This is the first of a series of posts about stability and equilibrium in trading networks. I will review and recall established results from network flows and point out how they immediately yield results about equilibria, stability and the core of matching markets with quasi-linear utility. It presumes familiarity with optimization and the recent spate of papers on matchings with contracts.

The simplest trading network one might imagine would involve buyers () and sellers () of a homogenous good and a set of edges between them. No edges between sellers and no edges between buyers. The absence of an edge in linking and means that and cannot trade directly. Suppose buyer has a constant marginal value of upto some amount and zero thereafter. Seller has a constant marginal cost of upto some capacity and infinity thereafter.

Under the quasi-linear assumption, the problem of finding the efficient set of trades to execute can be formulated as a linear program. Let for denote the amount of the good purchased by buyer from seller . Then, the following program identifies the efficient allocation:

subject to

This is, of course, an instance of the (discrete) transportation problem. The general version of the transportation problem can be obtained by replacing each coefficient of the objective function by arbitrary numbers . This version of the transportation problem is credited to the mathematician F. J. Hitchcock and published in 1941. Hitchcock’s most famous student is Claude Shannon.

The `continuous’ version of the transportation problem was formulated by Gaspard Monge and described in his 1781 paper on the subject. His problem was to split two equally large volumes (representing the initial location and the final location of the earth to be shipped) into infinitely many small particles and then `match them with each other so that the sum of the products of the lengths of the paths used by the particles and the volume of the particles is minimized. The ‘s in Monge’s problem have a property since called the Monge property, that is the same as submodularity/supermodularity. This paper describes the property and some of its algorithmic implications. Monge’s formulation was subsequently picked up by Kantorovich and the study of it blossomed into the specialty now called optimal transport with applications to PDEs and concentration of measure. That is not the thread I will follow here.

Returning to the Hitchcock, or rather discrete, formulation of the transportation problem let be the dual variables associated with the first set of constraints (the supply side) and the dual variables associated with the second or demand set of constraints. The dual is

subject to

We can interpret as the unit price of the good sourced from seller and as the surplus that buyer will enjoy at prices . Three things are immediate from the duality theorem, complementary slackness and dual feasibility.

- If is a solution to the primal and an optimal solution to the dual, then, the pair form a Walrasian equilibrium.
- The set of optimal dual prices, i.e., Walrasian prices live in a lattice.
- The dual is a (compact) representation of the TU (transferable utility) core of the co-operative game associated with this economy.
- Suppose the only
*bilateral*contracts we allow between buyer and seller are when . Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair . - Because the constraint matrix of the transportation problem is totally unimodular, the previous statements hold even if the goods are indivisible.

As these are standard, I will not reprove them here. Note also, that none of these conclusions depend upon the particular form of the coefficients in the objective function of the primal. We could replace by where we interpret to be the joint gain gains from trade (per unit) to be shared by buyer and seller .

Now, suppose we replace constant marginal values by increasing concave utility functions, and constant marginal costs by ? The problem of finding the efficient allocation becomes:

subject to

This is an instance of a concave flow problem. The Kuhn-Tucker-Karush conditions yield the following:

- If is a solution to the primal and an optimal Lagrangean, then, the pair form a Walrasian equilibrium.
- The set of optimal Lagrange prices, i.e., Walrasian prices live in a lattice.
- Suppose the only
*bilateral*contracts we allow between buyer and seller are when . Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair .

Notice, we lose the extension to indivisibility. As the objective function in the primal is now concave, an optimal solution to the primal may occur in the interior of the feasible region rather than at an extreme point. To recover `integrality’ we need to impose a stronger condition on and , specifically, they be -concave and convex respectively. This is a condition tied closely to the gross substitutes condition. More on this in a subsequent post.

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