Acemoglu ch20 — RICH 프리뷰: 구조변화와 인구

$$Y(t) = \bigl[\eta_A Y_A(t)^{\sigma} + \eta_M Y_M(t)^{\sigma}\bigr]^{1/\sigma}$$

$$Y_A = B_A\, F(K_A, X L_A),\ Y_M = B_M\, F(K_M, X L_M),\ Y_S = B_S\, F(K_S, X L_S)$$

$$\frac{\dot X(t)}{X(t)} = g$$

$$\frac{\partial \log c_A}{\partial \log Y} < 1$$

$$L_A(t) + L_M(t) + L_S(t) = L(t)$$

$$\dot K(t) + c_M(t)\, L(t) = Y_M(t)$$

$$c_A(t)\, L(t) = Y_A(t),\ c_S(t)\, L(t) = Y_S(t)$$

$$p_A(t)\, \frac{c_A(t) - \gamma_A}{\eta_A} = \frac{c_M(t)}{\eta_M}$$

$$p_S(t)\, \frac{c_S(t) + \gamma_S}{\eta_S} = \frac{c_M(t)}{\eta_M}$$

$$w(t) = \frac{\partial B_M F(K_M, X L_M)}{\partial L_M}$$

$$r(t) = \frac{\partial B_M F(K_M, X L_M)}{\partial K_M}$$

$$\{[K_A(t), K_M(t), K_S(t), L_A(t), L_M(t), L_S(t)]\}, \{p_A(t), p_M(t), p_S(t)\}$$

$$\frac{\dot c_M(t)}{c_M(t)} = \frac{1}{\theta}\, (r(t) - \rho)\quad \text{(Prop 20.2)}$$

$$\frac{B_M(c_A - \gamma_A)}{B_A\, \eta_A} = \frac{c_M}{\eta_M},\ \frac{B_M(c_S + \gamma_S)}{B_S\, \eta_S} = \frac{c_M}{\eta_M}$$

$$L_A(t) + L_M(t) = L(t)$$

$$Y(t) = F(Y_1(t), Y_2(t))$$

$$Y_1(t) = A_1(t)\, G_1(K_1(t), L_1(t))$$

$$Y_2(t) = A_2(t)\, G_2(K_2(t), L_2(t)),\ K_1 + K_2 = K,\ L_1 + L_2 = L$$

$$\frac{p_1(t)}{p_2(t)} = \frac{\partial F/\partial Y_1}{\partial F/\partial Y_2}$$

$$\{[p_1(t), p_2(t), w(t), r(t)]\}_{t=0}^{\infty}$$

$$g_1 k_1 \equiv \frac{G_1(K_1, L_1)}{L_1},\ g_2 k_2 \equiv \frac{G_2(K_2, L_2)}{L_2}$$

$$\frac{\dot p_1}{p_1} = \frac{\dot p_2}{p_2} = 0$$

$$r = p_1\, A_1\, g'_1(k_1) = p_2\, A_2\, g'_2(k_2)$$

$$w = p_1 A_1\, [g_1(k_1) - g'_1(k_1) k_1] = p_2 A_2\, [g_2(k_2) - g'_2(k_2) k_2]$$

$$\frac{\varepsilon\, g'_1\, \dot k_1}{k_1} = \frac{\varepsilon\, g'_2\, \dot k_2}{k_2}$$

$$\dot K(t) + L(t)\, c(t) \leq Y(t)$$

$$\frac{\dot A_1}{A_1} = a_1 > 0,\ \frac{\dot A_2}{A_2} = a_2 > 0$$

$$L_1(t) + L_2(t) = L(t)$$

$$K_1(t) + K_2(t) = K(t)$$

$$1 = \gamma^{\varepsilon}\, p_1(t)^{1-\varepsilon} + (1-\gamma)^{\varepsilon}\, p_2(t)^{1-\varepsilon}$$

$$\gamma\, (1-\alpha_1)\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{L_1(t)} = (1-\gamma)\, (1-\alpha_2)\, \left(\frac{Y(t)}{Y_2(t)}\right)^{1/\varepsilon}\, \frac{Y_2(t)}{L_2(t)}$$

$$\gamma\, \alpha_1\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{K_1(t)} = (1-\gamma)\, \alpha_2\, \left(\frac{Y(t)}{Y_2(t)}\right)^{1/\varepsilon}\, \frac{Y_2(t)}{K_2(t)}$$

$$w(t) = \gamma\, (1-\alpha_1)\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{L_1(t)}$$

$$r(t) = \gamma\, \alpha_1\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{K_1(t)}$$

$$\kappa(t) = 1 + \frac{\alpha_1}{\alpha_2}\, \frac{1-\alpha_2}{1-\alpha_1} \cdots$$

$$\lambda(t) = 1 + \frac{\alpha_1}{\alpha_2}\, \frac{1-\alpha_2}{1-\alpha_1}\, \frac{1-\kappa}{\kappa} \cdots$$

$$\frac{d \log \kappa}{d \log K} = -\frac{d \log \kappa}{d \log L} = \cdots$$

$$\frac{d \log \kappa}{d \log A_2} = -\frac{d \log \kappa}{d \log A_1} = (1-\varepsilon)\, \cdots$$

$$\sigma_K(t) \equiv \frac{r(t)\, K(t)}{Y(t)} = \gamma\, \alpha_1\, \left(\frac{Y_1(t)}{Y(t)}\right)^{(\varepsilon-1)/\varepsilon}$$

$$\frac{d \log(w/r)}{d \log L} = -\frac{d \log(w/r)}{d \log L} = \frac{1}{1 + (\alpha_2 - \alpha_1)/\cdots}$$

$$\frac{d \log(w/r)}{d \log A_2} = -\frac{d \log(w/r)}{d \log A_1} = -(1-\varepsilon)\, \cdots$$

$$\frac{d \log \sigma_K}{d \log K}\ \text{positive iff } (\alpha_2 - \alpha_1)(1-\varepsilon) > 0$$

$$\frac{d \log \sigma_K}{d \log A_2} = -\frac{d \log \sigma_K}{d \log A_1} < 0$$

$$\left(\frac{Y_1}{Y}\right)^{(\varepsilon-1)/\varepsilon} = \gamma + (1-\gamma)\, \cdots$$

$$\frac{d \log \sigma_K}{d \log A_2} = -\frac{d \log \sigma_K}{d \log A_1} = \frac{1-\sigma_K}{\sigma_K}\, \alpha_1\, \cdots$$

$$\frac{\dot c}{c} = \frac{1}{\theta}\, (r(t) - \rho)$$

$$\lim_{t\to\infty} K(t)\, e^{-\int_0^t r(\tau) d\tau} = 0$$

$$\varepsilon < 1: n_1 \gtrless n_2 \Leftrightarrow \cdots$$

$$\frac{1}{\varepsilon}\, g + \frac{\varepsilon-1}{\varepsilon}\, g_1 - z_1 = \frac{1}{\varepsilon}\, g + \frac{\varepsilon-1}{\varepsilon}\, g_2 - z_2$$

$$g^*_1, g^*_2\ \text{exist},\ \varepsilon < 1 \Rightarrow \min,\ \varepsilon > 1 \Rightarrow \max$$

$$\lim_{t\to\infty} \dot K/K = \min\{g^*_1, g^*_2\}\ \text{or}\ \max$$

$$a_1/(1-\alpha_1) < a_2/(1-\alpha_2)\ \text{and}\ \varepsilon < 1\ \text{(or symm)}$$

$$z^*_2 = n - (1-\varepsilon)\, a_2 + (1 + (1-\varepsilon)(1-\alpha_2))\, \frac{a_1}{1-\alpha_1}$$

$$g^*_2 = n + \varepsilon\, a_2 + (1-\varepsilon(1-\alpha_2))\, \frac{a_1}{1-\alpha_1} > g^*$$

$$n^*_1 = n,\ n^*_2 = n - \frac{(1-\varepsilon)(1-\alpha_2)\, a_2}{1-\alpha_2} - \frac{a_1}{1-\alpha_1}$$

$$z^*_1, z^*_2, m^*_1, m^*_2, g^*_1, g^*_2\ \text{satisfy TVC}$$

$$g^* = g^*_C = g^*_2 = z^*_2 = n + \frac{a_2}{1-\alpha_2},\ z^*_1 = n - (1-\varepsilon)\, a_1 + \cdots$$

$$\int_0^{\infty} e^{-\rho t}\, (c_A(t) - \gamma_A)^{\eta}\, c_M(t)^{1-\eta}\, dt$$

$$Y_M(t) = X(t)\, F(L_M(t)),\ Y_A(t) = B_A\, G(L_A(t))$$

$$L_M(t) + L_A(t) \leq 1,\ \dot X(t) = \kappa\, \cdots$$

$$B_A\, G'(1 - n(t)) = p(t)\, X(t)\, F'(n(t))$$

$$B_A\, G(1) > \gamma_A > 0$$

$$n^* = \varphi^{-1}(\gamma_A / B_A)$$

$$\frac{\dot \varphi(t)}{\varphi(t)} = \frac{1}{\theta}\, [\alpha_1\, \gamma\, \eta(t)^{1/\varepsilon}\, \lambda(t)^{1-\alpha_2}\, \cdots]$$

$$\eta(t) \equiv \gamma^{\varepsilon}\, \frac{\varepsilon-1}{\varepsilon}\, [1 + \alpha_2\, \cdots]$$

$$X(0)/B_A\to X^F(0)/B_F:\quad n^*(0)\ \text{satisfies threshold}$$