Mathematically, a three-period life-cycle model gives rise to a third- or higher order non linear difference for which there are, in general, no known analytical solutions. The problem becomes even more complicated in dealing with 55 generations. In the A-К Model agents consider factor prices and, thus, capital-labor ratios, over their entire lifetimes. Hence, the youngest agent at any point in time is considering, among other things, the factor prices and capital-labor ratio that will prevail 55 years in the future.
But this agent knows that the capital-labor ratio 55 years in the future will be determined, in part, by the labor supply of the youngest agent in the economy in that year. This youngest agent 55 years hence will, in turn, be considering factor prices and, thus, capital-labor ratios, over the following 55 years. In sum, the economic decisions of the youngest agent alive this year depend on 110 capital-labor ratio; i.e., the A-К Model entails, roughly speaking, a 110th order non linear difference equation for the economy’s key state variable — it’s ratio of capital to labor.
Now one could, as Laitner (1984) ultimately did, linearize such a high order non linear difference equation and apply standard solution techniques from the theory of linear difference equations. But in so doing, one would be studying an approximation to the actual transition path of the economy under investigation. Without solving for the exact transition path, one wouldn’t be able to tell the precise size of the approximation error.
Absent a method of solving for the economy’s exact transition path, public finance economists in the late 1970s turned to solving for economies’ long-run steady-states. The steady states of dynamic economies also entail an infinite set of agents and markets — but the same set over time. This feature provides the mathematical closure for computing where certain large-scale economies would end up even if one didn’t know precisely how they would get there. Kotlikoff (1979) (which appeared as a mimeo in 1977), which considered the long-run effects of pay-as-you-go social security, and Summers (1981) (which appeared as a mimeo in 1980), which examined the long-run effects of tax reform, are two examples of this line of research. Both of these studies considered 55-period life cycle models in which adult agents live from age 20 through age 75. Kotlikoff s study confirmed Feldstein’s contention that unfunded social security, of the scale established in the U.S., would have a significant deleterious impact on an economy’s ultimate capital stock and living standard. Summers’ study showed that the choice of the tax base used to finance government spending could have equally large long-run impacts on the economy.