\$TITLE Pure exchange model solved with EMP, SJM, and CGE (NEGISHI, SEQ=21) \$ontext We consider a pure exchange model in which a set of agents (i.e. consumers) are each endowed with a fixed quantity of goods. The agents can trade to maximize their utility. The solution consists of a consumption vector for each agent and a set of prices for each good such that: each agent maximizes her utility at this consumption level s.t. the budget constraint imposed by her endowment and the prices Utility is given by a Cobb-Douglas function of the form U(agent) = prod(good, C(good,agent)**alpha(good,agent)) This utility function implies that the income level is the Negishi weight. This cannot be solved as a single NLP model (see reference below) but there are a number of ways to solve this model: 1. Via EMP and a complementarity model, finding the weights directly 2. Via the SJM approach of Rutherford 3. Via a CGE approach (using the implicit demand functions) Negishi, T, Welfare Economics and the Existence of an Equilibrium for a Competitive Economy. Metroeconomics, Vol 12 (1960), 92-97. Editor: Steve Dirkse, August 2009 With contributions from Sherman Robinson and Michael Ferris \$offtext sets g goods / g1 * g3 / a utility-maximizing agents / a1 * a3 / table alpha(g,a) Cobb-Douglas elasticities sum to 1 for each agent a1 a2 a3 g1 .7 .4 .2 g2 .2 .3 .4 g3 .1 .3 .4 table endow(g,a) endowment a1 a2 a3 g1 10 g2 8 g3 3 Parameters RepY(a,*) income report RepP(g,*) price report RepC(g,a,*) consumption report; \$macro rep(style) RepY(a,'style') = Y.l(a); RepP(g,'style') = P.l(g); RepC(g,a,'style') = C.l(g,a); variables utility utility function C(g,a) consumption Y(a) income positive variables P(g) prices equations DefUtility utility definition balance(g) material balance: consumption <= endowment budget(a) budget constraint; defutility.. utility =E= sum{a, Y(a)*sum{g, alpha(g,a)*log(C(g,a))}}; balance(g).. sum{a, C(g,a)} =L= sum{a, endow(g,a)}; budget(a).. Y(a) =E= sum{g, endow(g,a)*P(g)}; C.lo(g,a) = 1e-6; C.l (g,a) = 5; model negishi / defutility, balance, budget /; * fix a numeraire y.l(a) = 1; y.fx('a1') = 1; ******************************************************************************* *** 0. This cannot be solved as a single NLP model ****************************************************************************** solve negishi maximizing utility using nlp; rep(WRONG) ******************************************************************************* *** 1. Via EMP and a complementarity model, finding the weights directly ****************************************************************************** file myinfo / '%emp.info%' /; put myinfo '* negishi model'; put / 'dualVar P balance'; putclose / 'dualEqu budget Y'; solve negishi maximizing utility using emp; rep(EMP) ******************************************************************************* *** 2. Via the SJM approach of Rutherford *** *** In the SJM (Sequential Joint Maximization) approach, we start with estimates *** for the Negishi weights and iterate: *** Repeat *** 1. Solve the NLP using the current weights *** 2. Update the weights based on the new prices, *** i.e. the marginals from the NLP solve *** 3. compute the error, i.e. | old weights - updated weights | *** until the error is small *** *** As the weights converge, the agents will move toward balanced budgets, where *** their incomes equal their expenditures. ******************************************************************************* model negishiA / defutility, balance /; set iters / iter1 * iter30 /; parameters err sum of changes from previous iterate / 1 / m damping factor / 0.9 / oldy(a) previous values of Y; y.fx(a) = 1; loop{iters\$[err > 1e-5], oldy(a) = y.l(a); solve negishiA using nlp maximizing utility; negishiA.solprint=2; y.fx(a) = (1-m)*y.l(a) + m*sum{g, endow(g,a)*balance.m(g)}; err = sum{a, abs(y.l(a) - oldy(a))} }; y.fx(a) = y.l(a)/y.l('a1'); rep(SJM) ******************************************************************************* *** 3. Via a CGE approach (using the implicit demand functions) ******************************************************************************* Equation negbalance(g) reorient balance equation to maintain convexity of MCP model, demand(g,a) implicit demand function ; negbalance(g).. sum{a, endow(g,a)} =G= sum{a, C(g,a)} ; demand(g,a).. p(g)*c(g,a) =E= alpha(g,a)*Y(a) ; model CGE / negbalance.p, demand.c, budget.Y / ; Y.lo(a) = -inf; Y.up(a) = inf; Y.fx("a1") = 1 ; cge.iterlim = 0; solve cge using mcp ; rep(CGE); ******************************************************************************* *** Now check for the same solutions ******************************************************************************* display RepY,RepP,RepC; Parameters DiffY(a) DiffP(g) DiffC(g,a); DiffY(a) = abs(RepY(a,'CGE')-RepY(a,'SJM')) + abs(RepY(a,'CGE')-RepY(a,'EMP')); abort\$[smax{a, DiffY(a)} > 1e-4] 'Incomes differ'; DiffP(g) = abs(RepP(g,'CGE')-RepP(g,'SJM')) + abs(RepP(g,'CGE')-RepP(g,'EMP')); abort\$[smax{g, DiffP(g)} > 1e-4] 'Prices differ'; DiffC(g,a) = abs(RepC(g,a,'CGE')-RepC(g,a,'SJM')) + abs(RepC(g,a,'CGE')-RepC(g,a,'EMP')); abort\$[smax{(g,a), DiffC(g,a)} > 1e-4] 'Consumptions differ';