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negishi.gms : Pure exchange model solved with EMP, SJM, and CGE

**Description**

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

**Small Model of Type : ** ECS

**Category :** GAMS EMP library

**Main file :** negishi.gms

```
$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';
```