$title Optimal Patterns of Growth and Aid (PAK,SEQ=34)

$onText
This model explores the use of external resources to accelerate
development by supplying additional resources to increase imports
and investment. Terminal conditions cannot reproduced from the text.
All units in billions of 1965 rupees.

Chenery, H B, and Macewan, A, Chapter 9: Optimal Patterns of Growth
and Aid. In Chenery, H B, Ed, Structural Change and Development
Policy. Oxford University Press, New York and Oxford, 1979.

Keywords: linear programming, patterns of growth, economic development, capital
          investments, economic growth model, capital inflows, financial investments,
          economic resources
$offText

$sTitle Basic Data
Set
   te    'extended planning period' / 1962*1985 /
   t(te) 'planning period'          / 1963*1985 /
   j     'sectors'                  / non-traded, traded /;

Scalar
   fbb   'foreign aid 1962'                   /  1.183 /
   sb    'saving 1962'                        /  3.381 /
   tib   'total investment 1962'              /  4.564 /
   mb    'imports 1962'                       /  3.743 /
   eb    'exports 1962'                       /  2.559 /
   gnpb  'gnp 1962'                           / 37.380 /
   cb    'consumption 1962'                   / 33.999 /
   rho   'discount rate'                      /   .08  /
   r     'post plan discount'                 /   .10  /
   g     'post plan growth'                   /   .073 /
   gama  'cost of foreign capital'            /  2.0   /
   d     'post plan weight'                   /  1.0   /
   alpha 'marginal savings rate'              /   .24  /
   mgnp  'marginal import rate on gnp'        /   .10  /
   mi    'marginal import rate on investment' /   .35  /
   p     'population growth'                  /   .025 /
   beta  'maximum growth of investment'       /   .13  /
   ee    'export growth'                      /   .049 /
   q     'aid ration'                         /   .5   /
   num   'years without aid'                  /  4     /;

Parameter
   k(j)    'capital output ratio' / non-traded 3.0, traded 4.5 /
   delt(t) 'discount factor'
   dis     'discounting for post horizon cons'
   vb(j)   'base year outputs'
   e(t)    'exports';

e(t)    = eb*(1 + ee)**ord(t);
display e;

delt(t) = (1 + rho)**(-ord(t));
display delt;

dis     = (1 + r)**(-card(t))*(1 - alpha)*(1 + g)/(r - g);
display dis;

vb("non-traded") = gnpb;

$sTitle Model Definition
Variable
   gnp(t)     'gross national product'
   v(t,j)     'net output'
   ti(te)     'total investment'
   i(te,j)    'investment'
   ks(te,j)   'capital stock'
   s(t)       'gross savings'
   f(t)       'net capital inflow'
   fb         'total discounted aid'
   m(t)       'traditional imports'
   c(te)      'consumption'
   w          'welfare';

Positive Variable v, i, s;

Equation
   gnpd(t)    'gnp definition'
   invd(t)    'investment definition'
   invt(te)   'investment totals'
   tgap(t)    'trade gap'
   incd(t)    'national income definition'
   capb(t,j)  'capacity balance'
   kbal(te,j) 'capital stock balance'
   savl(t)    'maximum savings'
   impl(t)    'minimum imports'
   invu(te)   'upper bound on investment'
   invl(te)   'lower bound on investment'
   conl(te)   'lower bound on consumption'
   fup (t)    'upper bound on f(t)'
   taid       'total aid definition'
   wdef       'welfare definition';

gnpd(t)..      gnp(t) =e= sum(j, v(t,j));

invd(t)..      ti(t)  =e= s(t) + f(t);

invt(te)..     ti(te) =e= sum(j, i(te,j));

tgap(t)..      f(t)   =e= m(t) - e(t) - v(t,"traded");

incd(t)..      gnp(t) =e= c(t) + ti(t) - f(t);

capb(t,j)..    v(t,j) =l= vb(j) + 1/k(j)*ks(t,j);

kbal(te+1,j).. ks(te+1,j) =e= ks(te,j) + i(te,j);

savl(t)..      s(t) =l= sb + alpha*(gnp(t) - gnpb);

impl(t)..      m(t) =g= mb + mgnp*(gnp(t) - gnpb) + mi*(ti(t) - tib);

invu(te+1)..   ti(te+1) =l= (1 + beta)*ti(te);

invl(te+1)..   ti(te+1) =g= ti(te);

conl(te+1)..   c(te+1)  =g= (1 + p)*c(te);

fup(t)..       f(t) =l= q*gnp(t);

taid..         fb   =e= sum(t, delt(t)*f(t));

wdef..         w    =e= sum(t, delt(t)*c(t)) - gama*fb + d*dis*gnp("1985");

ks.fx("1962",j) = 0;
i.fx("1962","non-traded") = tib;
i.fx("1962","traded")     = 0;
c.fx("1962") = cb;
f.up(t) = inf$(card(t) - ord(t) >= num);

Model pak1 / all /;

solve pak1 maximizing w using lp;

Parameter rep 'summary report (billions of rupees)';
rep(t,"f")   = f.l(t);
rep(t,"gnp") = gnp.l(t);
rep(t,"ti")  = ti.l(t);
rep(t,"s")   = s.l(t);
rep(t,"c")   = c.l(t);

display rep;