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), 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 j is divided among the households in proportion to how much stock each household holds for the producer 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."
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 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 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 \mathit{CPS}^i\subset \R_+^N. • Each household has a
preference relation \succeq^i over \mathit{CPS}^i. • With assumptions on \succeq^i (given in the next section), each preference relation is representable by a
utility function u^i: \mathit{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 \mathit{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 \mathit{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 to be well-defined at equilibrium price vectors.
producers • The producers are indexed as j\in J. • Each producer has a
Production Possibility Set \mathit{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 \mathit{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 \mathit{PPS}^j} \langle p, y^j\rangleIt may not be well-defined for all p \in \R^N_{++}. However, we will use enough assumptions to be well-defined at equilibrium price vectors. • The
profit is \Pi^j(p) := \langle p, S^j(p)\rangle = \max_{y^j\in \mathit{PPS}^j} \langle p, y^j\rangle
aggregates • aggregate consumption possibility set \mathit{CPS} = \sum_{i\in I}\mathit{CPS}^i. • aggregate production possibility set \mathit{PPS} = \sum_{j\in J}\mathit{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, \mathit{CPS}^i, \succeq^i, \mathit{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 (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 \mathit{CPS}^i, each y^j\in \mathit{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 \mathit{PPS}_r := \{y\in \mathit{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, iffZ(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 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 , 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. 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 \mathit{PPS}^j)_{j\in J}, if \sum_{j\in J} y^j + r \succeq 0, then \|y^j\| 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 \mathit{PPS}_r = \left\{\sum_{j\in J} y^j : y^j \in \mathit{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), \mathit{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 \mathit{PPS}_r^j := \{y^j \in \mathit{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), \mathit{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)\| , 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)\| , 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 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. {{Math proof|title=Proof sketch|proof= 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_n + \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.
The role of convexity leaves the point
(0,0) fixed but moves every point on the non–convex
unit circle. 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. 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: {{Math proof|title=Proof sketch|proof= 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 + \mathit{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 \mathit{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. }} 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. {{Math proof|title=Proof|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 \mathit{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 \mathit{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 \mathit{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 \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+\mathit{PPS}\rangle \leq c. Claim: p \succ 0. : Suppose not, then there exists some n\in 1:N such that p_n . Then \langle p, r + 0 - k e_n\rangle > c if k is large enough, but we also have r + 0 - k e_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+\mathit{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. 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. == Extensions ==