Fisher market

Suppose the utilities of all the buyers are homogeneous functions (this includes, in particular, utilities with constant elasticity of substitution).

Then, the equilibrium conditions in the Fisher model can be written as solutions to a convex optimization program called the Eisenberg-Gale convex program.[4] This program finds an allocation that maximizes the weighted geometric mean of the buyers' utilities, where the weights are determined by the budgets. Equivalently, it maximizes the weighted arithmetic mean of the logarithms of the utilities:

Maximize ${\displaystyle \sum _{i=1}^{n}\left(B_{i}\cdot \log {(u_{i})}\right)}$

Subject to:

Non-negative quantities: For every buyer ${\displaystyle i}$ and product ${\displaystyle j}$: ${\displaystyle x_{i,j}\geq 0}$

Sufficient supplies: For every product ${\displaystyle j}$: ${\displaystyle \sum _{i=1}^{n}x_{i,j}\leq 1}$

(since supplies are normalized to 1).

This optimization problem can be solved using the Karush–Kuhn–Tucker conditions (KKT). These conditions introduce Lagrangian multipliers that can be interpreted as the prices, ${\displaystyle p_{1},\dots ,p_{m}}$. In every allocation that maximizes the Eisenberg-Gale program, every buyer receives a demanded bundle. I.e, a solution to the Eisenberg-Gale program represents a market equilibrium.[5]^:141–142

A special case of homogeneous utilities is when all buyers have linear utility functions. This means that for every buyer ${\displaystyle i}$ and product ${\displaystyle j}$, there is a constant ${\displaystyle u_{i,j}}$ (utility of buyer ${\displaystyle i}$ for product ${\displaystyle j}$) such that the total utility that a buyer derives from a bundle is:

${\displaystyle u_{i}(x_{1},\dots ,x_{j})=\sum _{j=1}^{m}u_{i,j}\cdot x_{j}}$, where ${\displaystyle u_{i,j}\geq 0}$

We assume that each product has a potential buyer - a buyer that derives positive utility from that product. Under this assumption, market-clearing prices exist and are unique. The proof is based on the Eisenberg-Gale program. The KKT conditions imply that the optimal solutions (allocations ${\displaystyle x_{i,j}}$ and prices ${\displaystyle p_{j}}$) satisfy the following inequalities:

1. All prices are non-negative: ${\displaystyle p_{j}\geq 0}$.
2. If a product has a positive price, then all its supply is exhausted: ${\displaystyle p_{j}>0\implies \sum _{i=1}^{n}x_{i,j}=1}$.
3. The total utility-per-coin of a buyer (total utility divided by total budget) is weakly larger than his utility-per-coin from each single product: ${\displaystyle {\frac {\sum _{k=1}^{m}u_{i,k}\cdot x_{i,k}}{B_{i}}}\geq {\frac {u_{i,j}}{p_{j}}}}$
4. If a buyer buys a positive amount of a product, then his total utility-per-coin equals his utility-per-coin from that product: ${\displaystyle x_{i,j}>0\implies {\frac {\sum _{k=1}^{m}u_{i,k}\cdot x_{i,k}}{B_{i}}}={\frac {u_{i,j}}{p_{j}}}}$

Assume that every product ${\displaystyle j}$ has a potential buyer - a buyer ${\displaystyle i}$ with ${\displaystyle u_{i,j}>0}$. Then, inequality 3 implies that ${\displaystyle p_{j}>0}$, i.e, all prices are positive. Then, inequality 2 implies that all supplies are exhausted. Inequality 4 implies that all buyers' budgets are exhausted. I.e, the market clears.

Since the log function is a strictly concave function, if there is more than one equilibrium allocation then the utility derived by each buyer in both allocations must be the same (a decrease in the utility of a buyer cannot be compensated by an increase in the utility of another buyer). This, together with inequality 4, implies that the prices are unique.[2]^:107

There is a weakly polynomial-time algorithm for finding equilibrium prices and allocations in a linear Fisher market. The algorithm is based on condition 4 above. The condition implies that, in equilibrium, every buyer buys only products that give him maximum utility-per-coin. Let's say that a buyer "likes" a product, if that product gives him maximum utility-per-coin in the current prices. Given a price-vector, we can build a flow network in which the capacity of each edge represents the total money "flowing" through that edge. The network is as follows:

• There is a source node, s.
• There is a node for each product; there is an edge from s to each product j, with capacity ${\displaystyle p_{j}}$ (this is the maximum amount of money that can be expended on product j, since the supply is normalized to 1).
• There is a node for each buyer; there is an edge from a product to a buyer, with infinite capacity, iff the buyer likes the product (in the current prices).
• There is a target node, t; there is an edge from each buyer i to t, with capacity ${\displaystyle B_{i}}$ (the maximum expenditure of i).

The price-vector p is an equilibrium price-vector, if and only if the two cuts ({s},V\{s}) and (V\{t},{t}) are min-cuts. Hence, an equilibrium price-vector can be found using the following scheme:

• Start with very low prices, which are guaranteed to be below the equilibrium prices; in these prices, buyers have some budget left (i.e, the maximum flow does not reach the capacity of the nodes into t).
• Continuously increase the prices and update the flow-network accordingly, until all budgets are exhausted.

There is an algorithm that solves this problem in weakly polynomial time.[2]^:109–121