Fisher market

Fisher market is an economic model attributed to Irving Fisher. It has the following ingredients:^[1]

• A set of ${\displaystyle m}$ divisible products with pre-specified supplies (usually normalized such that the supply of each good is 1).
• A set of ${\displaystyle n}$ buyers.
• For each buyer ${\displaystyle i=1,\dots ,n}$, there is a pre-specified monetary budget ${\displaystyle B_{i}}$.

Each product ${\displaystyle j}$ has a price ${\displaystyle p_{j}}$; the prices are determined by methods described below. The price of a bundle of products is the sum of the prices of the products in the bundle. A bundle is represented by a vector ${\displaystyle x=x_{1},\dots ,x_{m}}$, where ${\displaystyle x_{j}}$ is the quantity of product ${\displaystyle j}$. So the price of a bundle ${\displaystyle x}$ is ${\displaystyle p(x)=\sum _{j=1}^{m}p_{j}\cdot x_{j}}$.

A bundle is affordable for a buyer if the price of that bundle is at most the buyer's budget. I.e, a bundle ${\displaystyle x}$ is affordable for buyer ${\displaystyle i}$ if ${\displaystyle p(x)\leq B_{i}}$.

Each buyer has a preference relation over bundles, which can be represented by a utility function. The utility function of buyer ${\displaystyle i}$ is denoted by ${\displaystyle u_{i}}$. The demand set of a buyer is the set of affordable bundles that maximize the buyer's utility among all affordable bundles, i.e.:

${\displaystyle {\text{Demand}}_{i}(p):=\arg \max _{p(x)\leq B_{i}}u_{i}(x)}$.

A competitive equilibrium (CE) is a price-vector ${\displaystyle p_{1},\dots ,p_{m}}$in which it is possible to allocate, to each agent, a bundle from his demand-set, such that the total allocation exactly equals the supply of products. The corresponding prices are called market-clearing prices. The main challenge in analyzing Fisher markets is finding a CE.^{[2]: 103–105}