An MPSGE Formulation
(1) Factor endowments over a 60-year lifecycle. These endowments are chose such income shares in any period amount to 3/16 for generations indexed 1-20, 12/16 for those in their the second 20 years, and 1/16 for those the final 20 years. (All calculations are conducted on an annual basis.)
(2) Intertemporal preferences characterized by a reference interest rate (5%) and consumption growth rate (1%). I have adopted this characterization of intertemporal preferences to facilitate comparison of alternative values for the intertemporal elasticity of substitution. In my analysis of the steady-state equilibrium, I consider intertemporal elasticities of 0.25, 0.5 and 0.75. In computations of the transition paths from perturbed initial conditions, I adopt an intertemporal elasticity equal to 0.5. The steady-state interest rate in this model is 4.6%.
(3) Agents born prior to the first year of the model have endowments during the periods of the model that they are alive. In addtion, they have initial wealth holdings corresponding to their net debt poistion relative to other "old" generations at the start of the model. In our computations, these Initial wealth endowments are perturbed from steady-state levels in order to evaluate computational efficiency and robustness. In the first scenario, intial wealth endowments for agents born prior to the first year of the model are reduced by 10%. This shock improves welfare for net creditors (younger generations) and decreases welfare for net debtors (older generations). In the second scenario, I perform a 5% random perturbation of initial wealth for all generations born prior to the first year of the model.
The second figure ("Income balance in steady-state") illustrates sensitivity of the steady-state growth rate to the intertemporal elasticity of substitution. When the intertemporal elasticity is small, there may be multiple equilibria.
Even though the model is ill-behaved for low intertemporal elasticities, it is perfectly robust for elasticities as low as 0.5. Here we present calculated transition paths for two different perturbations of the initial asset distributions. Both scenarios are computed in fewer than 5 Newton iterations, and a comparison of model results from alternative time horizons suggests that the transition values are virtually unaffected by the termination procedure.
The most significant advantage of this approach to overlapping generations models is compactness of representation. Here is the complete MPSGE code for this model:
$model:olg $commodities: p(g)$t(g) ! Output prices in period g $consumers: ra(g) ! Consumer income for generation g $demand:ra(g) s:sigma e:p(tfirst) q:e0(g) e:p(tlast) q:et(g) e:p(t) q:e(g,t) d:p(t) q:d(g,t) p:p0(t)Thomas F. Rutherford