7 weeks into a quarter, I am now allergic to the word “interest rate” “saddle path” and “EIGENVALUES”.
This note presents a continuous-time neoclassical growth model, deriving household and firm optimality conditions, characterizing the steady state, and analyzing the resulting saddle-path dynamics. On top of which, per-household productivity $A$ is left out, which makes it extendable.
Consider continuous time model $t\in[0, \infty)$.
(Representative) Household’s Problem $$ \begin{align} \max_{\lbrace c_t, x_t, k_t\rbrace_{t\ge 0}}& \int_0^\infty e^{-\rho t}U(c_t)\thinspace\text{d}t\cr \text{s.t.}& \ \dot k_t = x_t -\delta k_t, \forall t\cr & \int_0^\infty p_t(c_t + x_t)\thinspace\text{d}t = \int_0^\infty p_t(w_t + v_tk_t)\thinspace\text{d}t \cr&\text{where, } p_t :=\exp{(-\int_0^\infty r_{\tau}\thinspace\text{d}\tau)}. \end{align} $$
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