$title Ramsey Model of Optimal Economic Growth (RAMSEY,SEQ=63)
$onText
This formulation is described in 'GAMS/MINOS: Three examples'
by Alan S. Manne, Department of Operations Research, Stanford
University, May 1986.
Ramsey, F P, A Mathematical Theory of Saving. Economics Journal (1928).
Murtagh, B, and Saunders, M A, A Projected Lagrangian Algorithm and
its Implementation for Sparse Nonlinear Constraints. Mathematical
Programming Study 16 (1982), 84-117.
The optimal objective value is 2.4875
Keywords: nonlinear programming, economics
$offText
*---------------------------------------------------------------------
* The planning horizon covers the years from 1990 (TFIRST) to 2000
* (TLAST). The intervening asterisk indicates that this set includes
* all the integers between these two values. This first statement is
* the only one that needs to be changed if one wishes to examine a
* different planning horizon.
*---------------------------------------------------------------------
Set
t 'time periods' / 1990*2000 /
tfirst(t) 'first period'
tlast(t) 'last period';
*---------------------------------------------------------------------
* Data may also be entered in the form of SCALAR(S), as illustrated
* below.
*---------------------------------------------------------------------
Scalar
bet "discount factor" / .95 /
b "capital's value share" / .25 /
g "labor growth rate" / .03 /
ac "absorptive capacity rate" / .15 /
k0 "initial capital" / 3.00 /
i0 "initial investment" / .05 /
c0 "initial consumption" / .95 /
a "output scaling factor";
Parameter
beta(t) 'discount factor'
al(t) 'output-labor scaling vector';
*-----------------------------------------------------------------------
* The following statements show how we may avoid entering information
* about the planning horizon in more than one place. Here the symbol
* "$" means "such that"; "ORD" defines the ordinal position in a set;
* "CARD" defines the cardinality of the set. Thus, TFIRST is
* determined by the first member included in the set; and TLAST by the
* cardinality (the last member) of the set.
* This seems like a roundabout way to do things, but is useful if we
* want to be able to change the length of the planning horizon by
* altering a single entry in the input data. The same programming style
* is employed when we calculate the present-value factor BETA(T) and the
* output-labor vector AL(T).
*-----------------------------------------------------------------------
tfirst(t) = yes$(ord(t) = 1);
tlast(t) = yes$(ord(t) = card(t));
display tfirst, tlast;
beta(t) = bet**ord(t);
beta(tlast) = beta(tlast)/(1-bet);
*-----------------------------------------------------------------------
* BETA(TLAST), the last period's utility discount factor, is calculated
* by summing the infinite geometric series from the horizon date onward.
* Because of the logarithmic form of the utility function, the
* post-horizon consumption growth term may be dropped from the maximand.
*-----------------------------------------------------------------------
a = (c0+i0)/k0**b;
al(t) = a*(1+g)**((1-b)*(ord(t)-1));
display beta, al;
Variable
k(t) 'capital stock (trillion rupees)'
c(t) 'consumption (trillion rupees per year)'
i(t) 'investment (trillion rupees per year)'
utility;
*---------------------------------------------------------------------*
* Note that variables and equations cannot be identified by the same
* name. That is why the capital stock variables are called K(T), and
* the capital balance equations are KK(T).
*---------------------------------------------------------------------*
Equation
cc(t) 'capacity constraint (trillion rupees per year)'
kk(t) 'capital balance (trillion rupees)'
tc(t) 'terminal condition (provides for post-terminal growth)'
util 'discounted log of consumption: objective function';
*---------------------------------------------------------------------*
cc(t).. al(t)*k(t)**b =e= c(t) + i(t);
kk(t+1).. k(t+1) =e= k(t) + i(t);
tc(tlast).. g*k(tlast) =l= i(tlast);
util.. utility =e= sum(t, beta(t)*log(c(t)));
*-----------------------------------------------------------------------
* Instead of requiring that "ALL" of these constraints are to be
* included, we specify that the RAMSEY model consists of each of the
* four individual constraint types. If, for example, we omit TC, we can
* check the sensitivity of the solution to this terminal condition.
*-----------------------------------------------------------------------
Model ramsey / all /;
*-----------------------------------------------------------------------
* The following statements represent lower bounds on the individual
* variables K(T), C(T) and I(T); a fixed value for the initial period's
* capital stock, K(TFIRST); and upper bounds (absorptive capacity
* constraints) on I(T). Bounds are required for K and C because
* LOG(C(T)) and K(T)**B are defined only for positive values of C and K
*-----------------------------------------------------------------------
k.lo(t) = k0;
c.lo(t) = c0;
i.lo(t) = i0;
k.fx(tfirst) = k.lo(tfirst);
i.up(t) = i0*((1+ac)**(ord(t)-1));
*-----------------------------------------------------------------------
solve ramsey maximizing utility using nlp;