\$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) 'Equlibrium and iterative NLP solution not the same'; *Solve the EMP Equlibrium 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) Equlibrium and iterative NLP solution not the same'; display rep;