Acemoglu ch10 — RICH 프리뷰: 인적자본과 경제성장

$$\dot{h}(t) = G(t,\, h(t),\, s(t))$$

$$[\hat c, \hat s, \hat h]^T \text{ jointly optimal} \Leftrightarrow [\hat s, \hat h] \text{ maximize } W \text{ alone}$$

$$\max_S \eta(S)\, w(0)\, e^{-(r+\nu-g_w)S} \cdot \frac{1}{r+\nu - g_h - g_w}$$

$$\frac{\eta'(S^*)}{\eta(S^*)} = r + \nu - g_w$$

$$\log \eta(S^*) = \text{const} + (r+\nu - g_w)\, S^*$$

$$\log W(S^*,t) = \text{const} + (r+\nu-g_w)\, S^* + g_w\, t + g_h(t-S^*)$$

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

$$H(h,s,\mu) = (1-s(t))\, h(t) + \mu(t)[\varphi(s\, h) - \delta_h\, h]$$

$$1 = \mu(t)\, \varphi'(x(t))$$

$$x^* = \varphi'^{-1}(r + \nu + \delta_h)$$

$$\frac{\dot{x}(t)}{x(t)} = \frac{1}{\varepsilon\, \varphi'(x(t))}\, (r+\nu+\delta_h - \varphi'(x(t)))$$

$$c(t) + i_k(t) + i_h(t) \leq f(k(t),\, h(t))$$

$$f_k(k,h) - f_h(k,h) = \delta_k - \delta_h$$

$$\eta^{-\eta}(1-\eta)^{-(1-\eta)} c_i^{\eta}\, b_i^{1-\eta} - \gamma(e_i)$$

$$c_i(t) + b_i(t) \leq m_i(t) = w(t)\, h_i(t) + R(t)\, b_i(t-1)$$

$$Y(t) = F(K,H);\ H = \int_0^1 h_i\, di,\ K = \int_0^1 b_{i,t-1}\, di$$

$$c_i = \eta\, m_i,\quad b_i = (1-\eta)\, m_i$$

$$a\, w(t) = \gamma'(h_i/a)$$

$$\kappa(t+1)\, \gamma'^{-1}[a\, w(\kappa(t+1))] = (1-\eta)\, f(\kappa(t))\, \kappa(t)\, \gamma'^{-1}[a\, w(\kappa(t))]$$

$$\kappa^* = (1-\eta)\, f(\kappa^*)$$

$$\lambda\, a_i\, \frac{\partial F(\hat{k},\, \hat{h}_i(\hat{k}))}{\partial h} = \gamma'(\hat{h}_i/a_i)$$

$$(1-\lambda)[\chi\, F_k(\hat{k}, \hat{h}_1) + (1-\chi) F_k(\hat{k}, \hat{h}_2)] = R^*$$