Acemoglu ch11 — RICH 프리뷰: 1세대 내생성장 (AK + Romer 86)

$$\int_0^{\infty} e^{-(\rho-n)t}\, \frac{c(t)^{1-\theta} - 1}{1-\theta}\, dt$$

$$\dot{a}(t) = (r(t) - n)\, a(t) + w(t) - c(t)$$

$$\lim_{t\to\infty} a(t)\, \exp\!\left(-\int_0^t (r(s) - n)\, ds\right) \geq 0$$

$$\lim_{t\to\infty} a(t)\, \exp\!\left(-\int_0^t (r(s) - n)\, ds\right) = 0$$

$$\dot{k}(t) = (A - \delta - n)\, k(t) - c(t)$$

$$\lim_{t\to\infty} k(t)\, e^{-(A-\delta-n)\, t} = 0$$

$$c(t) = c(0)\, \exp\!\left(\frac{A - \delta - \rho}{\theta}\, t\right)$$

$$A > \rho + \delta > (1-\theta)(A-\delta) + \theta\, n + \delta$$

$$\dot{k}(t) = (A - \delta - n)\, k(t) - c(0)\, \exp(g_c\, t)$$

$$k(t) = \kappa\, e^{(A-\delta-n) t} + \frac{c(0)/\theta}{(A-\delta)(\theta-1)/\theta + \rho/\theta - n}\, e^{g_c t}$$

$$c(0) = [(A-\delta)(\theta-1)/\theta + \rho/\theta - n]\, k(0)$$

$$\dot{a}(t) = (1-\tau)\, r(t)\, a(t) + w(t) - c(t) - \tau_w$$

$$s = \frac{(1-\tau)A - \rho + \theta n - (1-\tau-\theta)\delta}{\theta A}$$

$$\dot{a}(t) = r(t)\, a(t) + w(t)\, h(t) - c(t) - i_h(t)$$

$$\dot{h}(t) = i_h(t) - \delta_h\, h(t)$$

$$R(t) = f'(k(t)),\quad w(t) = f(k(t)) - k(t)\, f'(k(t))$$

$$f'(k) - \delta_k = f(k) - k\, f'(k) - \delta_h$$

$$C(t) = B\, K_C(t)^{\alpha}\, L_C(t)^{1-\alpha}$$

$$K_C(t) = (1-\kappa(t))\, K(t),\ K_I(t) = \kappa(t)\, K(t)$$

$$r_C(t) = \frac{r_I(t)}{p_I(t)} + \frac{\dot{p}_I(t)}{p_I(t)} - \frac{\dot{p}_C(t)}{p_C(t)}$$

$$Y(t) = \tilde{f}(L)\, K(t)$$

$$\tilde{f}(L) - L\, \tilde{f}'(L) - \delta - \rho > 0$$

$$(1-\theta)(\tilde{f}(L) - L\, \tilde{f}'(L) - \delta) < \rho$$