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A small linear dynamic macroeconomic model of U.S. economy in which
both monetary and fiscal policy variables are used.

Linear Quadratic Riccati Equations are solved as a General Nonlinear
Programming Problem instead of the usual Matrix Recursion.

Please see:
Kendrick, D, Caution and Probing in a Macroeconomic Model.
Journal of Economic Dynamics and Control 4, 2 (1982) pp.149-170.
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Sets n states / consumpt, invest /
     m controls / gov-expend, money /
     k horizon / 1964-i, 1964-ii, 1964-iii, 1964-iv
                 1965-i, 1965-ii, 1965-iii, 1965-iv /
     ku(k) control horizon
     ki(k) initial period
     kt(k) terminal period ;

Alias (n,np), (m,mp) ;

ku(k) = yes$(ord(k) lt card(k));
ki(k) = yes$(ord(k) eq 1);
kt(k) = not ku(k);

Display k, ki, kt, ku;

Table a(n,np) state vector matrix

           consumpt     invest
consumpt    .914         -.016
invest      .097          .424

Table b(n,m) control vector matrix

          gov-expend    money
consumpt      .305       .424
invest       -.101      1.459

Table wk(n,np) penalty matrix for states - input

          consumpt      invest
consumpt   .0625
invest                    1

Table rk(m,mp) penalty matrix for controls

           gov-expend    money
gov-expend      1
money                    .444

Parameter c(n) constant term / consumpt -59.4, invest -184.7 /
          xinit(n) initial value / consumpt 387.9, invest 85.3 /
          uinit(m) initial controls / gov-expend 110.5, money 147.1 /
          xtilde(n,k) desired path for x
          utilde(m,k) desired path for u
          w(n,np,k) penalty matrix on states ;

w(n,np,ku) = wk(n,np);
w(n,np,kt) = 10000*wk(n,np);
xtilde(n,k) = xinit(n)*1.0075**(ord(k)-1);
utilde(m,k) = uinit(m)*1.0075**(ord(k)-1);

Display w, xtilde, utilde;

Variables x(n,k) state variable
          u(m,k) control variable
          j criterion

Equations criterion criterion definition
          stateq(n,k) state equation ;

criterion..
j =e= .5*sum((k,n,np),
         (x(n,k)-xtilde(n,k))*w(n,np,k)*(x(np,k)-xtilde(np,k)))
    + .5*sum((ku,m,mp),
         (u(m,ku)-utilde(m,ku))*rk(m,mp)*(u(mp,ku)-utilde(mp,ku)));

stateq(n,k+1)..

x(n,k+1) =e= sum(np, a(n,np)*x(np,k)) + sum(m, b(n,m)*u(m,k)) + c(n);

Model macro /all/;

x.l(n,k) = xinit(n);
u.l(m,k) = uinit(m);
x.fx(n,ki) = xinit(n);

Solve macro minimizing j using nlp;

Display x.l, u.l;

$ifThenI x%mode%==xbook
file res1 /macro.dat/;
put res1
loop(k, put xtilde('consumpt',k):10:5, put/)
loop(k, put xtilde('invest',k):10:5, put/)
loop(k, put x.l('consumpt',k):10:5, put/)
loop(k, put x.l('invest',k):10:5, put/)

loop(k, put utilde('gov-expend',k):10:5, put/)
loop(k, put utilde('money',k):10:5, put/)
loop(k, put u.l('gov-expend',k):10:5, put/)
loop(k, put u.l('money',k):10:5, put/)
$endIf

* End Macro