On the uniform limit condition for discrete-time infinite horizon problems

In this note, a simplified version of the four main results for discrete-time infinite horizon problems, theorems 4.2-4.5 from Stokey, Lucas and Prescott (1989) [SLP], is presented. A novel assumption on these problems is proposed—the uniform limit condition, which is formulated in terms of the data of the problem. It can be used for example before one has started to look for the optimal value function and for an optimal plan or if one cannot find them analytically: one verifies the uniform limit condition and then one disposes of criteria for optimality of the value function and a plan in terms of the functional equation and the boundedness condition. A comparison to [SLP] is made. The version in [SLP] requires one to verify whether a candidate optimal value function satisfies the boundedness condition; it is easier to check the uniform limit condition instead, as is demonstrated by examples. There is essentially no loss of strength or generality compared to [SLP]. The necessary and sufficient conditions for optimality coincide in the present paper but not in [SLP]. The proofs in the present paper are shorter than in [SLP]. An earlier attempt to simplify, in Acemoglu (2009) --here the limit condition is used rather than the uniform limit condition-- is not correct.