private lemma rpow_succ_pred {x : ℝ} (hx : 0 < x) (ρ : ℝ) :
x * x ^ (ρ - 1) = x ^ ρ := by
have h := rpow_add hx 1 (ρ - 1)
rw [rpow_one, show 1 + (ρ - 1) = ρ from by ring] at h
exact h.symmDirected Technical Change Extension (Acemoglu 2002)