Exercise 20.16. * In this exercise, you are first asked to provide an alternative proof of Proposition 20.11 and then characterize the local transitional dynamics in the neighborhood of the constant growth path. (1) Reexpress the equilibrium equations in terms of the following three variables ϕ(t) ≡ c (t) /A1 (t) 1/(1−α1) , χ (t) ≡ K (t) / ³ L(t) A1 (t) 1/(1−α1) ´ and κ (t). In particular, show that the following three differential equations, together with the appropriate transversality condition and initial values χ (0) and κ (0), characterize the dynamic equilibrium ϕ˙ (t) ϕ(t) = 1 θ h α1γη (t) 1/ε λ (t) 1−α1 κ (t) −(1−α1) χ (t) −(1−α1) − ρ i − a1 1 − α1 (20.80) , χ˙ (t) χ (t) = λ (t) 1−α1 κ (t) α1 χ (t) −(1−α1) η (t) − χ (t) −1 ϕ(t) − n − a1 1 − α1 , κ˙ (t) κ (t) = (1 − κ (t)) h (α2 − α1) χ˙ (t) χ(t) + a2 − 1−α2 1−α1 a1 i (1 − ε) −1 + (α2 − α1) (κ (t) − λ (t)) , where κ (t) and λ (t) are given by (20.46) and (20.47), and (20.81) η (t) ≡ γ ε ε−1 • 1 + α1 α2 µ1 − κ (t) κ (t) ¶¸ ε ε−1 . [Hint: use the Euler equation of the representative consumer and the resource constraint of the economy, rearrange these to express the laws of motion of ϕ(t) and χ (t) in terms of κ (t), λ (t) and η (t) as defined in (20.81), and then differentiate (20.46).] (2) State the appropriate transversality condition. (3) Show that if an allocation satisfies the three differential equations in (20.80) and the appropriate transversality condition, then it corresponds to an equilibrium path. (4) Show that in a CGP equilibrium ϕ(t) must be constant. Using this, show that the CGP requires that κ (t) → 1 and that χ (t) be constant. From these observations, derive an alternative proof of Proposition 20.11. (5) Now linearize these three equations around the CGP of Proposition 20.11 and show that the linearized system has two negative and one positive eigenvalues and using this fact conclude that the constant growth path is locally stable. [Hint: as part of this argument, explain why κ (t) should be considered a state variable with κ (0) taken as an initial value].