\$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, economic development, economic growth, theory of saving, investment planning \$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;