Approximating an Infinite Horizon Model in the presence of optimal experimentation

In an recent article Amman and Tucci (2020) make a comparison of the two dominant approaches for solving models with optimal experimentation in economics; the value function approach and an approximation approach. The approximation approach goes back to engineering literature in the 1970ties (cf. Tse & Bar-Shalom, 1973). Kendrick (1981) introduces this approach in economics. By using the same model and dataset as in Beck and Wieland (2002), Amman and Tucci conclude that differences may be small between the both approaches. In the previous paper we did not present the derivation of the approximation approach for this class of models. Hence, here we will present all derivations of the approximation approach for the case where there is an infinite horizon as is most common in economic models. By presenting the derivations, a better understanding and insight is obtained by the reader on how the value function is adequately approximated.