Stability Properties of the Rate-of-Return Regulation Process

This note provides provides proofs for several of the claims made in our paper “Dynamics of Rate-of-Return Regulation,” referenced from heron as Nezlobin, Rajan and Reichelstein (2011). The following two proofs provide details for the claims in Substeps B1 and C1 of Proposition 2 in Nezlobin, Rajan and Reichelstein (2011). Sub-Step B1. If state θt is feasible for some t, Pt (K (θ ′ t)) = p∞, and there exists a subsequence of s (t), s (ti), such that θt = (1 + μ) t lim i→∞ s (ti) . (1) then there exists an investment I ′ t such that state θ ′ t+1 is feasible, Pt+1 ( K ( θt+1 )) = p∞, and there exists a subsequence of s (t), s (ti), such that θt+1 = (1 + μ) t+1 lim i→∞ s (ti) . (2) Proof of Sub-Step B1 Consider a sequence {s (ti + 1)}i=1 which is comprised of elements of {s (t)} immediately following the elements of {s (ti)} from (1). The sequence {s (ti + 1)}i=1 must have a converging subsequence (since all vectors s (·) are drawn from a compact set). Let {s (ti)} ∞ i=1 denote this converging subsequence and let φ = (1 + μ) lim i→∞ s (ti) . To show that φ is a legitimate candidate for θt+1, we need to check that the first T − 1 elements of φ correspond to the last T − 1 elements of θt. Letting φτ denote the τ -th coordinate of φ, we have: φτ = (1 + μ) t+1 lim i→∞ sτ (t ′ i) . By construction, for τ < T sτ (t ′ i) = (1 + μ) −1 sτ+1 (t ′ i − 1) . Therefore, for τ < T , φτ = (1 + μ) t+1 lim i→∞ sτ (t ′ i) = (1 + μ) t lim i→∞ sτ+1 (t ′ i − 1) = (1 + μ) lim i→∞ sτ+1 (ti) = I ′ t−T+τ . Setting I ′ t = φT , we will have θt+1 = φ = (1 + μ) t+1 lim i→∞ s (ti) .