Negishi Method to Compute Competitive Equilibria

1. Solve the social planners problem for Pareto efficient allocations indexed
by Pareto weight α

2. Compute transfers, indexed by α , necessary to make the efficient allocation
affordable. As prices use Lagrange multipliers on the resource constraints
in the planner's ’problem.

3. Find the Pareto weight(s) α^ that makes the transfer functions 0.

4. The Pareto efficient allocations corresponding to α^ are equilibrium allocations; the supporting equilibrium prices are (multiples of) the Lagrange multipliers from the planning problem.

Dirk Krueger "Macroeconomic Theory" p.17
http://citeseer.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7768&rep=rep1&type=pdf