Arrow–Debreu model
In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions, 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 equilibrium, and it is used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954, and Lionel W. McKenzie independently in 1954, with later improvements in 1959.
The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium of an economy. In general, there may be many equilibria.
Arrow and Debreu were separately awarded the Nobel Prize in Economics for their development of the model. McKenzie, however, did not receive the award.
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, 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 is divided among the households in proportion to how much stock each household holds for the producer. 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. 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."
| 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
- if
- is the set of such that
- is the set of such that
- 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. Here is the number of commodities in the economy. It is a finite number.
- The price vector is a vector of length, with each coordinate being the price of a commodity. The prices may be zero or positive.
households
- The households are indexed as.
- Each household begins with an endowment of commodities.
- Each household begins with a tuple of ownerships of the producers. The ownerships satisfy.
- 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:
- Each household has a Consumption Possibility Set.
- Each household has a preference relation over.
- With assumptions on , each preference relation is representable by a utility function 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, written as.
- is the set of consumption plans at least as preferable as.
- The budget set is the set of consumption plans that it can afford:.
- For each price vector, the household has a demand vector for commodities, as. This function is defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.It may not be well-defined for all. However, we will use enough assumptions to be well-defined at equilibrium price vectors.
producers
- The producers are indexed as.
- Each producer has a Production Possibility Set. Note that the supply vector may have both positive and negative coordinates. For example, 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, written as.
- For each price vector, the producer has a supply vector for commodities, as. This function will be defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.It may not be well-defined for all. However, we will use enough assumptions to be well-defined at equilibrium price vectors.
- The profit is
aggregates
- aggregate consumption possibility set.
- aggregate production possibility set.
- aggregate endowment
- aggregate demand
- aggregate supply
- excess demand
the whole economy
- An economy is a tuple. 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 for the economy.
- A state of the economy is a tuple of price, consumption plans, and production plans for each household and producer:.
- A state is feasible iff each, each, and.
- The feasible production possibilities set, given endowment, is.
- Given an economy with distribution, the state corresponding to a price vector is.
- Given an economy with distribution, a price vector is an equilibrium price vector for the economy with initial distribution, iffThat 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.
- A state is an equilibrium state iff it is the state corresponding to an equilibrium price vector.
Assumptions
| assumption | explanation | can we relax it? |
| is strictly convex | diseconomies of scale | Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section. |
| is convex | no economies of scale | Yes, to nonconvexity, with Shapley–Folkman lemma. |
| contains 0. | Producers can close down for free. | |
| is a closed set | Technical assumption necessary for proofs to work. | No. It is necessary for the existence of supply functions. |
| is bounded | There is no arbitrarily large "free lunch". | No. Economy needs scarcity. |
| is bounded | The economy cannot reverse arbitrarily large transformations. |
Imposing an artificial restriction
The functions are not necessarily well-defined for all price vectors. For example, if producer 1 is capable of transforming units of commodity 1 into units of commodity 2, and we have, then the producer can create plans with infinite profit, thus, and is undefined.Consequently, we define "restricted market" to be the same market, except there is a universal upper bound, such that every producer is required to use a production plan. Each household is required to use a consumption plan. Denote the corresponding quantities on the restricted market with a tilde. So, for example, is the excess demand function on the restricted market.
is chosen to be "large enough" for the economy so that the restriction is not in effect under equilibrium conditions. In detail, is chosen to be large enough such that:
- For any consumption plan such that, 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, if, then for each. In other words, for any attainable production plan under the given endowment, each producer's individual production plan must lie strictly within the restriction.
- Define the set of attainable aggregate production plans to be, then under the assumptions for the producers given above, is bounded for any . Thus the first requirement is satisfiable.
- Define the set of attainable individual production plans to be then under the assumptions for the producers given above, is bounded for any . Thus the second requirement is satisfiable.
- At any price vector, if, then exists and is equal to. 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.
- If all, then the restricted and unrestricted households have the same budget. Now, if we also have, then exists and is equal to. 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.