Retaining Productive Units: A Dynamic Programming Model with a Steady State Solution

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Abstract
We consider a system consisting of productive units which age (the units could be men). As these units age, they tend to leave the system and must be replaced. We can retard this by spending more money on them. The system is required to maintain a constant production rate. The latter is a linear function of the number of units in the different age groups. The desideratum is an optimal retention policy. We formulate an infinite-horizon deterministic dynamic programming model and find a steady state policy (a policy under which the system moves directly to a target state) whose cost differs from the optimal cost by an amount whose bound is independent of the interest rate. We also find optimal policies for some special cases.

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