  The Fisher market is one of the most fundamental models for resource
allocation problems in economic theory, wherein agents spend a budget of
currency to buy goods that maximize their utilities, while producers sell
capacity constrained goods in exchange for currency. However, the consideration
of only two types of constraints, i.e., budgets of individual buyers and
capacities of goods, makes Fisher markets less amenable for resource allocation
settings when agents have additional linear constraints, e.g., knapsack and
proportionality constraints. In this work, we introduce a modified Fisher
market, where each agent may have additional linear constraints and show that
this modification to classical Fisher markets fundamentally alters the
properties of the market equilibrium as well as the optimal allocations. These
properties of the modified Fisher market prompt us to introduce a budget
perturbed social optimization problem (BP-SOP) and set prices based on the dual
variables of BP-SOP's capacity constraints. To compute the budget
perturbations, we develop a fixed point iterative scheme and validate its
convergence through numerical experiments.
  Since this fixed point iterative scheme involves solving a centralized
problem at each step, we propose a new class of distributed algorithms to
compute equilibrium prices. In particular, we develop an Alternating Direction
Method of Multipliers (ADMM) algorithm with strong convergence guarantees for
Fisher markets with homogeneous linear constraints as well as for classical
Fisher markets. In this algorithm, the prices are updated based on the
tatonnement process, with a step size that is completely independent of the
utilities of individual agents. Thus, our mechanism, both theoretically and
computationally, overcomes a fundamental limitation of classical Fisher
markets, which only consider capacity and budget constraints.
