$title	Liquidity-Constrained Lifecycle Expenditure Model (NLP and MCP formulations)



I am again. I guess that I already found the problem
in the formulation of the model (I mathematical error
with some sign in the previous calculus), and now the
program runs good.
The model is basisclly a Ramsey model. For speaking
more about it, it tries to represent the behavior of a
household with exogenous income and intertemporal
desicions in consumption and savings. 

But now I have another little problem. Why the final
level values of the desicions variables change
(sometimes little, others a lot) when you modify the
initial values for them and solve the model (of
course, after the calibration of the paramters)?.

Would you please help me with this problem?

I send you the program that I did in GAMS, with
another document in WORD that have the FOC equations
obtained by solving the model using KKT. 


The Original Model
set t /t1*t21/; alias(t,s); Variables c(t) d(t) p(t) r(t) y(t); equations euler(t) budget(t); parameter beta(T); parameter d0(t); euler(t) .. beta(t)/(p(t)*c(t))-(beta(t+1)*(1+r(t))/(p(t+1)*c(t+1)))$(ord(t)<>card(t))=g=0; budget(t).. p(t)*c(t)+d(t)=e=(1+r(t-1))*d(t-1)+y(t); y.fx(t)=200*(1.01)**(ord(t)-1); p.fx(t)=(0.9)**(ord(t)-1); r.fx(t)=0.03; c.lo(t)=1.e-5; d.lo(t)=0; d.fx('t21')=0; c.l(t)=150*(1.01)**(ord(t)-1); loop(t, d.l(t)=((1+r.l(t-1))*d.l(t-1))+y.l(t)-p.l(t)*c.l(t);); d0(t)=d.l(t); c.l('t21')=(p.l('t21')*c.l('t21')+d.l('t21'))/p.l('t21'); beta(t)=(prod(s$(ord(s)1)+1$(ord(t)=1); *This is something that I cant understand. Why the final variables change at differentes values of this initial value. c.l(t)=5000; model cos / euler.d budget.c /; option mcp = miles; solve cos using mcp; execute_unload 'full.gdx'; execute '=gdxviewer full.gdx' $offtext
An Alternative Approach
set t Time periods for a typical life-cycle /20*75/, tr(t) Retirement period /65*75/, tl(t) Labels for plots /20, 40, 60/, tlast(t) Last period; alias(t,tt); * Define which period is the last one: tlast(t) = yes$(ord(t)=card(t)); parameter r Interest rate on savings /0.03/, delta Intertemporal discount rate /0.05/, y(t) Income level in time period t pv(t) Present-value price index beta(t) Utility discount factor; * Assume a 10% market interest, a 3% rate of return on savings and future * income growth of 1% per annum: pv(t) = 1/(1+r)**(ord(t)-1); beta(t) = 1/(1+delta)**(ord(t)-1); * Assume a lifecycle income profile as in Auerbach and Kotlikoff (1987): y(t) = EXP(4.47 + 0.033*(ord(t)-1) - 0.00067 * (ord(t)-1)**2); y(tr) = 0; variables c(t) Consumption, b(t) Savings balance at the end of time period t, u Intertemporal utility; equations budget(t) Budget constraint in time period t utility Defines intertemporal utility; utility.. u =e= sum(t, beta(t) * log(c(t))); budget(t).. (1+r) * b(t-1) + y(t) =e= c(t) + b(t); model ramsey /all/; c.l(t) = beta(t) * sum(tt, pv(tt) * y(tt)) / pv(t); * Place a lower bound on C to avoid bad function calls: c.lo(t) = 0.1; * First solve without a liquidity constraint: b.lo(t)=-inf; b.fx(tlast) = 0; solve ramsey using nlp maximizing u; parameter results Consumption and Savings over the Lifecycle; results(t,"c*") = c.l(t); results(t,"b*") = b.l(t); * Then solve the liquidity-constrained problem: b.lo(t)=0; solve ramsey using nlp maximizing u; results(t,"c0") = c.l(t); results(t,"b0") = b.l(t); $setglobal domain t $setglobal labels tl $libinclude plot results variable mu(t) Shadow value on income balance in period t; equation foc_b(t) First-order condition on savings balance foc_c(t) First-order condition on consumption; foc_b(t).. mu(t) =g= (1+r) * mu(t+1); foc_c(t).. mu(t) * c(t) =e= beta(t); model mcp /budget.mu, foc_b.b, foc_c.c/; mu.l(t) = - budget.m(t); mcp.iterlim = 0; solve mcp using mcp; b.lo(t)=-inf; b.fx(tlast) = 0; mcp.iterlim = 10000; solve mcp using mcp;