Description
This model is a variant of T. Rutherford's model 'State Variable Targetting in an NLP Framework'. http://www.mpsge.org/nlptarget/ This program illustrates how to use recursive NLP methods for solving an infinite-horizon optimization model with minimal terminal effects. Thomas F. Rutherford December 1, 2005 It makes use of EMP's embedded complementarity system framework. Contributors: Jan-H. Jagla, January 2009 Michael Ferris, April 2010
Small Model of Type : ECS
Category : GAMS EMP library
Main file : target.gms
$title State Variable Targetting in EMPs ECS Framework (TARGET,SEQ=13)
$ontext
This model is a variant of T. Rutherford's model 'State Variable Targetting in
an NLP Framework'.
http://www.mpsge.org/nlptarget/
* This program illustrates how to use recursive NLP methods
* for solving an infinite-horizon optimization model with minimal
* terminal effects.
* Thomas F. Rutherford
* December 1, 2005
It makes use of EMP's embedded complementarity system framework.
Contributors: Jan-H. Jagla, January 2009
Michael Ferris, April 2010
$offtext
$if not set horizon $set horizon 2020
set t Time periods /2005*%horizon%/,
tlast(t) Last time period /%horizon%/
tfirst(t) First time period /2005/;
parameter
kvs Capital value share /0.3/
delta Capital depreciation rate /0.07/
r Baseline interest rate / 0.05/
g Growth rate /0.02/,
phi Production scale faster
L(t) Labor supply
kinit Initial capital stock
kterm Terminal capital stock
dfactor Discount factor;
L(t) = power(1+g, ord(t)-1);
kinit = 0.5 * kvs / (r + delta);
dfactor(t) = power(1/(1+r), ord(t)-1);
phi = 1 / kinit**kvs;
VARIABLES
C(t) Consumption trillion US dollars,
K(t) Capital stock trillion US dollars,
I(t) Investment trillion US dollars,
Y(t) Output net abatement and damage costs,
UTILITY Maximand;
POSITIVE VARIABLES Y, C, K, I;
EQUATIONS
UTIL Objective function
CC(t) Consumption
YY(t) Output
KK(t) Capital balance
TERMCAP Terminal capital stock constraint with terminal capital stock parameter;
UTIL.. UTILITY =E= SUM(t, 10 * dfactor(t) * L(t) * LOG(C(t)/L(t)));
CC(t).. C(t) =E= Y(t) - I(t);
YY(t).. Y(t) =E= phi * L(t)**(1-kvs) * K(t)**kvs;
KK(t).. K(t) =L= (1-delta)**10 * K(t-1) + 10 * I(t-1) + kinit$tfirst(t);
TERMCAP.. kterm =E= sum(tlast, (1-delta)**10 * K(tlast) + 10 * I(tlast));
model ramsey NLP Model using parameter kterm /all/;
C.L(t) = 1; C.LO(t) = 0.01;
K.L(t) = 1; K.LO(t) = 0.01;
I.L(t) = 1;
Y.L(t) = 1;
*Run iterative loop of ramsey NLP model
set iter /iter1*iter20/;
kterm = kinit * power(1+g,card(t));
parameter invest(t,iter) Investment in successive iterations
kt(iter) Terminal capital stock in successive iterations;
option solprint=off, limrow=0, limcol=0;
loop(iter,
kt(iter) = kterm;
solve ramsey maximizing UTILITY using NLP;
invest(t,iter) = I.L(t);
kterm = sum(tlast(t), K.L(tlast) * Y.L(t)/Y.L(t-1));
);
option solprint=on;
Parameter rep(*,*);
rep('NLP','KTERM') = kterm;
rep('NLP','iter') = na;
*-------------------------------------------------------------------------------
*Now we write this as an MCP
Variables
KTERMV Terminal capital stock,
uCC(t) Shadow value of market supply,
uKK(t) Shadow value of capital,
uYY(t) Shadow value of output,
uTERMCAPV ;
Negative Variables
uKK(t) Shadow value of capital;
Equations
dLdC(t) First-order-conditions from the NLP for variable C
dLdK(t) First-order-conditions from the NLP for variable K
dLdI(t) First-order-conditions from the NLP for variable I
dLdY(t) First-order-conditions from the NLP for variable Y
TERMCAPV Terminal capital stock constraint with terminal capital stock variable
SSTERM First-order-condition for terminal capital stock variable;
* Dual constraints (first-order-conditions from the NLP):
dLdC(t).. - 10 * dfactor(t) * L(t) / C(t)
- uCC(t) =N= 0;
dLdK(t).. [phi * L(t)**(1-kvs) * kvs * K(t)**(kvs-1)] * uYY(t)
- uKK(t)
+ [(1-delta)**10]* uKK(t+1)
+ ([(1-delta)**10] * uTERMCAPV)$tlast(t) =N= 0;
dLdI(t).. - uCC(t)
+ 10 * uKK(t+1)
+ 10 * uTERMCAPV$tlast(t) =N= 0;
dLdY(t).. + uCC(t)
- uYY(t) =N= 0;
*Substitute TERMCAP of NLP by TERMCAPV (using variable KTERMV instead of parameter kterm)
TERMCAPV.. KTERMV =E= sum(tlast, (1-delta)**10 * K(tlast) + 10 * I(tlast));
*First-order-condition for terminal capital stock variable
SSTERM.. sum(tlast(t),I(t)/I(t-1) - Y(t)/Y(t-1)) =E= 0;
*Use duals of primal variables
uKK.L(t) = KK.m(t);
uYY.L(t) = - YY.m(t);
uCC.L(t) = - CC.m(t);
uTERMCAPV.l = - TERMCAP.m;
model ramseymcp / CC.uCC, YY.uYY, KK.uKK, TERMCAPV.uTERMCAPV, dLdY.Y, dLdC.C, dLdI.I, dLdK.K, SSTERM.KTERMV /;
*-------------------------------------------------------------------------------
*Solve the MCP and use NLP solution as starting point
solve ramseymcp using mcp;
rep('MCP','KTERM') = KTERMV.l;
rep('MCP','iter') = ramseymcp.iterusd;
*One may be able to decrease tolerance if running more iteration for the nlp
scalar tol /1e-3/;
*Compare terminal capital stock of iterative NLP Framework with the obtained
*solving the hand-written MCP
abort$(abs(kterm-KTERMV.l)>tol) 'MCP and iterative NLP solution differ';
*Solve the MCP again but now start from the same starting point as the
*NLP iterative process
C.L(t) = 1;
K.L(t) = 1;
I.L(t) = 1;
Y.L(t) = 1;
KK.m(t) = 0;
YY.m(t) = 0;
CC.m(t) = 0;
TERMCAPV.m = 0;
solve ramseymcp using mcp;
rep('MCP_2','KTERM') = KTERMV.l;
rep('MCP_2','iter') = ramseymcp.iterusd;
abort$(abs(kterm-KTERMV.l)>tol) '(Start Over) MCP and iterative NLP solution differ';
*Now we use EMP's Embedded Complementarity System (ECS)
model ramseyemp /UTIL,CC,YY,KK,TERMCAPV,SSTERM/;
$onecho > "%emp.info%"
dualequ SSTERM KTERMV
$offecho
solve ramseyemp maximizing UTILITY using emp;
rep('ECS','KTERM') = KTERMV.l;
rep('ECS','iter') = ramseyemp.iterusd;
abort$(abs(kterm-KTERMV.l)>1e-3) 'ECS and iterative NLP solution not the same';
*Solve the EMP ECS model again but now start from the same starting point
*as the NLP iterative process
C.L(t) = 1;
K.L(t) = 1;
I.L(t) = 1;
Y.L(t) = 1;
KK.m(t) = 0;
YY.m(t) = 0;
CC.m(t) = 0;
TERMCAPV.m = 0;
solve ramseyemp maximizing UTILITY using emp;
rep('ECS_2','KTERM') = KTERMV.l;
rep('ECS_2','iter') = ramseyemp.iterusd;
abort$(abs(kterm-KTERMV.l)>1e-3) '(Start over) ECS and iterative NLP solution not the same';
*Now we use an EMP Equilibrium
file e / '%emp.info%' /;
put e 'equilibrium' /;
put 'max utility C K I Y util termcapv CC KK YY' /;
putclose 'vi SSTERM KTERMV' /;
solve ramseyemp using emp;
rep('Equil','KTERM') = KTERMV.l;
rep('Equil','iter') = ramseyemp.iterusd;
abort$(abs(kterm-KTERMV.l)>1e-3) 'Equilibrium and iterative NLP solution not the same';
*Solve the EMP Equilibrium model again but now start from the same starting
*point as the NLP iterative process
C.L(t) = 1;
K.L(t) = 1;
I.L(t) = 1;
Y.L(t) = 1;
KK.m(t) = 0;
YY.m(t) = 0;
CC.m(t) = 0;
TERMCAPV.m = 0;
solve ramseyemp using emp;
rep('Equil_2','KTERM') = KTERMV.l;
rep('Equil_2','iter') = ramseyemp.iterusd;
abort$(abs(kterm-KTERMV.l)>1e-3) '(Start over) Equilibrium and iterative NLP solution not the same';
display rep;