\text{ and }}\sum _{j\in J}y^{j}+r\succeq 0\right\}}$, then under the assumptions for the producers given above (especially the "no arbitrarily large free lunch" assumption), ${\displaystyle PPS_{r}}$ is bounded for any ${\displaystyle r\succeq 0}$ (proof omitted). Thus the first requirement is satisfiable.
• Define the set of attainable individual production plans to be ${\displaystyle PPS_{r}^{j}:=\{y^{j}\in PPS^{j}:y^{j}{\text{ is a part of some attainable production plan under endowment }}r\}}$then under the assumptions for the producers given above (especially the "no arbitrarily large transformations" assumption), ${\displaystyle PPS_{r}^{j}}$ is bounded for any ${\displaystyle j\in J,r\succeq 0}$ (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 ${\displaystyle p}$, if ${\displaystyle \|{\tilde {S}}^{j}(p)\|<C}$, then ${\displaystyle S^{j}(p)}$ exists and is equal to ${\displaystyle {\tilde {S}}^{j}(p)}$. 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 ${\displaystyle C}$.
• If all ${\displaystyle S^{j}(p)={\tilde {S}}^{j}(p)}$, then the restricted and unrestricted households have the same budget. Now, if we also have ${\displaystyle \|{\tilde {D}}^{i}(p)\|<C}$, then ${\displaystyle D^{i}(p)}$ exists and is equal to ${\displaystyle {\tilde {D}}^{i}(p)}$. 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 ${\displaystyle C}$.

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

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

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

Walras's law can be interpreted on both sides:

• On the side of the households, it is said 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.

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

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.

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.^[9] Ross M. Starr (1969) proved the existence of economic equilibria when some consumer preferences need not be convex.^[9] 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.^[10]

(Uzawa, 1962)^[11] 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.^[12]

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 ${\displaystyle r}$ for the entire society, the planner must pick a feasible master plan of production and consumption plans ${\displaystyle ((x^{i})_{i\in I},(y^{j})_{j\in J})}$. 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 ${\displaystyle ((x^{i})_{i\in I},(y^{j})_{j\in J})}$ by ${\displaystyle ((x^{i})_{i\in I},(y^{j})_{j\in J})\succeq ((x'^{i})_{i\in I},(y'^{j})_{j\in J})}$ iff ${\displaystyle x^{i}\succeq ^{i}x'^{i}}$ for all ${\displaystyle i\in I}$.

Then, we say that a plan is Pareto-efficient with respect to a starting endowment ${\displaystyle r}$, 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 ${\displaystyle r}$.

With the set up, we have two fundamental theorems of welfare economics:^[13]

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.