Over- and under-investment according to different benchmarks

In a two-stage oligopoly, with investment in the first stage and quantity or price competition in the second stage, there is a kind of Folk Theorem: We find (i) over-investment if the goods are substitutes and competition is in strategic substitutes, (ii) under-investment if we have either complements instead of substitutes or strategic complements instead of strategic substitutes, and (iii) again over-investment if both attributes change. The existing literature, however, lacks a proof of this theorem and, in particular, it lacks a systematic comparison of the different benchmarks for over-and under-investment. A "naive" benchmark is the efficient investment with respect to the subgame perfect (closed loop) equilibrium quantities. Alternative benchmarks (which are more often proposed) are the open loop equilibrium investment or the welfare maximizing investment. The chosen benchmark is critical because the Folk Theorem applies (under certain conventional conditions) only for the naïve benchmark. The other two benchmarks require additional assumptions or the distinction of subcases.