How To Solve Equilibrium Problems

How To Solve Equilibrium Problems-71
Objective variables (if and when they exist) belong to individual agents, not to the model as a whole.Consider the following example from Kim & Ferris (2017) [143]. sum(i, p(i)*x(i)) =l= sum(i, p(i)*b(i)); model m / mkt, profit, udef, budget /; file empinfo /'%emp.info%'/; putclose empinfo 'equilibrium' / ' max', u, 'x', udef, budget / ' vi profit y' / ' vi mkt p' / ; * the second commodity is used as a numeraire p.fx('2') = 1; x.l(i) = 1; solve m using EMP; is maximized) and the activities of the producer and the price-setting market are expressed as VI.

Objective variables (if and when they exist) belong to individual agents, not to the model as a whole.Consider the following example from Kim & Ferris (2017) [143]. sum(i, p(i)*x(i)) =l= sum(i, p(i)*b(i)); model m / mkt, profit, udef, budget /; file empinfo /'%emp.info%'/; putclose empinfo 'equilibrium' / ' max', u, 'x', udef, budget / ' vi profit y' / ' vi mkt p' / ; * the second commodity is used as a numeraire p.fx('2') = 1; x.l(i) = 1; solve m using EMP; is maximized) and the activities of the producer and the price-setting market are expressed as VI.

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In this economic equilibrium problem there are three agents: one profit-maximizing producer, one utility-maximizing consumer and a market that determines the price of three commodities based on production and demand. As there are three commodities, the first VI actually generates three VI functions, one for each commodity.

The problem data include a technology matrix \(A\), where the entry \(a_ set i 'commodities' / 1*3 /; variable u 'consumer utility'; positive variables y 'activity of the producer' x(i) 'Marshallian demand of the consumer' p(i) 'prices' ; parameters A(i) 'technology matrix' / 1 1, 2 -1, 3 -1 / s(i) 'budget share' / 1 0.9, 2 0.1, 3 0 / b(i) 'endowment' / 1 0, 2 5, 3 3 / ; equations profit 'profit of activity' mkt(i) 'constraint on excess demand' udef 'Cobb-Douglas utility function' budget 'budget constraint' ; profit.. Thus there are three agents and four VI functions in the equilibrium problem.

The latter demonstrates that there are equilibrium problems where the optimization problems of the individual agents are solvable, but the overall equilibrium problem does not have a solution.

In many applications equilibrium problems come with a twist: the dual variable associated with a constraint in the problem of one agent appears in the problem of another agent.

This is reflected in the EMP Summary in the listing file: In the GAMS EMP Library there are several models that have a similar form, e.g.

Scarf's activity analysis model [SCARFEMP-DEM] and the simple equilibrium problem [SIMPEQUIL].

We typically assume that each variable and each equation is controlled by or belongs to exactly one agent.

Variables that are controlled by one agent but appear in the equations of a second agent are regarded as fixed or exogenous variables by that second agent: when taking first-order conditions, the second agent won't take derivatives wrt these exogenous variables.

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages.

We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems.

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