★ 베타 수렴 회귀 (Convergence regression)
$$g_{i,t,t-1} = \alpha\, \log y_{i,t-1} + X^T_{i,t-1}\, \beta + \varepsilon_{i,t}$$
총 생산함수
$$Y(t) = F\bigl(K(t),\, L(t),\, A(t)\bigr)$$
동차성 스케일링 (보조정리)
$$\lambda^{m}\, g(x, y, z) = g(\lambda x,\, \lambda y,\, z)$$
노동시장 청산 (내부해)
$$L(t) = \bar{L}(t)$$
노동시장 청산 (보완성 형태)
$$L(t) \le \bar{L}(t),\quad w(t) \ge 0,\quad \bigl(L(t) - \bar{L}(t)\bigr)\,w(t) = 0$$
정태 이윤 극대화
$$\max_{K \ge 0,\, L \ge 0}\;\; F(K,\, L,\, A(t)) - R(t)\,K - w(t)\,L$$
임금 = 노동의 한계생산
$$w(t) = F_L\bigl(K(t),\, L(t),\, A(t)\bigr)$$
자본 임대료 = 자본의 한계생산
$$R(t) = F_K\bigl(K(t),\, L(t),\, A(t)\bigr)$$
자본축적 운동방정식 (이산시간)
$$K(t+1) = (1-\delta)\, K(t) + I(t)$$
국민소득 항등식
$$Y(t) = C(t) + I(t)$$
고정 저축률 가정
$$S(t) = s\, Y(t)$$
소비 = (1-s) 소득
$$C(t) = (1-s)\, Y(t)$$
Solow 성장 모형의 운동방정식 (정식)
$$K(t+1) = s\, F\bigl(K(t),\, L(t),\, A(t)\bigr) + (1-\delta)\, K(t)$$
1인당 자본 (자본-노동 비율)
$$k(t) \equiv \frac{K(t)}{L}$$
1인당 산출 — intensive form 생산함수
$$y(t) = F\!\left(\frac{K(t)}{L},\, 1,\, A\right) \equiv f\bigl(k(t)\bigr)$$
intensive form에서 요소가격
$$R(t) = f'\!\bigl(k(t)\bigr) > 0 \quad\text{and}\quad w(t) = f\bigl(k(t)\bigr) - k(t)\, f'\!\bigl(k(t)\bigr) > 0$$
Cobb-Douglas 생산함수
$$Y(t) = A\,K(t)^{\alpha}\,L(t)^{1-\alpha},\quad 0 < \alpha < 1$$
per-capita Solow 차분방정식
$$k(t+1) = s\,f\bigl(k(t)\bigr) + (1-\delta)\,k(t)$$
정상상태 자본-노동 비율 조건
$$\frac{f(k^{*})}{k^{*}} = \frac{\delta}{s}$$
정상상태 1인당 산출
$$y^{*} = f(k^{*})$$
정상상태 1인당 소비
$$c^{*} = (1-s)\,f(k^{*})$$
유일성 증명 — 단조성 도함수
$$\frac{\partial \bigl(f(k)/k\bigr)}{\partial k} = \frac{f'(k)\,k - f(k)}{k^{2}} = \frac{-w}{k^{2}} < 0$$
Golden rule 도출 — c* 의 s 에 대한 미분
$$\frac{\partial c^{*}(s)}{\partial s} = \bigl[f'(k^{*}(s)) - \delta\bigr]\,\frac{\partial k^{*}}{\partial s}$$
Phelps Golden Rule 조건
$$f'\bigl(k^{*}_{\text{gold}}\bigr) = \delta$$
비선형 자율 차분방정식 일반형
$$x(t+1) = G\bigl(x(t)\bigr)$$
선형 차분방정식 시스템
$$x(t+1) = A\,x(t) + b$$
비선형 시스템 — 국소 안정성 (Theorem 2.3)
$$x(t+1) = G\bigl(x(t)\bigr),\quad x^{*} = G(x^{*}),\ A := DG(x^{*})$$
Solow를 일반 g 함수형으로
$$k(t+1) = g\bigl(k(t)\bigr)$$
Steady state — g 의 fixed point
$$k^{*} = g(k^{*})$$
Strict concavity 부등식
$$f(k) > f(0) + k\,f'(k) = k\,f'(k)$$
이산 시간 차분 형태
$$x(t+1) - x(t) = g\bigl(x(t)\bigr)$$
연속 시간 미분방정식
$$\dot x(t) \equiv \lim_{\Delta t \to 0} \frac{x(t+\Delta t) - x(t)}{\Delta t} \simeq g\bigl(x(t)\bigr)$$
지수 인구성장
$$L(t) = e^{nt}\,L(0)$$
Solow 모형 연속시간 운동방정식
$$\frac{\dot k(t)}{k(t)} = s\,\frac{f(k(t))}{k(t)} - (n + \delta)$$
연속시간 Solow steady state
$$\frac{f(k^{*})}{k^{*}} = \frac{n + \delta}{s}$$
선형 미분방정식 시스템 안정성 (Theorem 2.4)
$$\dot x(t) = A\,x(t) + b$$
비선형 ODE 국소 안정성 (Theorem 2.5)
$$\dot x(t) = G\bigl(x(t)\bigr)$$
Elasticity of Substitution
$$\sigma \equiv -\frac{\partial \log(F_K/F_L)}{\partial \log(K/L)} \biggl|_{\text{ratio}}$$
CES 생산함수
$$F(K, L, A) = \bigl[\alpha (A_K K)^{(\sigma-1)/\sigma} + (1-\alpha)(A_H L)^{(\sigma-1)/\sigma}\bigr]^{\sigma/(\sigma-1)}$$
AK 모형
$$F(K(t), L(t), A(t)) = A\,K(t)$$
AK + BL 일반 선형 CRS
$$F(K(t), L(t), A(t)) = A\,K(t) + B\,L(t)$$
Capital·Labor Augmenting Tech
$$F\bigl(A_K(t)K(t),\ A_L(t)L(t)\bigr)$$
Uzawa 정리 도출 — 시점 T
$$Y(t) = F\bigl(K(t),\ L(t),\ \tilde A(T)\bigr)$$
Uzawa Derivation — Partial Derivatives
$$\tilde F_K(K, L, \tilde A) = F_K(K, AL),\quad \tilde F_L(K, L, \tilde A) = A\,F_L(K, AL)$$
Uzawa Derivation — R* and w*
$$R^* = \tilde F_K(K, L, \tilde A),\quad e^{g(t-T)}\,w^* = \tilde F_L(K, L, \tilde A)$$
Euler 정리 적용
$$\hat F(K, AL) \equiv \hat F_1(K, AL)\,K + \hat F_2(K, AL)\,AL$$
외생 기술진보율
$$\frac{\dot A(t)}{A(t)} = g > 0$$
자본축적 (기술진보 포함)
$$\dot K(t) = s\,F\bigl(K(t),\ A(t)L(t)\bigr) - \delta\,K(t)$$
Effective Capital-Labor Ratio
$$k(t) \equiv \frac{K(t)}{A(t)L(t)}$$
k 성장률 분해
$$\frac{\dot k(t)}{k(t)} = \frac{\dot K(t)}{K(t)} - g - n$$
1인당 산출 (기술 포함)
$$y(t) = A(t)\,f\bigl(k(t)\bigr)$$
Solow with All 3 — Final ODE
$$\frac{\dot k(t)}{k(t)} = s\,\frac{f(k(t))}{k(t)} - (\delta + g + n)$$
Solow Final Steady State
$$\frac{f(k^*)}{k^*} = \frac{\delta + g + n}{s}$$
총 생산함수 (재기술)
$$Y(t) = F\bigl(K(t),\, L(t),\, A(t)\bigr)$$
Y의 시간 도함수 분해
$$\frac{\dot Y}{Y} = \frac{F_A A}{Y}\,\frac{\dot A}{A} + \frac{F_K K}{Y}\,\frac{\dot K}{K} + \frac{F_L L}{Y}\,\frac{\dot L}{L}$$
근본 성장회계 방정식
$$x = g - \alpha_K g_K - \alpha_L g_L$$
TFP 시점별 추정량
$$\hat x(t) = g(t) - \alpha_K(t)\,g_K(t) - \alpha_L(t)\,g_L(t)$$
이산시간 TFP 추정량 (평균 분배몫)
$$\hat x_{t+1,t} = g_{t+1,t} - \bar\alpha_{K,t+1,t}\,g_{K,t+1,t} - \bar\alpha_{L,t+1,t}\,g_{L,t+1,t}$$
1인당 산출 (노동 augmenting)
$$y(t) = A(t)\, f\bigl(k(t)\bigr)$$
Solow 최종 ODE (재기술)
$$\frac{\dot k(t)}{k(t)} = \frac{s\, f(k(t))}{k(t)} - (\delta + g + n)$$
y 성장률 = g + ε_k · k 성장률
$$\frac{\dot y(t)}{y(t)} = g + \varepsilon_k\bigl(k(t)\bigr)\, \frac{\dot k(t)}{k(t)}$$
1인당 생산함수의 산출 탄력성
$$\varepsilon_k\bigl(k(t)\bigr) \equiv \frac{f'(k(t))\, k(t)}{f(k(t))} \in (0,\, 1)$$
수렴 방정식 (convergence equation)
$$\frac{\dot y(t)}{y(t)} \approx g - \bigl(1 - \varepsilon_k(k^*)\bigr)(\delta + g + n)\,\bigl(\log y(t) - \log y^*(t)\bigr)$$
Cobb-Douglas 수렴 방정식
$$\frac{\dot y(t)}{y(t)} \approx g - (1-\alpha)(\delta + g + n)\,\bigl(\log y(t) - \log y^*(t)\bigr)$$
이산시간 성장 회귀
$$g_{i,t,t-1} = b_0 + b_1\,\log y_{i,t-1} + \varepsilon_{i,t}$$
국가별 절편 (조건부 수렴)
$$g_{i,t,t-1} = b_0^i + b_1\,\log y_{i,t-1} + \varepsilon_{i,t}$$
Barro 조건부 수렴 회귀
$$g_{i,t,t-1} = X_{i,t}^{\top}\,\beta + b_1\,\log y_{i,t-1} + \varepsilon_{i,t}$$
Country-FE + Year-FE 패널 회귀
$$\log y_{i,t} = \alpha\,\log y_{i,t-1} + X_{i,t}^{\top}\,\beta + \delta_i + \mu_t + \varepsilon_{i,t}$$
인적자본 augmented 생산함수
$$Y = F(K,\, H,\, AL)$$
k̇ = 0 locus (물적자본 정상상태 곡선)
$$s_k\, f(k^*, h^*) - (\delta_k + g + n)\, k^* = 0$$
ḣ = 0 locus (인적자본 정상상태 곡선)
$$s_h\, f(k^*, h^*) - (\delta_h + g + n)\, h^* = 0$$
k̇ = 0 locus 의 기울기 (Implicit Function)
$$\left.\frac{dh}{dk}\right|_{\dot k = 0} = \frac{(\delta_k + g + n) - s_k\, f_k(k^*, h^*)}{s_k\, f_h(k^*, h^*)}$$
ḣ = 0 locus 의 기울기 (Implicit Function)
$$\left.\frac{dh}{dk}\right|_{\dot h = 0} = \frac{s_h\, f_k(k^*, h^*)}{(\delta_h + g + n) - s_h\, f_h(k^*, h^*)}$$
Augmented Cobb-Douglas 생산함수 (Example 3.2)
$$Y(t) = K(t)^{\alpha}\, H(t)^{\beta}\, \bigl(A(t) L(t)\bigr)^{1-\alpha-\beta}$$
Augmented k* (효과적 자본 정상상태)
$$k^* = \left[\left(\frac{s_k}{n+g+\delta_k}\right)^{1-\beta}\left(\frac{s_h}{n+g+\delta_h}\right)^{\beta}\right]^{\frac{1}{1-\alpha-\beta}}$$
Augmented ŷ* (효과적 산출 정상상태)
$$\hat y^* = \left(\frac{s_k}{n+g+\delta_k}\right)^{\frac{\alpha}{1-\alpha-\beta}} \left(\frac{s_h}{n+g+\delta_h}\right)^{\frac{\beta}{1-\alpha-\beta}}$$
국가별 BGP 1인당 소득 (MRW)
$$y_j^*(t) = A_j(t)\,\left(\frac{s_{k,j}}{n_j+g_j+\delta_k}\right)^{\frac{\alpha}{1-\alpha-\beta}}\left(\frac{s_{h,j}}{n_j+g_j+\delta_h}\right)^{\frac{\beta}{1-\alpha-\beta}}$$
MRW log-linear cross-country 회귀
$$\log y_j^*(t) = \log \bar A_j + gt + \frac{\alpha}{1-\alpha-\beta}\log\!\left(\frac{s_{k,j}}{n_j+g+\delta_k}\right) + \frac{\beta}{1-\alpha-\beta}\log\!\left(\frac{s_{h,j}}{n_j+g+\delta_h}\right)$$
MRW augmented 회귀 — 분해 형태
$$\log y_j^* = \text{const} + \frac{\alpha}{1-\alpha-\beta}\log(s_{k,j}) - \frac{\alpha}{1-\alpha-\beta}\log(n_j+g+\delta_k) + \frac{\beta}{1-\alpha-\beta}\log(s_{h,j}) - \frac{\beta}{1-\alpha-\beta}\log(n_j+g+\delta_h) + \varepsilon_j$$
Mincer 임금 회귀
$$\log w_i = X_i^{\top}\,\gamma + \phi\, S_i + u_i$$
자본의 한계생산성 (firm-level)
$$R_j = \alpha\,\left(\frac{K_f}{A_j H_f}\right)^{-(1-\alpha)}$$
Hall-Jones Cobb-Douglas with embedded H
$$Y_j = K_j^{\alpha}\,(A_j H_j)^{1-\alpha}$$
Cross-Country Growth Accounting (Caselli)
$$\hat x_{j,j+1} = g_{j,j+1} - \bar\alpha_{K,j,j+1}\, g_{K,j,j+1} - \bar\alpha_{L,j,j+1}\, g_{H,j,j+1}$$
Naive cross-country 회귀
$$\log Y_j = \alpha\, \log K_j + (1-\alpha)\, \log H_j + \alpha\, \log A_j$$
Trefler 순요소수출 (Heckscher-Ohlin augmented)
$$X_j^K = A_j^k K_j - \gamma_j^C \sum_{i=1}^{J} A_i^k K_i \quad \text{and} \quad X_j^H = A_j^h H_j - \gamma_j^C \sum_{i=1}^{J} A_i^h H_i$$
조건부 인자가격 균등화 (Conditional Factor Price Equalization)
$$\frac{R_j}{A_j^k} = \frac{R_{j'}}{A_{j'}^k} \quad \text{and} \quad \frac{w_j}{A_j^h} = \frac{w_{j'}}{A_{j'}^h}$$
지식 축적 방정식 (인구 비례)
$$\dot A(t) = \lambda\, L(t)$$
맬서스적 인구–소득 관계
$$L(t) = \varphi\, Y(t)$$
결합 방정식 — 지식의 자가 비례 성장
$$\dot A(t) = \lambda\, \varphi^{1/(1-\alpha)}\, A(t)$$
지수 폭발 — 결합 모형의 해
$$A(t) = \exp\!\bigl(\lambda\, \varphi^{1/(1-\alpha)}\, t\bigr)\, A(0)$$
쌍곡선 특이점 — Kremer 모형의 폭발해
$$A(t) = \frac{1}{A(0)^{-1} - \lambda\, \varphi^{1/(1-\alpha)}\, t}$$
유한지평 평생 효용 (가구별)
$$U^h\bigl(c^h(1), c^h(2), \ldots, c^h(T)\bigr) = \sum_{t=0}^{T} (\beta^h)^t\, u^h\bigl(c^h(t)\bigr)$$
무한지평 평생 효용 (대표가구)
$$\sum_{t=0}^{\infty} \beta^t\, u\bigl(c(t)\bigr)$$
Gorman 폴라 형식 (간접 효용)
$$v^h(p, w^h) = a^h(p) + b(p)\, w^h$$
CES 직접 효용 (가구별, Stone-Geary 형식)
$$U^h\bigl(x_1^h, \ldots, x_N^h\bigr) = \left[\,\sum_{j=1}^{N} \bigl(x_j^h - \xi_j^h\bigr)^{(\sigma-1)/\sigma}\,\right]^{\sigma/(\sigma-1)}$$
CES 간접 효용 (가구별)
$$v^h(p, w^h) = \frac{-\sum_{j=1}^{N} p_j\, \xi_j^h + w^h}{\bigl[\sum_{j=1}^{N} p_j^{1-\sigma}\bigr]^{1/(1-\sigma)}}$$
CES 직접 효용 (대표가구, 집계 결과)
$$U(x_1, \ldots, x_N) = \left[\,\sum_{j=1}^{N} (x_j - \xi_j)^{(\sigma-1)/\sigma}\,\right]^{\sigma/(\sigma-1)}$$
Pareto 최적 — 가구 가중 효용 합 극대화
$$\max_{\{y_j\}, \{w^h\}} \sum_{h \in \mathcal{H}} \alpha^h\, v^h(p, w^h) = \max_{\{y_j\}, \{w^h\}} \sum_{h \in \mathcal{H}} \alpha^h\bigl(a^h(p) + b(p) w^h\bigr)$$
Pareto 최대화 — 단순화 형태
$$\max_{\{y_j\}_{j=1}^{N},\, p,\, \{w^h\}} \sum_{h \in \mathcal{H}} \bigl(a^h(p) + b(p)\, w\bigr)$$
최대 Pareto weight 외 가구의 소득 영(0)
$$w^{h*} = 0 \quad \text{for all } h \notin \mathcal{H}^M$$
비교 부등식 — Pareto 최적의 strict 우위
$$\sum_{h \in \mathcal{H}} a^h + b(p^*)\, w^* > \sum_{h \in \mathcal{H}} a^h + b(p^{**}_\alpha)\, w^{**}_\alpha \;;\quad b(p^*)\, w^* > b(p^{**}_\alpha)\, w^{**}_\alpha$$
Pareto 가중 부등식 — (5.7) 가설로부터의 함의
$$\sum_{h \in \mathcal{H}} \alpha^h\, b(p^{**}_\alpha)\, w^{h**}_\alpha \geq \sum_{h \in \mathcal{H}} \alpha^h\, b(p^*)\, w^{h*}$$
Yaari 영생 모형 — 사망확률 ν 의 효용 합산
$$U_0\bigl(c(0), c(1), \ldots\bigr) = \sum_{t=0}^{\infty} \bigl(\hat\beta(1-\nu)\bigr)^t\, u(c(t)) \equiv \sum_{t=0}^{\infty} \beta^t\, u(c(t)), \quad \beta \equiv \hat\beta(1-\nu)$$
대표 기업 — 집계 생산가능집합
$$Y = \left\{\,\sum_{f \in \mathcal{F}} y^f \;:\; y^f \in Y^f \text{ for each } f \in \mathcal{F}\,\right\}$$
대표 기업 — 이윤 부등식 (정리 5.4 증명 단계)
$$p \cdot \hat y \geq p \cdot \sum_{f \in \mathcal{F}} \hat y^f$$
이산시간 무한지평 효용 (대표가구, §5.5 재기술)
$$\sum_{t=0}^{\infty} \beta^t\, u\bigl(c(t)\bigr)$$
연속시간 무한지평 효용 (대표가구)
$$\int_{0}^{\infty} \exp(-\rho t)\, u\bigl(c(t)\bigr)\, dt$$
제1후생정리 — 예산제약 부등식 (가구별)
$$p^* \cdot \tilde x^h \geq p^* \cdot x^{h*} = p^* \cdot \left(\omega^h + \sum_{f \in \mathcal{F}} \theta^h_f\, y^{f*}\right)$$
제1후생정리 — strict 우위 가구의 strict 부등식
$$p^* \cdot \tilde x^h > p^* \cdot \left(\omega^h + \sum_{f \in \mathcal{F}} \theta^h_f\, y^{f*}\right)$$
제1후생정리 — 가구합산 strict 부등식
$$p^* \cdot \sum_{h \in \mathcal{H}} \tilde x^h > p^* \cdot \sum_{h \in \mathcal{H}} \left(\omega^h + \sum_{f \in \mathcal{F}} \theta^h_f\, y^{f*}\right) = p^* \cdot \left(\sum_{h\in\mathcal{H}} \omega^h + \sum_{f\in\mathcal{F}} y^{f*}\right)$$
제1후생정리 — 이윤 극대화 부등식 (모순 도출)
$$p^* \cdot \sum_{f \in \mathcal{F}} y^{f*} \geq p^* \cdot \sum_{f \in \mathcal{F}} y^f \quad \text{for any } \{y^f\}_{f\in\mathcal{F}} \text{ with } y^f \in Y^f$$
제2후생정리 — Hahn-Banach 분리 부등식
$$\phi(y) \leq \phi(x^*) \leq \phi(x) \quad \text{for all } y \in Y' \text{ and all } x \in P$$
제2후생정리 — 무한차원 함수 극한 표현
$$\bar\phi(x) = \lim_{T \to \infty} \phi(x[T])$$
Sequential trading 예산제약 (Arrow 채권 보유)
$$\sum_{j=1}^{N} p^{**}_{j,t}\, x^h_{j,t} \leq \sum_{j=1}^{N} p^{**}_{j,t}\, \omega^h_{j,t} + b^h_t$$
최적 성장 문제 — 이산시간 (Ramsey-Cass-Koopmans)
$$\max_{[c(t),\, k(t)]_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t\, u\bigl(c(t)\bigr)$$
이산시간 자본 축적 — Ramsey 모형 자원 제약
$$k(t+1) = f\bigl(k(t)\bigr) + (1-\delta)\, k(t) - c(t)$$
이산시간 최적 성장 — 분권화된 가구 자산 동학
$$a(t+1) = \bigl(1 + r(t)\bigr)\, a(t) - c(t) + w(t)$$
최적 성장 문제 — 연속시간 (Ramsey-Cass-Koopmans)
$$\max_{[c(t),\, k(t)]_{t=0}^{\infty}} \int_{0}^{\infty} \exp(-\rho t)\, u\bigl(c(t)\bigr)\, dt$$
연속시간 자본 축적 — Ramsey 자원 제약
$$\dot k(t) = f\bigl(k(t)\bigr) - c(t) - \delta\, k(t)$$
Bellman 방정식 (정상 동적계획)
$$V(x) = \sup_{y \in G(x)} \{U(x, y) + \beta\, V(y)\} \quad \text{for all } x \in X$$
Bellman 방정식 (최적 trajectory 형)
$$V\bigl(x^*(t)\bigr) = U\bigl(x^*(t), x^*(t+1)\bigr) + \beta\, V\bigl(x^*(t+1)\bigr) \quad \text{for all } t = 0, 1, \ldots$$
Principle of Optimality — V* 형태
$$V^*\bigl(x^*(t)\bigr) = U\bigl(x^*(t), x^*(t+1)\bigr) + \beta\, V^*\bigl(x^*(t+1)\bigr)$$
Envelope 정리 — 가치함수 미분 = 한계 효용
$$DV(x) = D_x U\bigl(x, \pi(x)\bigr)$$
수축 사상 부등식 — Cauchy 단계
$$d(z_{n+1}, z_n) \leq \beta^n\, d(z_1, z_0), \quad n = 1, 2, \ldots$$
수축 사상 — m, n 격차 부등식
$$d(z_m, z_n) \leq \sum_{k=n}^{m-1} \beta^k\, d(z_1, z_0)$$
Cauchy 한계 부등식
$$d(z_m, z_n) \leq \frac{\beta^n}{1 - \beta}\, d(z_1, z_0)$$
Blackwell 충분조건 — 단조성·discounting 검증
$$\int_0^z f(g(x))\, dx - \int_0^z f(x)\, dx \leq \cdots$$
Blackwell 정리 적용 — 단조 연산자
$$\text{If } g(x) \leq f(x) + \|g - f\|, \text{ then } (Tg)(x) \leq (Tf)(x) + \beta \|g - f\|$$
Bellman 연산자 단조성
$$(Tg)(x) \leq (Tf)(x) + \beta\, \|g - f\|$$
최적 가치 함수 — 가능 sequence 의 sup
$$V^*(x(0)) = \sup_{x \in \Phi(x(0))} \bar U(x)$$
V* 상한 정의
$$V^*(x(0)) \geq \bar U(x) \quad \text{for all } x \in \Phi(x(0))$$
V* sup 근사 — ε 부등식
$$\exists\, x' \in \Phi(x(0)) \text{ such that } V^*(x(0)) \leq \bar U(x') + \varepsilon$$
Bellman 부등식 — 한 단계 후 V
$$V\bigl(x(0)\bigr) \geq U\bigl(x(0), y\bigr) + \beta\, V(y'),$$
Bellman 상한 — ε 근사
$$V(x(0)) \leq U(x(0), y') + \beta\, V(y') + \varepsilon$$
최적 sequence 의 supremum 도달
$$x^*_t \text{ attains sup starting from } x^*(t), \quad \bar U(x^*_t) = V^*(x^*(t))$$
V* tail 등치
$$V^*(x^*(t)) = \bar U(x^*_t)$$
Bellman 연산자 정의
$$(TV)(x) = \max_{y \in G(x)} \{U(x, y) + \beta\, V(y)\}$$
정책함수 — Bellman max 의 argmax
$$\pi(x) = \arg\max_{y \in G(x)} \{U(x, y) + \beta\, V(y)\}$$
Bellman 연산자 비교 부등식
$$TV(x') = U(x', y') + \beta V(y'), \quad TV(x'') = U(x'', y'') + \beta V(y'')$$
Theorem 6.6 증명 — V 미분 가능성
$$V^*(x) = V(x) = \bar U(x^*), \quad V(x + \tilde\varepsilon) \cdots$$
응용 — V 상하한 sandwich
$$V^-_k(x) \leq U_k(x, x') \leq V^+_k(x)$$
Sandwich 단조 수렴
$$V^-_k(x) \leq V^+_k(x), \quad V^-_k(x) \geq V^+_{k-1}(x)$$
응용 setup — 미분 가능 효용
$$\text{(setup equation for Section 6.6)}$$
FOC — Bellman 의 한계 조건 (DP form)
$$D_y U(x, y^*) + \beta\, DV(y^*) = 0$$
Envelope Theorem 결과 — DV(x)
$$DV(x) = D_x U(x, y^*)$$
FOC + Envelope 결합
$$D_y U(x, y^*) + \beta\, D_x U(y^*, \pi(y^*)) = 0$$
FOC — 부분 미분 표기
$$\frac{\partial U(x, y^*)}{\partial y} + \beta\, V'(y^*) = 0$$
Envelope — 부분 미분 표기
$$V'(x) = \frac{\partial U(x, y^*)}{\partial x}$$
Euler 방정식 (DP 도출)
$$\frac{\partial U(x(t), x^*(t+1))}{\partial y} + \beta\, \frac{\partial U(x^*(t+1), x^*(t+2))}{\partial x} = 0$$
Transversality 조건 (DP form)
$$\lim_{t \to \infty} \beta^t\, D_x U(x^*(t), x^*(t+1)) \cdot x^*(t) = 0$$
Transversality — 부분 미분 표기
$$\lim_{t \to \infty} \beta^t\, \frac{\partial U(x^*(t), x^*(t+1))}{\partial x} \cdot x^*(t) = 0$$
Transversality 부등식 — strict 위반
$$-\varepsilon\, \limsup_{T \to \infty} \beta^T D_x U(x^*(T), x^*(T+1)) \cdot x^*(T) \leq 0$$
Perturbation 한계 — ε→0
$$\lim_{\varepsilon \to 0} \limsup_{T \to \infty} \sum_{t=0}^{T} \beta^t\, o(\varepsilon, T, t) = 0$$
Perturbation 한계 — T 상한
$$\text{For } T > \bar T: \quad \lim_{T \to \infty} \limsup_{t=0}^{T} \beta^t\, o(\varepsilon, T, t)$$
Envelope 적용 — Example 6.4 풀이
$$\frac{1}{x^\alpha - y} = \beta\, V'(y)$$
Example 6.4 해 — 자본 동학
$$k(t+1) = \beta\, \alpha\, k(t)^\alpha$$
Consumption Euler — Bellman 도출
$$u'((1+r)a + w - a') = u'(c) = \beta\, V'(a')$$
Consumption Euler — 시간 비교
$$u'(c) = \beta(1+r)\, u'(c')$$
Consumption growth — r vs 1/β-1
$$\dot c/c \begin{cases} = 0 & \text{if } r = \beta^{-1} - 1 \\ > 0 & \text{if } r > \beta^{-1} - 1 \\ < 0 & \text{if } r < \beta^{-1} - 1 \end{cases}$$
비정상 Euler 방정식
$$\frac{\partial U(x^*(t), x^*(t+1))}{\partial y} + \beta\, \frac{\partial U(x^*(t+1), x^*(t+2))}{\partial x} = 0$$
비정상 transversality 도출 단계
$$\text{(perturbation step for nonstationary case)}$$
비정상 transversality 조건
$$\lim_{t \to \infty} \beta^t\, D_x U(t, x^*(t), x^*(t+1)) \cdot x^*(t) = 0$$
비정상 응용 setup
$$\text{(setup for Section 6.7.3 application)}$$
RCK 재기술 — 무한지평 최적 성장
$$\text{(restatement of (5.24)-(5.25) RCK problem)}$$
RCK Bellman 표현 — sequence form
$$\max_{\{k(t), c(t)\}_{t=0}^{\infty}} \sum_{t=0}^{\infty} \beta^t\, u(c(t))$$
RCK 자원 제약
$$k(t+1) = f(k(t)) + (1-\delta)\, k(t) - c(t), \quad k(0) > 0 \text{ given}$$
RCK Bellman 함수 방정식
$$V(k) = \max_{s \in G(k)} \{u(f(k) + (1-\delta)k - s) + \beta\, V(s)\}$$
Concavity proof step
$$u(f(k) + (1-\delta)k - s(k)) - u(f(k) + (1-\delta)k - s'(k')) \geq \cdots$$
RCK Euler 방정식 — 표준 형식
$$u'(c(t)) = \beta\, [f'(k(t+1)) + (1-\delta)]\, u'(c(t+1))$$
RCK Transversality
$$\lim_{t \to \infty} \beta^t \bigl[f'(k(t)) + (1-\delta)\bigr]\, u'(c(t))\, k(t) = 0$$
Modified Golden Rule
$$\beta\, [f'(k^*) + (1-\delta)] = 1$$
Competitive Equilibrium 정의
$$\text{Definition 6.3 (CE for RCK)}$$
분권화 가구 문제 (CE)
$$\max_{\{c(t), a(t)\}} \sum \beta^t u(c), \quad w(t) = f(k(t)) - k(t) f'(k(t))$$
CE Euler — 분권화 형식
$$a(t) = k(t), \quad u'(c(t)) = \beta(1+r(t+1))\, u'(c(t+1))$$
CE = SP 결합 결과
$$\beta\, [f'(k(t+1)) + (1-\delta)] = 1$$
Bolza 최적제어 문제 (objective functional)
$$\max_{x(t), y(t), x_1} W(x(t), y(t)) \equiv \int_0^{t_1} f(t, x(t), y(t))\, dt$$
상태 방정식 (state equation)
$$\dot x(t) = g(t, x(t), y(t))$$
feasibility — 제약 집합
$$x(t) \in X, \quad y(t) \in Y$$
Variation 변분 setup
$$\eta(t) \in C^1, \quad \eta(0) = 0$$
Perturbed feasibility
$$x(t, \varepsilon) = \hat x(t) + \varepsilon\, \eta(t)$$
변분 적분 W(ε)
$$W(\varepsilon) = \int_0^{t_1} f(t, x(t, \varepsilon), y(t, \varepsilon))\, dt$$
Lagrangian 도입 — costate λ
$$W(\varepsilon) = \int_0^{t_1} \{f(t, x, y) + \lambda(t)[g(t, x, y) - \dot x(t)]\}\, dt$$
변분 정리 — 부분적분
$$W(\varepsilon) = \int_0^{t_1} [f + \lambda g]\, dt + [\lambda x]_0^{t_1} - \int_0^{t_1} \dot \lambda x\, dt$$
1차 최적성 조건 W'(0)=0
$$W'(0) = 0 \quad \text{for all } \eta(t)$$
Costate (adjoint) ODE — Pontryagin
$$\dot \lambda(t) = -\bigl[f_x(t, \hat x(t), \hat y(t)) + \lambda(t)\, g_x(t, \hat x(t), \hat y(t))\bigr]$$
Pontryagin 1차 조건 (control variable)
$$f_y(t, \hat x(t), \hat y(t)) + \lambda(t)\, g_y(t, \hat x(t), \hat y(t)) = 0 \quad \forall t \in [0, t_1]$$
단순화된 자유 종단 시간 문제
$$\max_{x(t), y(t)} W(x(t), y(t)) \equiv \int_0^{t_1} f(t, x(t), y(t))\, dt$$
Example 7.1 — costate
$$\exp(-\rho t)\, u'(\hat c(t)) = \lambda(t)$$
Example 7.1 — costate ODE
$$\dot \lambda(t) = -r\, \lambda(t)$$
Maximum Principle — 종단 자유 형태
$$x(t_1) = x_1 \text{ free}, \quad \lambda(t_1) = 0$$
Simplified Maximum Principle (Hamiltonian form)
$$H_y(t, \hat x(t), \hat y(t), \lambda(t)) = 0 \quad \forall t \in [0, t_1]$$
Costate ODE (Hamiltonian form)
$$\dot \lambda(t) = -H_x(t, \hat x(t), \hat y(t), \lambda(t))$$
State ODE (Hamiltonian form)
$$\dot x(t) = H_\lambda(t, \hat x(t), \hat y(t), \lambda(t))$$
최대화된 Hamiltonian (M function)
$$M(t, x(t), \lambda(t)) \equiv \max_{y \in Y} H(t, x(t), y, \lambda(t))$$
M concavity 단계
$$\int_0^{t_1} M\, dt \leq \int_0^{t_1} M(\hat x)\, dt + \int_0^{t_1} M_x \cdot (x - \hat x)\, dt$$
Envelope — M_x = H_x
$$M_x(t, \hat x, \lambda) = H_x(t, \hat x, \hat y, \lambda)$$
충분조건 증명 단계
$$\int_0^{t_1} [M(t, x, \lambda) - M(\hat x, \lambda)]\, dt \leq \int_0^{t_1} M_x \cdot (x - \hat x)\, dt$$
충분조건 마무리 단계
$$\int [\lambda(g - g(\hat x))]\, dt - \int \dot \lambda (x - \hat x)\, dt = [\lambda(x - \hat x)]_0^{t_1}$$
★ Maximum Principle — 일반 형태 (벡터)
$$\max_{x(t), y(t), x_1} W(x, y) \equiv \int_0^{t_1} f(t, x, y)\, dt$$
벡터 상태 방정식
$$\dot x(t) = G(t, x(t), y(t))$$
벡터 feasibility
$$x(t) \in X \subset \mathbb{R}^{K_x}, \quad y(t) \in Y \subset \mathbb{R}^{K_y}$$
★ Generalized Hamiltonian
$$H(t, x, y, \lambda) \equiv f(t, x, y) + \lambda(t) \cdot G(t, x, y)$$
★ MP — 통제 FOC (벡터)
$$D_y H(t, \hat x, \hat y, \lambda) = 0 \quad \forall t \in [0, t_1]$$
★ MP — costate 동학 (벡터)
$$\dot \lambda(t) = -D_x H(t, \hat x, \hat y, \lambda)$$
★ MP — 상태 동학 (벡터)
$$\dot x(t) = D_\lambda H(t, \hat x, \hat y, \lambda)$$
무한지평 OC — 목적
$$\max_{x(t), y(t)} W(x, y) \equiv \int_0^{\infty} f(t, x, y)\, dt$$
무한지평 OC — 상태
$$\dot x(t) = g(t, x(t), y(t))$$
Stationarity — 종단 조건
$$\lim_{t \to \infty} b(t)\, x(t) \geq x_1$$
Value Function — 무한지평
$$V(t_0, x(t_0)) = \sup_{(x(t), y(t)) \in X \times Y} \int_{t_0}^{\infty} f(t, x, y)\, dt$$
V ≥ admissible 부등식
$$V(t_0, x(t_0)) \geq \int_{t_0}^{\infty} f(t, x, y)\, dt \quad \text{for any admissible}$$
V = optimal 적분
$$V(t_0, x(t_0)) = \int_{t_0}^{\infty} f(t, \hat x, \hat y)\, dt$$
Lemma 7.1 — V 단조성/연속성
$$\text{Lemma 7.1: } V \text{ continuous and concave under regularity}$$
Hamiltonian 최대성 — y* 최적
$$H(t, \hat x, \hat y, \lambda) \geq H(t, \hat x, y, \lambda)$$
Costate 동학 (무한지평)
$$\dot \lambda(t) = -H_x(t, \hat x, \hat y, \lambda)$$
State 동학 (무한지평)
$$\dot x(t) = H_\lambda(t, \hat x, \hat y, \lambda), \quad \lim_{t \to \infty} b(t) x(t) \geq x_1$$
HJB 도출 — Heuristic
$$V(t, x(t)) = \max_{y} \int_t^{t+\Delta t} f\, ds + V(t + \Delta t, x(t + \Delta t))$$
★ Stationary HJB — 가치 함수 형태
$$V(t, x(t)) = \exp(-\rho t)\, v(x(t)) \quad \forall t$$
★ ★ ★ Hamilton-Jacobi-Bellman 방정식
$$\rho\, v(\hat x(t)) = f(\hat x, \hat y) + \dot v(\hat x(t))$$
HJB 응용 표기
$$\rho\, v(\hat x(t)) - \dot v(\hat x(t)) = f(\hat x, \hat y) + v'(\hat x) \dot x$$
Hamiltonian 적분 형식
$$\int_0^{t_1} H(t, \hat x, y, \lambda)\, dt = \int_0^{t_1} [f + \lambda g]\, dt$$
HJB — 한 단계 후 표현
$$\rho\, v(\hat x(t)) = \max_y [f(\hat x, y) + v'(\hat x) g(\hat x, y)]$$
Perturbation — V 시간 차이
$$\frac{V(t_0+t, x_\delta(t_0+t)) - V(t_0, \hat x(t_0))}{t} \leq -\frac{1}{t} \int_{t_0}^{t_0+t} f\, dt$$
Perturbation — t→0 한계
$$\lim_{t \to 0} \frac{1}{t}\int_{t_0}^{t_0+t} f\, dt = f(t_0, x_\delta(t_0), y_\delta(t_0))$$
Perturbation — V 미분 한계
$$\lim_{t \to 0} \frac{V(t_0+t, x_\delta) - V(t_0, \hat x)}{t} = \partial_t V + \partial_x V \cdot \dot x_\delta$$
Perturbation — 부등식 결합
$$f(t_0, x_\delta, y_\delta) + \partial_t V + \partial_x V \cdot g(t_0, x_\delta, y_\delta) \leq 0$$
Perturbation — 최적 trajectory 등식
$$f(t_0, \hat x, \hat y) + \partial_t V + \partial_x V \cdot g(\hat x, \hat y) = 0$$
Perturbation — y free max
$$f(t_0, \hat x, \hat y) + \partial_t V + \partial_x V \cdot g(\hat x, \hat y) = \max_y \cdots$$
Costate ↔ V 연결
$$\lambda(t_0) = \partial_x V(t_0, \hat x(t_0))$$
Hamiltonian 무한지평 한계
$$\lim_{t \to \infty} H(t, \hat x, \hat y, \lambda) = 0$$
Theorem 7.6 증명 setup
$$\partial_t V(t, x) + \max_y H(t, x, y, V_x) = 0$$
Example 7.3 — 할인 무한지평
$$\int_0^{\infty} \exp(-\rho t)\, u(y(t))\, dt, \quad \dot x(t) = \cdots$$
Example 7.3 — costate 정상
$$\dot \lambda(t) = 0$$
Example 7.3 — 통제 풀이
$$\dot x(t) = -u'^{-1}[\exp(\rho t)\, \lambda(0)]$$
할인 OC — 일반 setup
$$\dot x(t) = g(t, x, y)$$
할인 OC — 종단 조건
$$\lim_{t \to \infty} b(t) x(t) \geq x_1$$
할인 형태 — current vs present
$$\text{(setup for current-value Hamiltonian)}$$
★ Current-Value Hamiltonian
$$\hat H(t, x, y, \mu) \equiv f(x, y) + \mu(t)\, g(t, x, y)$$
★ Current-value MP — 통제 FOC
$$\hat H_y(t, \hat x, \hat y, \mu) = 0 \quad \forall t \in \mathbb{R}_+$$
★ Current-value MP — costate
$$\rho\, \mu(t) - \dot \mu(t) = \hat H_x(t, \hat x, \hat y, \mu) \quad \forall t \in \mathbb{R}_+$$
★ Strong Transversality
$$\lim_{t \to \infty} \exp(-\rho t)\, \mu(t) \cdot \hat x(t) = 0$$
★ Transversality — 강 형식
$$\lim_{t \to \infty} \exp(-\rho t)\, \mu(t)\, \hat x(t) = 0$$
Transversality — Hamiltonian 형태
$$\lim_{t \to \infty} \exp(-\rho t)\, f(\hat x, \hat y) = 0$$
Transversality 연결
$$\lim_{t \to \infty} \exp(-\rho t)\, \mu(t)\, g(t, \hat x, \hat y) = \lim_{t \to \infty} \exp(-\rho t) \cdots$$
FOC — current-value 통제 풀이
$$f_y(\hat x, \hat y) + \mu(t)\, g_y(t, \hat x, \hat y) = 0$$
Sufficiency — concavity 결합
$$\text{(sufficiency proof step)}$$
충분조건 — W 비교
$$W(x(t), y(t)) \leq W(\hat x, \hat y)$$
Theorem 7.13 일반 결과
$$\text{(Theorem 7.13 statement)}$$
Constraint 형식 일반화
$$\max_{x(t), y(t)} W(x, y) \equiv \int_0^{\infty} \cdots$$
Vector state 진화
$$\dot x(t) = G(t, x(t), y(t))$$
Vector — 종단 조건
$$\lim_{t \to \infty} x(t) \geq x_1$$
할인 OC 마무리 — example
$$\int_0^{\infty} \exp(-\rho t)\, \cdots$$
★ RCK — OC 형식
$$\max_{[k(t), c(t)]} \int_0^{\infty} \exp(-\rho t)\, u(c(t))\, dt$$
★ RCK — OC FOC (소비)
$$\hat H_c(k, c, \mu) = u'(c(t)) - \mu(t) = 0$$
★ RCK — OC costate 진화
$$\hat H_k(k, c, \mu) = \mu(t)[f'(k(t)) - \delta] = \rho\, \mu(t) - \dot \mu(t)$$
★ RCK — OC Transversality
$$\lim_{t \to \infty} \exp(-\rho t)\, \mu(t)\, k(t) = 0$$
기업 가치 q 도입
$$\text{(setup for q-theory of investment)}$$
★ q-theory Hamiltonian
$$\hat H(K, I, q) \equiv [f(K(t)) - I(t) - \varphi(I(t))] + q(t)[I(t) - \delta K(t)]$$
★ q-theory FOC — 투자 한계
$$q(t) = 1 + \varphi'(I(t))$$
q-theory 시간 진화
$$\dot q(t) = \varphi''(I(t))\, \dot I(t)$$
★ Investment Law of Motion
$$\dot I(t) = \frac{1}{\varphi''(I(t))}[(r + \delta)(1 + \varphi'(I)) - f'(K)]$$
Linear ODE — 투자 편차
$$\dot x(t) = A\, x(t) + b$$
비선형 → 선형 근사
$$\text{(linearization step around steady-state)}$$
★ V'(K) = q — 가치-가격 정합성
$$V'(K(t)) = q(t)$$
신고전 가계 평생 효용
$$U_0 = \int_0^{\infty} e^{-\rho t}\, u(c(t))\, dt$$
인구 지수 성장
$$L(t) = \exp(n t)$$
1인당 효용 — 효과적 할인율
$$\int_0^{\infty} \exp(-(\rho - n) t)\, u(c(t))\, dt$$
가계 budget constraint
$$\dot a(t) = r(t) a(t) + w(t) - c(t) - n a(t)$$
★ 자본 한계 산출 — 이자율
$$R(t) = F_K(K, L) = f'(k(t))$$
★ 노동 한계 산출 — 임금
$$w(t) = F_L(K, L) = f(k(t)) - k(t)\, f'(k(t))$$
총 자산 진화
$$\dot A(t) = r(t)\, A(t) + w(t) L(t) - c(t) L(t)$$
1인당 자산 진화 — 인구 희석
$$\dot a(t) = (r(t) - n)\, a(t) + w(t) - c(t)$$
Equilibrium — 자산 = 자본
$$a(t) = k(t)$$
Hamiltonian (Cass-Koopmans)
$$H = e^{-\rho t} u(c) + \mu \bigl[r a + w - c - n a\bigr]$$
Natural debt limit — 평생 임금 현재가치
$$a(t) \geq -\int_t^{\infty} w(s)\, \exp\!\Bigl(-\int_t^s (r(z)-n)\, dz\Bigr)\, ds$$
Natural debt limit — 무한 한계
$$\lim_{t \to \infty} a(t) \geq \hat a \equiv -\lim_{t \to \infty} \int_t^{\infty} w(s)\, e^{-\int (r-n)} ds$$
Euler equation (소비 동학)
$$\frac{\dot c(t)}{c(t)} = \frac{1}{\theta}\bigl[r(t) - \rho\bigr]$$
No-Ponzi 조건
$$\lim_{t \to \infty} a(t)\, \exp\!\Bigl(-\int_0^t (r(s)-n)\, ds\Bigr) \geq 0$$
Lifetime budget — 적분
$$\int_0^T c(t) L(t)\, e^{-\int_0^t r\, ds}\, dt + a(0)\, e^{...} = \int_0^T w L\, e^{...}\, dt$$
★ Strong Transversality
$$\lim_{t \to \infty} a(t)\, e^{-\int_0^t (r-n) ds} = 0$$
★ FOC — 통제 (소비)
$$\hat H_c = u'(c(t)) - \mu(t) = 0$$
★ Costate 진화 — NGM
$$\hat H_a = \mu(r(t) - n) = -\dot\mu(t) + (\rho - n)\mu(t)$$
Transversality — costate 형태
$$\lim_{t \to \infty} e^{-(\rho-n)t}\, \mu(t)\, a(t) = 0$$
Transversality 조건 (TVC)
$$\lim_{t \to \infty} e^{-\rho t} \mu(t) a(t) = 0$$
FOC — μ = u'(c)
$$u'(c(t)) = \mu(t)$$
★ Consumption Euler — 일반 효용
$$\frac{\dot c(t)}{c(t)} = \frac{1}{\varepsilon_u(c(t))}\, (r(t) - \rho)$$
Coefficient of Relative Risk Aversion (CRRA)
$$\varepsilon_u(c) \equiv -\frac{u''(c)\, c}{u'(c)}$$
Costate 명시 풀이
$$\mu(t) = \mu(0)\, e^{-\int_0^t (r(s) - \rho) ds} = u'(c(0))\, e^{-\int (r-\rho)}$$
Transversality 조건 풀이형
$$\lim_{t \to \infty} e^{-(\rho-n)t}\, a(t)\, u'(c(0))\, e^{-\int (r-\rho)} = 0$$
Transversality — k 형태
$$\lim_{t \to \infty} k(t)\, e^{-\int_0^t (r(s) - n)\, ds} = 0$$
★ 시장 청산 — 이자율 = 한계 산출 - 감가
$$r(t) = f'(k(t)) - \delta$$
★ NGM 동적 시스템 — k 동학
$$\frac{\dot c}{c} = \frac{1}{\varepsilon_u}(f'(k) - \delta - \rho)$$
Transversality — k 형태 (averaged)
$$\lim_{t \to \infty} k(t)\, e^{-\int (f' - \delta - n)} = 0$$
신고전 modified golden rule
$$f'(k^*) = \rho + \delta + \theta g$$
평균 이자율 정의
$$\bar r(t) = \frac{1}{t}\int_0^t r(s)\, ds$$
Transversality — 평균 이자율 형식
$$\lim_{t \to \infty} e^{-(\bar r(t) - n) t}\, a(t) = 0$$
Consumption — 명시 풀이
$$c(t) = c(0)\, e^{\int_0^t \frac{r(s) - \rho}{\varepsilon_u(c(s))} ds}$$
FOC + costate 결합
$$\hat H_c(k, c, \mu) = 0 = u'(c(t)) - \mu(t)$$
Steady-State 정의
$$\dot c = 0, \quad \dot k = 0$$
★ Steady-state 이자율
$$r^* = f'(k^*) - \delta > n$$
★ Steady-state 소비
$$c^* = f(k^*) - (n + \delta)\, k^*$$
★ Steady-state 저축률
$$s^* = \frac{(n + \delta)\, k^*}{f(k^*)}$$
★ NGM 동적 시스템 — k 동학
$$\dot k(t) = f(k(t)) - (n + \delta) k(t) - c(t)$$
★ NGM 동적 시스템 — c 동학
$$\frac{\dot c}{c} = \frac{1}{\varepsilon_u}\, [f'(k) - \delta - \rho]$$
Transversality 결합
$$\lim_{t \to \infty} k(t)\, e^{-\int (f'(k) - \delta - n)} = 0$$
Discrete-time NGM — 평생 효용
$$\sum_{t=0}^{\infty} \beta^t u(c(t)) \text{ s.t. } a(t+1) = (1+r)a(t) + w - c$$
Discrete No-Ponzi
$$\lim_{t \to \infty} a(t) \prod_{s=1}^{t-1} \frac{1}{1+r(s)} \geq 0$$
★ Discrete Consumption Euler
$$u'(c(t)) = \beta(1 + r(t+1))\, u'(c(t+1))$$
Labor-augmenting 기술
$$Y(t) = F(K(t), A(t) L(t))$$
효과적 노동 단위 산출
$$\hat y(t) \equiv \frac{Y(t)}{A(t) L(t)} = f(\hat k(t)), \quad \hat k = \frac{K}{AL}$$
★ CRRA Utility
$$u(c) = \frac{c^{1-\theta} - 1}{1 - \theta}, \quad R = \theta$$
Budget — discrete prices
$$\sum_{j=1}^N p_j c_j \leq y$$
★ CRRA + 인구 — 평생 효용
$$\int_0^{\infty} e^{-(\rho-n)t}\, \frac{c(t)^{1-\theta} - 1}{1-\theta}\, dt$$
★ CRRA Consumption Euler
$$\frac{\dot c}{c} = \frac{1}{\theta}(r(t) - \rho)$$
Effective consumption 정의
$$\tilde c(t) \equiv \frac{C}{AL} = \frac{c(t)}{A(t)}$$
Transversality — effective form
$$\lim_{t \to \infty} \hat k(t)\, e^{-\int (f'(\hat k) - \delta - n - g)} = 0$$
★ ★ ★ Modified Golden Rule with Growth
$$f'(k^*) = \rho + \delta + \theta g$$
★ Effective Steady-State
$$\tilde c^* = f(\hat k^*) - (n + g + \delta)\, \hat k^*$$
Cobb-Douglas 명시 동학
$$f(k) = k^\alpha, \quad \frac{d\tilde c/dt}{\tilde c} = \frac{1}{\theta}[\alpha k^{\alpha-1} - \delta - \rho]$$
z = c̃/k 동학
$$\frac{\dot z}{z} = \frac{d\tilde c/dt}{\tilde c} - \frac{\dot k}{k}$$
조세 + 이자 — Distortion
$$r(t) = (1-\tau)(f'(k) - \delta), \quad \frac{d\tilde c/dt}{\tilde c} = \frac{1}{\theta}[(1-\tau)(f'(k)-\delta) - \rho]$$
Multi-country NGM
$$\int_0^{\infty} e^{-\rho t}\, \frac{C_j(t)^{1-\theta} - 1}{1-\theta}\, dt$$
다국가 Cobb-Douglas
$$Y_j(t) = K_j(t)^\alpha (A H_j(t))^{1-\alpha}$$
다국가 자본 축적
$$\dot K_j(t) = I_j(t) - \delta K_j(t)$$
다국가 예산 제약
$$(1+\tau_j)\, I_j(t) + C_j(t) \leq Y_j(t)$$
★ 다국가 Steady-state 자본
$$K_j^* = \left[\frac{\alpha}{(1+\tau_j)(\rho+\delta)}\right]^{1/(1-\alpha)}$$
★ 다국가 산출 비율
$$\frac{Y(\tau)}{Y(\tau')} = \left(\frac{1+\tau'}{1+\tau}\right)^{\alpha/(1-\alpha)}$$
★ 2-기간 OLG 효용
$$U_t(c_1(t), c_2(t+1)) = u(c_1(t)) + \beta\, u(c_2(t+1))$$
인구 진화 — 1+n 비율
$$L(t) = (1+n)^t\, L(0)$$
★ 생산함수 + 이자율
$$Y(t) = F(K(t), L(t)), \quad 1 + r(t) = R(t) = f'(k(t))$$
임금 — 노동 한계 산출
$$w(t) = f(k(t)) - k(t)\, f'(k(t))$$
★ ★ 2-기간 Euler — 청년-노년 소비
$$u'(c_1(t)) = \beta\, R(t+1)\, u'(c_2(t+1))$$
저축 함수
$$s(t) = s(w(t), R(t+1))$$
총 자본 = 청년 저축
$$S(t) = s(t)\, L(t), \quad K(t+1) = L(t)\, s(w(t), R(t+1))$$
★ OLG 자본 동학 — 1인당
$$k(t+1) = \frac{s(w(t), R(t+1))}{1+n} = \frac{s(f(k(t)) - k(t)f'(k(t)), f'(k(t+1)))}{1+n}$$
★ OLG 정상상태
$$k^* = \frac{s(f(k^*) - k^* f'(k^*), f'(k^*))}{1+n}$$
CRRA OLG 효용
$$U_t(c_1, c_2) = \frac{c_1^{1-\theta} - 1}{1-\theta} + \beta\, \frac{c_2^{1-\theta} - 1}{1-\theta}$$
CRRA Euler — OLG
$$s(t)^{-\theta}\, \beta\, R(t+1)^{1-\theta} = (w(t) - s(t))^{-\theta}$$
CRRA 저축 — 비례형
$$s(t) = w(t)\, \psi(t+1), \quad \psi = \frac{1}{1 + (\beta R)^{-1/\theta}}$$
CRRA 자본 동학
$$k(t+1) = \frac{s(t)}{1+n}$$
CRRA 자본 — 명시
$$k(t+1) = \frac{w(t)}{(1+n)\, \psi(t+1)} = \frac{f(k(t)) - k(t) f'(k(t))}{(1+n)\psi(t+1)}$$
CRRA 정상상태
$$k^* = \frac{f(k^*) - k^* f'(k^*)}{(1+n)[1 + \beta^{-1/\theta} f'(k^*)^{-(1-\theta)/\theta}]}$$
CRRA 정상상태 — 이자율 형식
$$(1+n)\, [1 + \beta^{-1/\theta}\, R^*(\theta - 1)/\theta] = R^*$$
Cobb-Douglas OLG
$$k(t+1) = \frac{(1-\alpha)\, k(t)^\alpha}{(1+n)[1 + \beta^{-1/\theta}\, \alpha\, k(t+1)^{\alpha-1}]^{...}}$$
★ Log Utility OLG
$$U_t(c_1, c_2) = \log c_1(t) + \beta\, \log c_2(t+1)$$
★ Log OLG — 상수 저축률
$$\frac{c_2(t+1)}{c_1(t)} = \beta\, R(t+1), \quad s(t) = \frac{\beta}{1+\beta}\, w(t)$$
Log OLG — 자본 동학
$$k(t+1) = \frac{s(t)}{1+n} = \frac{\beta}{(1+\beta)(1+n)}\, w(t)$$
Bequest motive — 효용
$$\log(c_i(t)) + \beta\, \log(b_i(t))$$
Bequest — 가구 max
$$\max_{c_i, b_i} \log(c_i(t)) + \beta\, \log(b_i(t))$$
Bequest — 예산
$$c_i(t) + b_i(t) \leq y_i(t) \equiv w(t) + R(t)\, b_i(t-1)$$
Bequest — 임금
$$w(t) = f(k(t)) - k(t) f'(k(t))$$
Bequest — 이자율
$$R(t) = f'(k(t))$$
Bequest — 자본 시장
$$k(t+1) = \int_0^1 b_i(t)\, di$$
Bequest — 정책 함수
$$b_i(t) = \frac{\beta}{1+\beta}\, [w(t) + R(t)\, b_i(t-1)]$$
Bequest — 자본 동학
$$k(t+1) = \frac{\beta}{1+\beta}\, [w(t) + R(t)\, k(t)]$$
Bequest — 산출 등식
$$w(t) + R(t)\, k(t) = f(k(t))$$
★ Bequest 정상상태
$$b^* = \frac{\beta\, w^*}{1 + \beta(1 - R^*)}$$
★ ★ Yaari Perpetual Youth — 유한 효용
$$\sum_{t=0}^{\infty} (\beta(1-\nu))^t\, u(c(t))$$
Yaari — 가구 자산 흐름
$$a(t+1 | \tau) = \frac{1+r(t) + \nu}{1-\nu}\, a(t|\tau) - c(t|\tau) + w(t)$$
Yaari — annuity rate
$$\pi(a, t) = -(1-\nu)\, z(a) + \nu\, a, \quad z(a(t)) = \frac{\nu}{1-\nu}\, a(t)$$
Yaari 인구 진화
$$L(t+1) = (1 + n - \nu)\, L(t)$$
Yaari 자산 동학
$$a(t+1|\tau) = \frac{1+r(t)+\nu}{1-\nu}\, a(t|\tau) - c(t|\tau) + w(t)$$
★ Yaari Euler — 사망확률 포함
$$u'(c(t|\tau)) = \beta\, [(1+r(t+1))(1-\nu) + \nu]\, u'(c(t+1|\tau))$$
★ Continuous Yaari — 평생 효용
$$\int_0^{\infty} \exp(-(\rho+\nu)t)\, \log c_i(t)\, dt$$
Continuous 인구 진화
$$\dot L(t) = (n - \nu)\, L(t)$$
Cohort survival distribution
$$L(t|\tau) = n\, \exp(-\nu(t-\tau))$$
Continuous Yaari — 자산
$$\dot a(t|\tau) = (r(t) + \nu)\, a(t|\tau) - c(t|\tau) + w(t)$$
Continuous Yaari — 가격
$$R(t) = f'(k(t)), \quad w(t) = f(k(t)) - k(t) f'(k(t))$$
★ Continuous Yaari — 자본 동학
$$\dot k(t) = f(k(t)) - (n - \nu + \delta)\, k(t) - c(t)$$
Aggregate consumption
$$c(t) = \frac{\int_{-\infty}^t c(t|\tau) L(t|\tau)\, d\tau}{\int_{-\infty}^t L(t|\tau)\, d\tau}$$
Yaari Transversality
$$\lim_{t\to\infty} e^{-\int (r-\rho-\nu) ds}\, a(t|\tau) = 0$$
Cohort consumption — 명시
$$c(t|\tau) = (\rho + \nu)\, [a(t|\tau) + \omega(t)]$$
★ Aggregate consumption — 명시
$$c(t) = (\rho + \nu)\, [a(t) + \omega(t)]$$
★ Aggregate ċ — 동학
$$\dot c(t) = (\rho + \nu)\, [\dot a(t) + \dot \omega(t)]$$
★ Aggregate ċ/c — 명시
$$\frac{\dot c}{c} = f'(k) - \delta - \rho - \frac{(\rho+\nu)\nu\, k}{c}$$
★ Yaari 정상상태 비율
$$\frac{c^*}{k^*} = \frac{(\rho+\nu)\, n}{f'(k^*) - \delta - \rho}$$
Yaari 정상상태 결합
$$\frac{f(k^*)}{k^*} - (n - \nu + \delta) = \frac{(\rho+\nu)\, n}{f'(k^*) - \delta - \rho}$$
Yaari — 코호트 소비 풀이
$$c(t|\tau) = (\rho + \nu)\, [\text{wealth at } \tau]$$
Yaari — 인적 부 동학
$$\frac{\dot c(t)}{c(t)} = f'(k(t)) - \delta - \rho + \zeta(t)$$
Ben-Porath 평생효용
$$\max \int_0^T e^{-(\rho+\nu)t}\, u(c(t))\, dt$$
★ 인적자본 진화방정식
$$\dot{h}(t) = G(t,\, h(t),\, s(t))$$
스콜링 시간 제약
$$s(t) \in S(t) \subset [0,\, 1]$$
평생 임금 수익
$$W(t) = w(t)\, [1-s(t)]\, [h(t) + \omega(t)]$$
★ Ben-Porath 분리정리
$$[\hat c, \hat s, \hat h]^T \text{ jointly optimal} \Leftrightarrow [\hat s, \hat h] \text{ maximize } W \text{ alone}$$
Linear h growth
$$\dot{h}(t) = g_h\, h(t)$$
Linear w growth
$$\dot{w}(t) = g_w\, w(t)$$
전이성 조건
$$g_w + g_h < r + \nu$$
★ 스콜링 PV 식
$$\max_S \eta(S)\, w(0)\, e^{-(r+\nu-g_w)S} \cdot \frac{1}{r+\nu - g_h - g_w}$$
★ Mincer FOC
$$\frac{\eta'(S^*)}{\eta(S^*)} = r + \nu - g_w$$
★ ★ Mincer log-linear
$$\log \eta(S^*) = \text{const} + (r+\nu - g_w)\, S^*$$
★ Mincer wage cross-section
$$\log W(S^*,t) = \text{const} + (r+\nu-g_w)\, S^* + g_w\, t + g_h(t-S^*)$$
★ BP 인적자본 진화
$$\dot{h}(t) = \varphi(s(t)\, h(t)) - \delta_h\, h(t)$$
★ Hamiltonian (BP)
$$H(h,s,\mu) = (1-s(t))\, h(t) + \mu(t)[\varphi(s\, h) - \delta_h\, h]$$
★ FOC: μφ' = 1
$$1 = \mu(t)\, \varphi'(x(t))$$
Costate dynamics
$$\frac{\dot{\mu}(t)}{\mu(t)} = r + \nu + \delta_h - \varphi'(x(t))$$
★ BP steady-state x*
$$x^* = \varphi'^{-1}(r + \nu + \delta_h)$$
BP steady-state h*
$$h^* = \frac{\varphi(x^*)}{\delta_h}$$
★ BP transition path
$$\frac{\dot{x}(t)}{x(t)} = \frac{1}{\varepsilon\, \varphi'(x(t))}\, (r+\nu+\delta_h - \varphi'(x(t)))$$
Two-sector 평생효용
$$\int_0^{\infty} e^{-\rho t}\, u(c(t))\, dt$$
Physical capital evolution
$$\dot{k}(t) = i_k(t) - \delta_k\, k(t)$$
Human capital evolution (2-sector)
$$\dot{h}(t) = i_h(t) - \delta_h\, h(t)$$
★ Resource constraint (2-sector)
$$c(t) + i_k(t) + i_h(t) \leq f(k(t),\, h(t))$$
Hamiltonian (2-sector)
$$H = u(f(k,h) - i_h - i_k) + \mu_k(i_k - \delta_k k) + \mu_h(i_h - \delta_h h)$$
★ K-H separation FOC
$$f_k(k,h) - f_h(k,h) = \delta_k - \delta_h$$
Tax distortion + Cobb-Douglas
$$c + (1+\tau)(i_k + i_h) \leq f(k,h);\quad Y = K^{\alpha} H^{\beta} L^{1-\alpha-\beta}$$
★ Loury 효용 (CES)
$$\eta^{-\eta}(1-\eta)^{-(1-\eta)} c_i^{\eta}\, b_i^{1-\eta} - \gamma(e_i)$$
Loury 효용 (h, b)
$$\eta^{-\eta}(1-\eta)^{-(1-\eta)} c_i^{\eta}\, b_i^{1-\eta} - \gamma\, h_i^a$$
★ Budget constraint (Loury)
$$c_i(t) + b_i(t) \leq m_i(t) = w(t)\, h_i(t) + R(t)\, b_i(t-1)$$
★ Aggregate Y, H, K (Loury)
$$Y(t) = F(K,H);\ H = \int_0^1 h_i\, di,\ K = \int_0^1 b_{i,t-1}\, di$$
Per-effective y
$$y(t) = Y/H = f(\kappa),\ \kappa = K/H$$
Factor prices (Loury)
$$R(t) = f'(\kappa(t)),\quad w(t) = f(\kappa) - \kappa\, f'(\kappa)$$
★ Loury optimal (c, b)
$$c_i = \eta\, m_i,\quad b_i = (1-\eta)\, m_i$$
Reduced utility (m)
$$V_i = m_i - \gamma\, h_i^a$$
★ Schooling FOC
$$a\, w(t) = \gamma'(h_i/a)$$
Equal h_i
$$h_i(t) = h(t) = a\, [\gamma'^{-1}(a\, w(t))]$$
★ OLG dynamic equation
$$\kappa(t+1)\, \gamma'^{-1}[a\, w(\kappa(t+1))] = (1-\eta)\, f(\kappa(t))\, \kappa(t)\, \gamma'^{-1}[a\, w(\kappa(t))]$$
★ Steady-state κ*
$$\kappa^* = (1-\eta)\, f(\kappa^*)$$
Capital allocation
$$\max\, (1-\lambda) \int_0^1 F(k_j, h_i)\, di - R\, k_j$$
Capital FOC (k)
$$(1-\lambda) \int_0^1 \frac{\partial F(\hat{k},\, \hat{h}_i(\hat{k}))}{\partial k}\, di = R^*$$
★ Human capital FOC
$$\lambda\, a_i\, \frac{\partial F(\hat{k},\, \hat{h}_i(\hat{k}))}{\partial h} = \gamma'(\hat{h}_i/a_i)$$
★ Talent-mixed FOC
$$(1-\lambda)[\chi\, F_k(\hat{k}, \hat{h}_1) + (1-\chi) F_k(\hat{k}, \hat{h}_2)] = R^*$$
Talent-specific h FOC
$$\lambda\, a_i\, F_h(\hat{k}, \hat{h}_i) = \gamma'(\hat{h}_i/a_i),\ i=1,2$$
Closing R&D / TFP
$$Y(t) = A(t)\, L$$
AK 평생효용 (CRRA)
$$\int_0^{\infty} e^{-(\rho-n)t}\, \frac{c(t)^{1-\theta} - 1}{1-\theta}\, dt$$
AK 가계 예산 (자산)
$$\dot{a}(t) = (r(t) - n)\, a(t) + w(t) - c(t)$$
No-Ponzi
$$\lim_{t\to\infty} a(t)\, \exp\!\left(-\int_0^t (r(s) - n)\, ds\right) \geq 0$$
★ Consumption Euler (CRRA)
$$\frac{\dot{c}(t)}{c(t)} = \frac{1}{\theta}\, (r(t) - \rho)$$
전이성 조건 (TVC)
$$\lim_{t\to\infty} a(t)\, \exp\!\left(-\int_0^t (r(s) - n)\, ds\right) = 0$$
★ AK 생산함수
$$Y(t) = A\, K(t),\quad y(t) = A\, k(t)$$
★ Inada 위배 + R=A-δ
$$\lim_{k\to\infty} f'(k) = A > 0,\quad R(t) = A - \delta$$
AK 자본 동학
$$\dot{k}(t) = (A - \delta - n)\, k(t) - c(t)$$
★ AK 소비 성장률
$$\frac{\dot{c}(t)}{c(t)} = g_c = \frac{1}{\theta}\, (A - \delta - \rho)$$
Transversality (k 형식)
$$\lim_{t\to\infty} k(t)\, e^{-(A-\delta-n)\, t} = 0$$
AK 소비 명시
$$c(t) = c(0)\, \exp\!\left(\frac{A - \delta - \rho}{\theta}\, t\right)$$
★ AK 균형 안정성 조건
$$A > \rho + \delta > (1-\theta)(A-\delta) + \theta\, n + \delta$$
AK 자본 ODE
$$\dot{k}(t) = (A - \delta - n)\, k(t) - c(0)\, \exp(g_c\, t)$$
AK 자본 일반해
$$k(t) = \kappa\, e^{(A-\delta-n) t} + \frac{c(0)/\theta}{(A-\delta)(\theta-1)/\theta + \rho/\theta - n}\, e^{g_c t}$$
AK k(t) 균형 (κ=0)
$$k(t) = \frac{c(0)/\theta}{(A-\delta)(\theta-1)/\theta + \rho/\theta - n}\, e^{g_c t}$$
AK c(0) 명시
$$c(0) = [(A-\delta)(\theta-1)/\theta + \rho/\theta - n]\, k(0)$$
★ AK 저축률
$$s = \frac{\dot{K}/K + \delta}{A} = \frac{A - \rho + \theta n + (\theta-1)\delta}{\theta A}$$
Government distortion 가계
$$\dot{a}(t) = (1-\tau)\, r(t)\, a(t) + w(t) - c(t) - \tau_w$$
★ Distorted growth rate
$$g = \frac{(1-\tau)(A-\delta) - \rho}{\theta}$$
Distorted saving rate
$$s = \frac{(1-\tau)A - \rho + \theta n - (1-\tau-\theta)\delta}{\theta A}$$
★ 2-자본 신고전 생산
$$Y(t) = F(K(t), H(t))$$
Lucas 가계 예산
$$\dot{a}(t) = r(t)\, a(t) + w(t)\, h(t) - c(t) - i_h(t)$$
Lucas 인적자본 진화
$$\dot{h}(t) = i_h(t) - \delta_h\, h(t)$$
Factor prices (Lucas)
$$R(t) = f'(k(t)),\quad w(t) = f(k(t)) - k(t)\, f'(k(t))$$
★ Lucas 자본 평형
$$\mu_a(t) = \mu_h(t) = \mu(t),\ w(t) = R(t) - \delta_k + \delta_h$$
Lucas 균형 조건
$$f'(k) - \delta_k = f(k) - k\, f'(k) - \delta_h$$
Cobb-Douglas C sector
$$C(t) = B\, K_C(t)^{\alpha}\, L_C(t)^{1-\alpha}$$
★ I sector (AK)
$$\dot{K}(t) = I(t) - \delta\, K(t),\quad I(t) = A\, K_I(t)$$
Capital allocation
$$K_C(t) = (1-\kappa(t))\, K(t),\ K_I(t) = \kappa(t)\, K(t)$$
★ Investment good price dynamics
$$\frac{\dot{p}_I(t)}{p_I(t)} = -(1-\alpha)\, g_K$$
★ 2-sector Fisher equation
$$r_C(t) = \frac{r_I(t)}{p_I(t)} + \frac{\dot{p}_I(t)}{p_I(t)} - \frac{\dot{p}_C(t)}{p_C(t)}$$
★ C 성장률 (Rebelo 2-sector)
$$g_C = \frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta}\, (A - \delta - (1-\alpha)\, g_K - \rho)$$
★ K 균형 성장률
$$g_K^* = \frac{A - \delta - \rho}{1 - \alpha(1-\theta)}$$
★ C 균형 성장률
$$g_C^* = \alpha\, \frac{A - \delta - \rho}{1 - \alpha(1-\theta)}$$
★ Romer 86 firm production
$$Y_i(t) = F(K_i(t),\, A(t)\, L_i(t))$$
★ Romer 86 spillover
$$A(t) = B\, K(t)$$
Aggregate Y (Romer 86)
$$Y(t) = \tilde{f}(L)\, K(t)$$
★ Romer 86 R = const
$$R = \tilde{f}(L) - L\, \tilde{f}'(L)$$
★ Romer 86 g* (decentralized)
$$g_C^* = \frac{1}{\theta}\, (\tilde{f}(L) - L\, \tilde{f}'(L) - \delta - \rho)$$
Positive growth condition
$$\tilde{f}(L) - L\, \tilde{f}'(L) - \delta - \rho > 0$$
Boundedness condition (Romer 86)
$$(1-\theta)(\tilde{f}(L) - L\, \tilde{f}'(L) - \delta) < \rho$$
★ Innovation surplus SI
$$S^I = \int_{\psi/\lambda^{-1}}^{\psi} D(p)\, dp - \mu = \int [D(p) - D(\psi)]\, dp + D(\psi)\, \lambda^{-1}(\lambda - 1)\, \psi - \mu$$
Monopoly price (CES)
$$p^M \equiv \lambda^{-1}\psi (1-\varepsilon)\, D(p^M)^{-1}$$
★ Monopoly profit (innov)
$$\hat{\pi}^I_1 = D(p^M)(p^M - \lambda^{-1}\psi) - \mu$$
Innovation profit floor
$$\pi^I_1 = D(\psi)\, \lambda^{-1}(\lambda - 1)\, \psi - \mu < \hat{\pi}^I_1$$
★ Social innovation surplus
$$S^I_1 = D(p^M)(p^M - \lambda^{-1}\psi) + \int_{p^M}^{\psi} D(p)\, dp - \mu$$
Monopoly price closed-form
$$\hat{p}^M \equiv \frac{\varepsilon}{\varepsilon - 1}\, \lambda^{-1}\psi$$
★ DS 효용 함수
$$U(c_1, \ldots, c_N, y) = u(C, y)$$
★ ★ CES aggregator
$$C \equiv \left(\sum_{i=1}^N c_i^{(\varepsilon-1)/\varepsilon}\right)^{\varepsilon/(\varepsilon-1)}$$
Budget constraint
$$\sum_{i=1}^N p_i\, c_i + y \leq m$$
★ DS 수요 — 단일 재화
$$\frac{c_i}{C} \cdot \varepsilon^{-1} = \frac{p_i}{P}$$
★ ★ DS 가격지수
$$P \equiv \left(\sum_{i=1}^N p_i^{1-\varepsilon}\right)^{1/(1-\varepsilon)}$$
Two-stage budget
$$u(C, y)\ \text{s.t.}\ P\, C + y \leq m$$
★ Two-stage FOC
$$\frac{\partial u/\partial y}{\partial u/\partial C} = \frac{1}{P},\quad y = g(P, m),\ C = (m - g(P, m))/P$$
★ 기업 가격 결정
$$\max_{p_i \geq 0} \left(\frac{p_i}{P}\right)^{-\varepsilon} C\, (p_i - \psi)$$
★ ★ DS 등가 가격
$$p_i = p = \frac{\varepsilon}{\varepsilon - 1}\, \psi\quad \forall i = 1, \ldots, N$$
★ DS 가격지수 (대칭)
$$P = N^{-1/(\varepsilon-1)} \cdot \frac{\varepsilon}{\varepsilon - 1}\, \psi$$
★ Continuum DS price index
$$P = \left(\int_0^N p_i^{1-\varepsilon}\, di\right)^{1/(1-\varepsilon)}$$
Romer 1990 expanding-variety
$$Y(t) = \frac{1}{1-\beta}\, \int_0^{N(t)} x(\nu, t)^{1-\beta}\, d\nu \cdot L^{\beta}$$
Romer 생산 (alt)
$$Y(t) = \frac{1}{1-\beta}\, \int_0^{N(t)} x(\nu, t)^{1-\beta}\, d\nu \cdot L^{\beta}$$
Romer 자원제약
$$C(t) + X(t) + Z(t) \leq Y(t)$$
R&D 진화 (Romer)
$$\dot N(t) = \eta\, Z(t)$$
R&D production function
$$\dot N(t) = \eta\, Z(t)$$
Romer 중간재 수요
$$x(\nu, t) = p_x(\nu, t)^{-1/\beta}\, L$$
Variety value (general)
$$V(\nu, t) = \int_t^{\infty} \exp\!\left(-\int_t^s r(s')\, ds'\right)\, \pi(\nu, s)\, ds$$
Asset pricing (V)
$$r(t)\, V(\nu, t) - \dot V(\nu, t) = \pi(\nu, t)$$
13.1.2 헤더
$$\text{Equilibrium characterization (placeholder)}$$
Monopolist optimal price
$$p^x = \frac{\psi}{1 - \beta}$$
Romer 독점이윤
$$\pi(\nu, t) = \beta\, L$$
Romer aggregate Y
$$Y(t) = \frac{1}{1-\beta}\, N(t)\, L$$
Romer 임금
$$w(t) = \frac{\beta}{1-\beta}\, N(t)$$
★ R&D 자유진입 조건
$$\eta\, V(\nu, t) \leq 1,\ Z(\nu, t) \geq 0,\ (\eta\, V - 1)\, Z = 0$$
Innovation value (V)
$$V(\nu, t) = \int_t^{\infty} e^{-\int_t^s r(z) dz}\, \pi(\nu, s)\, ds$$
Consumption Euler
$$\frac{\dot C(t)}{C(t)} = \frac{1}{\theta}\, (r(t) - \rho)$$
TVC (Romer)
$$\lim_{t\to\infty} \exp\!\left(-\int_0^t r(s)\, ds\right) \int_0^{N(t)} V(\nu, t)\, d\nu = 0$$
균형 변수 정의
$$[C(t), X(t), Z(t), N(t)]_{t=0}^{\infty}\ \text{equilibrium tuple}$$
★ Romer no-arbitrage
$$\frac{\eta\, \beta\, L}{r^*} = 1$$
Romer growth (alt form)
$$g^* = \frac{1}{\theta}\, (\eta\, \beta\, L - \rho)$$
Romer wellposedness
$$\eta\, \beta\, L > \rho \quad \text{and} \quad (1-\theta)\, \eta\, \beta\, L < \rho$$
Romer SP problem
$$C(t) + Z(t) \leq \frac{1}{1-\beta}\, \int_0^{N(t)} x^{1-\beta}\, d\nu \cdot L^{\beta} - \int_0^{N(t)} \psi\, x\, d\nu$$
★ SP growth rate
$$\frac{\dot{C}^S(t)}{C^S(t)} = \frac{1}{\theta}\, (\eta\, (1-\beta)^{-1/\beta}\, \beta\, L - \rho)$$
Pareto markup formula
$$p_x = \gamma\, \psi$$
BGP growth rate (Romer)
$$g^* = \frac{\eta\, L\, \beta - \rho}{\theta}$$
Labor allocation (R&D + production)
$$L_R(t) + L_E(t) \leq L,\ Y(t) = \frac{1}{1-\beta}\, N(t)\, L_E(t)$$
Profit (variant)
$$\pi(t) = \beta\, L_E(t)$$
Labor R&D condition
$$\eta\, N(t)\, V(\nu, t) = w(t)$$
η N β L_E condition
$$\frac{\eta\, N(t)\, \beta\, L_E(t)}{r^*} = \frac{\beta}{1-\beta}\, N(t)$$
★ Lab equipment growth
$$\frac{\dot{C}}{C} = \frac{1}{\theta}\, ((1-\beta)\, \eta\, L^*_E - \rho) \equiv g^*$$
★ L*_E 균형
$$L^*_E = \frac{\theta\, \eta\, L + \rho}{(1-\beta)\, \eta + \theta\, \eta}$$
Lab equipment wellposed
$$(1-\theta)(1-\beta)\, \eta\, L^*_E < \rho < (1-\beta)\, \eta\, L^*_E$$
Jones lifetime utility
$$\int_0^{\infty} e^{-(\rho-n)t}\, \frac{c(t)^{1-\theta} - 1}{1-\theta}\, dt$$
★ ★ Jones R&D 함수
$$\dot N(t) = \eta\, N(t)^{\varphi}\, L_R(t)$$
Jones labor
$$L_E(t) + L_R(t) \leq L(t)$$
Jones no-arbitrage
$$\frac{\eta\, N(t)^{\varphi}\, \beta\, L_E(t)}{r^* - n} = w(t)$$
★ Jones g*_N
$$g^*_N \equiv \frac{\dot N(t)}{N(t)} = \frac{n}{1-\varphi}$$
★ Jones g*_C
$$g^*_C = g^*_N = \frac{n}{1-\varphi}$$
GH lifetime utility
$$\int_0^{\infty} e^{-\rho t}\, \log C(t)\, dt$$
★ GH CES consumption
$$C(t) \equiv \left(\int_0^{N(t)} c(\nu, t)^{(\varepsilon-1)/\varepsilon}\, d\nu\right)^{\varepsilon/(\varepsilon-1)}$$
GH 생산함수 (단순)
$$y(\nu, t) = l(\nu, t)$$
GH R&D function
$$\dot N(t) = \eta\, N(t)\, L_R(t)$$
Resource (GH)
$$\int_0^{N(t)} l(\nu, t)\, d\nu + L_R(t) \leq L$$
★ GH 수요 (CES)
$$c(\nu, t) = p_c(\nu, t)^{-\varepsilon}\, \left(\int p_c(\nu')^{1-\varepsilon}\, d\nu'\right)^{-\varepsilon/(1-\varepsilon)}$$
GH normalization
$$\left(\int_0^{N(t)} p_c(\nu)^{1-\varepsilon}\, d\nu\right)^{1/(1-\varepsilon)} = 1$$
GH consumption Euler
$$\frac{\dot C}{C} = r(t) - \rho$$
GH symmetric solution
$$p_c = \frac{\varepsilon}{\varepsilon - 1}\, w(t),\ c(\nu, t) = l(\nu, t) = \frac{L_E}{N(t)}$$
GH 이윤
$$\pi(\nu, t) = \frac{1}{\varepsilon - 1}\, \frac{L_E(t)}{N(t)}\, w(t)$$
GH 통합 V
$$V(t) = N(t)\, \frac{\varepsilon}{\varepsilon-1}\, c(t) = L_E(t)\, \frac{N(t)}{\varepsilon-1}$$
GH free entry
$$\eta\, N(t)\, V(t) = w(t)$$
GH π = (1/(ε-1)) η V
$$\pi(t) = \frac{1}{\varepsilon - 1}\, L_E(t)\, \eta\, V(t)$$
GH BGP V
$$V(t) = \frac{\pi(t)}{r^* - g^* + g_N}$$
★ ★ GH L*_R
$$L^*_R = \frac{\eta\, L - (\varepsilon - 1)\, \rho}{\eta\, \varepsilon}$$
★ ★ ★ GH 성장률
$$g^* = \frac{g_N}{\varepsilon - 1} = \frac{\eta\, L - (\varepsilon - 1)\, \rho}{(\varepsilon - 1)\, \varepsilon}$$
Aghion-Howitt creative destruction
$$Y(t) = A(t)\, x(t)^{\alpha}\, L^{1-\alpha}$$
균형 변수 (AH)
$$[C(t), X(t), Z(t), N(t)]_{\nu, t=0}^{\infty}$$
Quality jump (innovation)
$$A(t) = \lambda^{n(t)}$$
Limit pricing
$$p_x(\nu, t \mid q) = q(\nu, t)$$
수요 = L
$$x(\nu, t \mid q) = L$$
이윤 (general q)
$$\pi(\nu, t \mid q) = (1-\beta)\, q(\nu, t)\, L$$
Aggregate Y (Q form)
$$Y(t) = \frac{1}{1-\beta}\, Q(t)\, L$$
Poisson innovation arrival
$$\Pr(\text{innovation in } [t, t+dt)) = \mu\, Z(t)\, dt$$
Aggregate X
$$X(t) = (1-\beta)\, Q(t)\, L$$
Aggregate w
$$w(t) = \frac{\beta}{1-\beta}\, Q(t)$$
★ HJB-style asset eq
$$r(t)\, V(\nu, t \mid q) - \dot V = \pi(\nu, t \mid q) - z(\nu, t \mid q)\, V(\nu, t \mid q)$$
Free-entry (AH)
$$\eta\, V(\nu, t \mid q) \leq \lambda^{-1}\, q(\nu, t)$$
Consumption Euler
$$\frac{\dot C}{C} = \frac{1}{\theta}\, (r(t) - \rho)$$
TVC (AH)
$$\lim_{t\to\infty} \exp\!\left(-\int_0^t r(s)\, ds\right) \int_0^1 V(\nu, t \mid q)\, d\nu = 0$$
V proportional to q
$$V(\nu, t \mid q) = q(\nu, t)\, v(t)$$
★ V BGP form
$$V(\nu, t \mid q) = \frac{\beta\, q(\nu, t)\, L}{r^* + z^*}$$
★ AH no-arb
$$r^* + z^* = \lambda\, \eta\, \beta\, L$$
Schumpeterian BGP
$$g^* = \mu\, \frac{Z^*}{L}\, \ln \lambda$$
Y growth = Q growth
$$\frac{\dot Y(t)}{Y(t)} = \frac{\dot Q(t)}{Q(t)}$$
★ g* = (λ-1) z*
$$g^* = (\lambda - 1)\, z^*$$
★ ★ AH g*
$$g^* = \frac{\lambda\, \eta\, \beta\, L - \rho}{\theta + (\lambda - 1)^{-1}}$$
Pareto x
$$x^S(\nu, t \mid q) = \psi^{-1/\beta}\, L = (1-\beta)^{-1/\beta}\, L$$
Pareto Y - X
$$\tilde{Y}^S(t) \equiv Y^S(t) - X^S(t) = (\text{cleaner expr})$$
★ Pareto Q evolution
$$\dot Q^S(t) = \eta\, (\lambda - 1)\, Z^S(t)$$
Pareto Hamiltonian
$$\hat H(Q^S, C^S, \mu^S) = \frac{(C^S)^{1-\theta} - 1}{1-\theta} + \mu^S\, [\eta(\lambda - 1)(1-\beta)^{-1/\beta} \cdots]$$
Quality ladder Y (single)
$$Y(t) = \frac{1}{1-\beta}\, x(t \mid q)^{1-\beta}\, q(t)^{\beta}\, L^{\beta}$$
★ AH L*_R
$$L^*_R = \frac{\lambda(1-\beta)\, \eta\, L - \rho}{\cdots}$$
Cleansing recessions
$$\text{Recessions} \Rightarrow \uparrow Z(t) \Rightarrow \uparrow A(t+\tau)$$
AH → modern firm dynamics
$$\eta(L_R(q))\, V(\lambda q) = w(q)$$
Two-firm Aghion duopoly
$$\eta(L^1_R)\, V_2(\lambda q) = w(q),\ \eta(L^2_R)\, V_1(\lambda q) = w(q)$$
★ Aghion Y (continuum)
$$Y(t) = \frac{1}{1-\beta}\, \int_0^1 q(\nu, t)^{\beta}\, x(\nu, t \mid q)^{1-\beta}\, d\nu \cdot L^{\beta}$$
η decreasing returns
$$\lim_{z \to \infty} \eta(z) = 0,\ \lim_{z \to 0} \eta(z) = \infty$$
Total Z
$$Z(t) = \int_0^1 [z(\nu, t) + \hat z(\nu, t)]\, d\nu$$
Demand (Aghion)
$$x(\nu, t \mid q) = p_x(\nu, t \mid q)^{-1/\beta}\, q(\nu, t)\, L$$
Step-by-step κ
$$\kappa \geq \frac{1}{1-\beta}\, \frac{1}{1-\beta} \cdots$$
Limit price
$$p_x(\nu, t \mid q) = 1$$
x = q L
$$x(\nu, t \mid q) = q\, L$$
★ Aghion HJB
$$r(t)\, V(\nu, t \mid q) - \dot V = \max_{z \geq 0} \{\pi - z V\}$$
Free-entry (general)
$$\eta(\hat z)\, V(\nu, t \mid \kappa q) \leq q(\nu, t)$$
Innovator value diff
$$\varphi[V(\nu, t \mid \lambda q) - V(\nu, t \mid q)] \leq q(\nu, t)$$
Innovator equality
$$\varphi[V(\lambda q) - V(q)] = q$$
Aghion firm description
$$\text{firm with quality } q,\ \text{leader value } V(q),\ \text{follower } V(q/\lambda)$$
Aghion BGP V
$$V(q) = \frac{q}{\kappa\, \eta(\hat z)}$$
Aghion BGP V (alt)
$$V(q) = \frac{\beta L\, q}{r^* + \hat z^*\, \eta(\hat z^*)}$$
★ Aghion no-arb
$$r^* = \varphi(\lambda - 1)\, \beta L - \hat z^*\, \eta(\hat z^*)$$
★ ★ Aghion g*
$$g^* = \frac{1}{\theta}\, [\varphi(\lambda - 1)\, \beta L - \hat z^*\, \eta(\hat z^*) - \rho]$$
Q evolution (transition)
$$Q(t + dt) = \lambda\, \varphi z(t)\, dt\, Q(t) + \kappa\, \hat z(t)\, \eta(\hat z(t))\, dt\, Q(t)$$
z-bar definition
$$z(t) \equiv \frac{1}{Q(t)}\, \int_0^1 z(\nu, t \mid q)\, q(\nu, t)\, d\nu$$
★ Q growth (Aghion)
$$\frac{\dot Q}{Q} = (\lambda - 1)\, \varphi z(t) + (\kappa - 1)\, \hat z(t)\, \eta(\hat z(t))$$
★ Aghion BGP g*
$$g^* = (\lambda - 1)\, \varphi z^* + (\kappa - 1)\, \hat z^*\, \eta(\hat z^*)$$
Aghion wellposed
$$\varphi(\lambda - 1)\beta L - (\theta(\kappa - 1) + 1)\, \hat z^*\, \eta(\hat z^*) > \rho$$
Quality jump prob
$$x(\nu, t + dt \mid q) = \begin{cases} \lambda x & \text{w/p } \varphi z\, dt \\ x & \text{else} \end{cases}$$
Patent fee + V
$$\eta(\hat z^*)\, V(\kappa q) = (1 + \tau_e)\, q,\ V(q) = \frac{q\, (1 + \tau_e)}{\kappa\, \eta(\hat z^*)}$$
Tax distortion (1+τe)/(1+τi)
$$\frac{\varphi(\lambda - 1)\, (1 + \tau_e)}{\kappa\, \eta(\hat z^*)\, (1 + \tau_i)} = 1$$
KK 평생효용
$$\int_0^{\infty} e^{-\rho t}\, \log C(t)\, dt$$
★ Log Euler g(t)=r(t)-ρ
$$g(t) \equiv \frac{\dot C}{C} = \frac{\dot Y}{Y} = r(t) - \rho$$
★ Cobb-Douglas Y (KK)
$$Y(t) = \exp\!\left(\int_0^1 \log y(\nu, t)\, d\nu\right)$$
Demand y_i
$$y(\nu, t) = \frac{Y(t)}{p_y(\nu, t)}$$
Production y_i = q l_i
$$y_i(\nu, t) = q_i(\nu, t)\, l_i(\nu, t)$$
Marginal cost
$$MC_i(\nu, t) = \frac{w(t)}{q_i(\nu, t)}$$
★ KK Limit price
$$p^y_i(\nu, t) = \frac{w(t)}{q_{-i}(\nu, t)}$$
★ KK firm output
$$y_i(\nu, t) = \frac{q_{-i}(\nu, t)}{w(t)}\, Y(t)$$
Innovation prob = h
$$z_i(\nu, t) = (h_i(\nu, t))$$
Innovation cost h_bar
$$\bar h \text{ defined as average R\&D}$$
★ KK quality ladder
$$q_{-i}(\nu, t) = \lambda^{n_{-i}(\nu, t)}$$
Markov n evolution
$$n(\nu, t + dt) = \begin{cases} n + 1 & \text{w/p } z_i\, dt \\ 0 & \text{w/p } z_{-i}\, dt + \kappa \end{cases}$$
Profit (KK)
$$\pi_i(\nu, t) = (p^y_i - MC_i)\, y_i = \frac{w(t)}{q_{-i}} \cdot \frac{Y(t)}{p^y_i}$$
ξ definition
$$\xi_n(t) \equiv z_n(t),\ p^y_i,\ y_i,\ \xi_{-n}(t) \equiv z_{-n}(t)$$
★ Aggregate R&D
$$h_n(t) = G(z_n(t)) + G(z_{-n}(t))$$
Resource constraint
$$1 \geq \sum_{n=0}^{\infty} \mu_n(t)\, [\omega(t)\, \lambda^{-n} + G(z_n) + G(z_{-n})]$$
Wage share ω
$$\omega(t) \equiv \frac{w(t)}{Y(t)}$$
log Q definition
$$\log Q(t) \equiv \int_0^1 \log q(\nu, t)\, d\nu$$
Wage formula
$$w(t) = Q(t)\, \lambda^{-\sum n\, \mu_n(t)}$$
★ KK steady-state HJB
$$r(t)\, V_n(t) - \dot V_n = \pi_n + z_n[V_{n+1} - V_n] + [z_{-n} + \kappa][V_0 - V_n]$$
v_n definition
$$v_n(t) \equiv V_n(t)/Y(t)$$
v_n max problem
$$\rho v_n = \max_{z_n} \{(1 - \lambda^{-n}) - \omega^*\, G(z_n) + z_n[v_{n+1} - v_n] - [z^*_{-n} + \kappa][v_n - v_0]\}$$
v_0 max
$$\rho v_0 = \max_{z_0} \{-\omega^*\, G(z_0) + z_0[v_1 - v_0] + z^*_0[v_{-1} - v_0]\}$$
v_-1 max
$$\rho v_{-1} = \max_{z_{-1}} \{-\omega^*\, G(z_{-1}) + [z_{-1} + \kappa][v_0 - v_{-1}]\}$$
z*_n FOC
$$z^*_n = \max\!\left\{G'^{-1}\!\left(\frac{v_{n+1} - v_n}{\omega^*}\right), 0\right\}$$
z*_-1 FOC
$$z^*_{-1} = \max\!\{G'^{-1}((v_0 - v_{-1})/\omega^*), 0\}$$
z*_0 FOC
$$z^*_0 = \max\!\{G'^{-1}((v_1 - v_0)/\omega^*), 0\}$$
Stationary distribution n+1
$$(z^*_{n+1} + z^*_{-1} + \kappa)\, \mu^*_{n+1} = z^*_n\, \mu^*_n$$
Stationary distribution n=1
$$(z^*_1 + z^*_{-1} + \kappa)\, \mu^*_1 = 2\, z^*_0\, \mu^*_0$$
Stationary distribution n=0
$$2\, z^*_0\, \mu^*_0 = z^*_{-1} + \kappa$$
Resource binding
$$1 \geq \sum_{n=0}^{\infty} \mu^*_n\, [\omega^*\, \lambda^{-n} + G(z^*_n) + G(z^*_{-n})]$$
★ ★ KK growth rate
$$g^* = \log \lambda\, \cdot \left(2\mu^*_0\, z^*_0 + \sum_{n} \mu^*_n\, z^*_n\right)$$
z* monotone
$$z^*_{n+1} \leq z^*_n\quad \forall n$$
Aux: ρ-bar Bellman
$$\bar\rho\, v_n = \max_{z_n} \{(1 - \lambda^{-n}) - \omega^*\, G(z_n) + z_n[v_{n+1} - v_n]\}$$
ρ-bar definition
$$\bar\rho \equiv \rho + z^*_{-1} + \kappa$$
ρ-bar δ inequality
$$\bar\rho\, \delta_{n+1} \leq \lambda^{-n}(1 - \lambda^{-1}) + z^*_{n+1}(\delta_{n+2} - \delta_{n+1})$$
Closing convexity check
$$v_0 > 0,\ v_{-1} + v_1 - 2\, v_0 > 0$$
한계생산성 비 (skill premium)
$$\frac{MP_H}{MP_L} = \gamma\, \left(\frac{A_H(t)}{A_L(t)}\right)^{\frac{\sigma-1}{\sigma}}\, \left(\frac{H(t)}{L(t)}\right)^{-\frac{1}{\sigma}}$$
DTC 평생효용
$$\int_0^{\infty} e^{-\rho t}\, \frac{C(t)^{1-\theta} - 1}{1-\theta}\, dt$$
CES 통합 산출 (skilled + unskilled)
$$Y(t) = \bigl[\gamma_L\, Y_L(t)^{\frac{\varepsilon-1}{\varepsilon}} + \gamma_H\, Y_H(t)^{\frac{\varepsilon-1}{\varepsilon}}\bigr]^{\frac{\varepsilon}{\varepsilon-1}}$$
자원제약
$$C(t) + X(t) + Z(t) \leq Y(t)$$
Skilled good production (expanding-variety)
$$Y_L(t) = \frac{1}{1-\beta}\, \int_0^{N_L(t)} x_L(\nu, t)^{1-\beta}\, d\nu \cdot L^{\beta}$$
Y_H 생산
$$Y_H(t) = \frac{1}{1-\beta}\, \int_0^{N_H(t)} x_H(\nu, t)^{1-\beta}\, d\nu \cdot H^{\beta}$$
Markets H sector
$$x^p_H(\nu, t)\ \forall \nu \in [0, N_H(t)]\ \text{markup pricing}$$
Variety value V_f
$$V_f(\nu, t) = \int_t^{\infty} \exp\!\left(-\int_t^s r(s')\, ds'\right)\, \pi_f(\nu, s)\, ds$$
Profit π_f
$$\pi_f(\nu, t) \equiv p^x_f\, x_f - \psi\, x_f$$
Price 정규화 (CES)
$$\bigl[\gamma_L^{\varepsilon}\, p_L(t)^{1-\varepsilon} + \gamma_H^{\varepsilon}\, p_H(t)^{1-\varepsilon}\bigr]^{\frac{1}{1-\varepsilon}} = 1$$
Equilibrium tuple
$$[C(t), X(t), Z(t), N_L(t), N_H(t)]_{t=0}^{\infty}$$
H Sector Profit Max
$$\max_{H,\, x_H} p_H(t)\, Y_H(t) - w_H(t)\, H - \int x_H \cdot p^x_H\, d\nu$$
L Sector demand
$$x_L(\nu, t) = \left(\frac{p_L(t)}{p^x_L}\right)^{1/\beta}\, L$$
H Sector demand
$$x_H(\nu, t) = \left(\frac{p_H(t)}{p^x_H}\right)^{1/\beta}\, H$$
Limit pricing both
$$p^x_L = p^x_H = 1,\ x_L = p_L^{1/\beta}\, L,\ x_H = p_H^{1/\beta}\, H$$
Y_L closed
$$Y_L(t) = \frac{1}{1-\beta}\, p_L(t)^{(1-\beta)/\beta}\, N_L(t)\, L$$
Y_H closed
$$Y_H(t) = \frac{1}{1-\beta}\, p_H(t)^{(1-\beta)/\beta}\, N_H(t)\, H$$
★ Relative price p
$$p(t) \equiv \frac{p_H(t)}{p_L(t)} = \gamma\, \left(\frac{Y_H}{Y_L}\right)^{-1/\varepsilon}$$
★ Skill premium ω
$$\omega(t) \equiv \frac{w_H(t)}{w_L(t)} = p(t)^{1/\beta}\, \frac{N_H}{N_L} = \gamma^{\varepsilon\sigma}\, \cdots$$
Equilibrium R&D allocation (skill bias)
$$\frac{N_H(t)}{N_L(t)} = \eta^*\, \left(\frac{H}{L}\right)^{\sigma}$$
Free-entry both H/L
$$\eta_H\, V_H \leq 1,\ Z_H \geq 0,\ \eta_H V_H = 1\ \text{if } Z_H > 0$$
Consumption Euler
$$\frac{\dot C}{C} = \frac{1}{\theta}\, (r(t) - \rho)$$
TVC
$$\lim_{t\to\infty} \exp\!\left(-\int_0^t r(s)\, ds\right) (N_L V_L + N_H V_H) = 0$$
BGP V_L, V_H
$$V_L = \frac{\beta\, p_L^{1/\beta}\, L}{r^*},\ V_H = \frac{\beta\, p_H^{1/\beta}\, H}{r^*}$$
★ V_H/V_L ratio
$$\frac{V_H}{V_L} = \gamma^{\varepsilon\sigma}\, \left(\frac{N_H}{N_L}\right)^{-1/\sigma}\, \left(\frac{H}{L}\right)^{\sigma}$$
σ vs ε relationship
$$\sigma > 1 \Leftrightarrow \varepsilon > 1\ \text{(strong induced innovation)}$$
★ ★ N_H/N_L equilibrium
$$\left(\frac{N_H}{N_L}\right)^* = \eta^{\sigma}\, \gamma^{\varepsilon}\, \left(\frac{H}{L}\right)^{\sigma - 1}$$
★ DTC growth rate
$$g^* = \frac{1}{\theta}\, [\beta\, \gamma^{\varepsilon}\, H(\eta_H H)^{\sigma-1} + \gamma^{\varepsilon}\, L(\eta_L L)^{\sigma-1} - \rho]$$
Aggregate growth
$$g^* = \frac{1}{\theta}\, [\beta\, \gamma^{\varepsilon}\, H(\eta_H H)^{\sigma-1} + \gamma^{\varepsilon}\, L(\eta_L L)^{\sigma-1} - \rho]$$
Skilled wage premium (long-run)
$$\frac{w_H}{w_L} = \eta\, \left(\frac{H}{L}\right)^{\sigma-2}$$
★ Spillover R&D function
$$\dot N_L = \eta_L\, N_L^{(1+\delta)/2}\, N_H^{(1-\delta)/2}\, S_L(t)$$
Scientist constraint
$$S_L(t) + S_H(t) \leq S$$
Free-entry L (spillover)
$$\eta_L\, N_L^{(1+\delta)/2}\, N_H^{(1-\delta)/2}\, V_L \leq w_S(t)$$
Free-entry H (spillover)
$$\eta_H\, N_L^{(1-\delta)/2}\, N_H^{(1+\delta)/2}\, V_H \leq w_S(t)$$
Profit ratio identity
$$\eta_L\, N_L^{\delta}\, \pi_L = \eta_H\, N_H^{\delta}\, \pi_H$$
★ N_H/N_L (spillover)
$$\left(\frac{N_H}{N_L}\right)^* = \eta^{\sigma/(1-\delta\sigma)}\, \gamma^{(1-\delta)\varepsilon/(1-\delta\sigma)}\, \cdots$$
★ ω* spillover
$$\omega^* \equiv \frac{w_H^*}{w_L^*} = \eta^{(\sigma-1)/(1-\delta\sigma)}\, \gamma^{(1-\delta)\varepsilon/\beta(1-\delta\sigma)}\, \cdots$$
S* allocation
$$\eta_H\, N_H^{\delta - 1}\, S_H = \eta_L\, N_L^{\delta - 1}\, S_L,\ \eta^{(1-\sigma)/(1-\delta\sigma)}\, \cdots$$
Wellposed
$$(1 - \theta)\, \eta_L \eta_H (N_H/N_L)^{\delta} < \rho$$
★ Jones-style spillover
$$\dot N_L = \eta_L\, N_L^{\lambda}\, S_L,\ \dot N_H = \eta_H\, N_H^{\lambda}\, S_H$$
★ Population-driven g*
$$g^* = \frac{n}{1 - \lambda}$$
Innovation balance
$$\eta_L\, N_L^{\lambda}\, \pi_L = \eta_H\, N_H^{\lambda}\, \pi_H$$
★ N_H/N_L (Jones)
$$\left(\frac{N_H}{N_L}\right)^* = \eta^{\sigma/(1-\lambda\sigma)}\, \gamma^{\varepsilon/(1-\lambda\sigma)}\, (H/L)^{\sigma-1}/(1-\lambda\sigma)$$
★ ω* (Jones)
$$\omega^* = \eta^{(\sigma-1)/(1-\lambda\sigma)}\, \gamma^{\varepsilon/\beta(1-\lambda\sigma)}\, (H/L)^{(\sigma-2)/(1-\lambda\sigma)}$$
Closing prop
$$N_H/N_L\ \text{rises with } H,\ \omega^*\ \text{can rise or fall}$$
K + L augmenting R&D
$$\frac{\dot N_L}{N_L} = \eta_L\, S_L,\ \frac{\dot N_K}{N_K} = \eta_K\, S_K,\ r(t)\, K(t)\, w(t) = \cdots$$
★ Differential growth
$$\frac{\dot N_L}{N_L} - \frac{\dot N_K}{N_K} = s_K$$
★ Interest rate (K-L)
$$r(t) = \beta\, \gamma_K\, N_K\, [\gamma_L\, (N_L L / (N_K K))^{(\sigma-1)/\sigma} + \gamma_K]^{1/\sigma}$$
Leontief firm
$$Y_i(t) = \min\{b_i\, K(t),\, a_i\, L(t)\}$$
★ Pareto draw G(b,a)
$$G(b, a) \equiv \Pr(a_i \geq a, b_i \geq b) = b^{-\beta/\gamma_b}\, a^{-\alpha/\gamma_a}$$
Output distribution H(y)
$$H(y) \equiv \Pr[\tilde Y_i \leq y] = 1 - \Pr[a_i\, L \geq y, b_i\, K \geq y]$$
Y aggregator
$$\tilde Y(t; N(t)) \equiv \max_{i = 1, \ldots, N(t)} \min\{b_i\, K(t),\, a_i\, L(t)\}$$
★ Aggregate distribution
$$\Pr[\tilde Y \leq y] = H(y)^{N(t)}$$
★ Frechet normalization
$$n(t) \equiv (\gamma\, N(t)\, K^{\beta}\, L^{\alpha})^{1/(\alpha+\beta)}$$
★ ★ Frechet limit
$$\lim_{N\to\infty} \Pr[\tilde Y \leq (\gamma\, N\, K^{\beta}\, L^{\alpha})^{1/(\alpha+\beta)}\, y] = e^{-y^{-(\alpha+\beta)}}$$
★ CD macro emergent
$$\tilde Y(t; N(t)) \approx \varepsilon(t)\, \gamma^{1/(\alpha+\beta)}\, N(t)^{1/(\alpha+\beta)}\, K^{\beta/(\alpha+\beta)}\, L^{\alpha/(\alpha+\beta)}$$
Stochastic capital dyn
$$k(t+1) = f(k(t), z(t)) + (1-\delta)\, k(t) - c(t)$$
History tilde k
$$\tilde k[z^t] = f(\tilde k[z^{t-1}], z(t)) + (1-\delta)\, \tilde k[z^{t-1}] - c[z^t]$$
Sequence problem (stochastic)
$$\max_{\{\tilde k[z^t]\}} \mathbb{E}_t \sum_{t=0}^{\infty} \beta^t\, U(\tilde k[z^{t-1}], \tilde k[z^t], z(t))$$
Markov policy
$$k(t+1) = \pi(k(t), z(t))$$
★ NGM Bellman
$$V(k, z) = \sup_{y \in [0, f(k,z) + (1-\delta)k]} u(f(k, z) + (1-\delta)k - y) + \beta\, \mathbb{E}[V(y, z') \mid z]$$
★ ★ Stochastic Bellman general
$$V(x, z) = \sup_{y \in G(x, z)} U(x, y, z) + \beta\, \mathbb{E}[V(y, z') \mid z]$$
Bellman with policy
$$V(x, z) = U(x, \pi(x, z), z) + \beta\, \mathbb{E}[V(\pi(x, z), z') \mid z]$$
Principle of Optimality (header)
$$\text{Theorem 16.2: V is unique fixed point of } T$$
Envelope (stochastic)
$$D_x V(x', z) = D_x U(x', \pi(x', z), z)$$
Assumption 16.6
$$\text{(continuity, compactness, monotonicity)}$$
ε-supremum 1
$$\forall \varepsilon > 0,\ \exists x'\ s.t.\ V^*(x(0), z(0)) - V^*(x', z') < \varepsilon$$
ε-supremum 2
$$\forall \varepsilon > 0,\ \exists y' \in G\ s.t.\ V(x(0), z(0)) - U(x(0), y') - \beta\, \mathbb{E}[V(y')] < \varepsilon$$
Optimality equation
$$\bar U(x^*_t \mid \tilde x^*[z^{t-1}], z(t)) = V^*(\tilde x^*[z^{t-1}], z(t))$$
Optimality recursion
$$V^*(\tilde x^*[z^{t-1}], z(t)) = \bar U(x^*_t \mid \tilde x^*[z^{t-1}], z(t))$$
Sub-optimality elimination
$$\mathbb{E}[\bar U(x^*_{t+1} \mid \tilde x^*[z^t], z(t+1)) \mid z(t)] \geq \mathbb{E}[\bar U(x_{t+1} \mid \tilde x^*[z^t], z(t+1)) \mid z(t)]$$
Theorem 16.3 statement
$$V^*(x, z) \leq V(x, z)\ \text{always}$$
★ ★ Stochastic Euler (FOC)
$$D_y U(x, y^*, z) + \beta\, \mathbb{E}[D_x V(y^*, z') \mid z] = 0$$
Envelope (FOC form)
$$D_x V(x, z) = D_x U(x, y^*, z)$$
★ ★ Stochastic Euler (combined)
$$D_y U(x, \pi(x, z), z) + \beta\, \mathbb{E}[D_x U(\pi(x, z), \pi(\pi(x, z), z'), z') \mid z] = 0$$
★ Stochastic TVC
$$\lim_{t\to\infty} \beta^t\, \mathbb{E}[D_x U(\tilde x^*[z^{t-1}], \tilde x^*[z^t], z(t)) \cdot \tilde x^*[z^{t-1}] \mid z(0)] = 0$$
PIH header
$$\text{Permanent Income Hypothesis}$$
★ ★ Hall random-walk consumption
$$\beta^t\, u'(\tilde c[w^t]) = \frac{1}{(1+r)^t}\, \tilde \lambda[w^t]$$
★ Bellman PIH
$$u'(c(t)) = \beta\, \mathbb{E}_t\!\left[\frac{\partial V(a(t+1), w(t+1))}{\partial a}\right]$$
Envelope PIH
$$\frac{\partial V(a(t), w(t))}{\partial a} = u'(c(t))$$
★ Quadratic utility PIH
$$u(c) = \varphi c - \frac{1}{2}c^2,\quad c(t) = (1-\kappa)\, [a(t) + \mathbb{E}_t \sum w(t+s)/(1+r)^s]$$
★ McCall accept-reject Bellman
$$V(a') = \max\!\left\{V_{\text{accept}}(a'),\ \beta\, \mathbb{E}V\right\}$$
Expected V
$$\mathbb{E}V = \int_0^{\bar a} V(a)\, dH(a)$$
★ McCall Bellman expanded
$$V(a') = \max\!\left\{\frac{a'}{1-\beta},\ \beta\, \int_0^{\bar a} V(a)\, dH(a)\right\}$$
Reservation R = β E V
$$\frac{R}{1-\beta} = \int_0^{\bar a} \beta\, V(a)\, dH(a)$$
★ Reservation wage equation
$$\frac{R}{1-\beta} = \beta\, \frac{R\, H(R)}{1-\beta} + \int_R^{\bar a} \frac{a}{1-\beta}\, dH(a)$$
★ R closed-form
$$\frac{R}{1-\beta} = \beta\!\left(\frac{R\, H(R)}{1-\beta} + \int_R^{\bar a} \frac{a}{1-\beta}\, dH(a)\right)$$
★ Stochastic NGM utility
$$\max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t\, u(c(t))$$
Stochastic capital constraint
$$k(t+1) = f(k(t), z(t)) + (1-\delta)\, k(t) - c(t),\ k(t) \geq 0$$
★ ★ Brock-Mirman Bellman
$$V(k, z) = \max_{k' \in [0, f(k,z) + (1-\delta)k]} u(\cdot) + \beta\, \mathbb{E}[V(k', z') \mid z]$$
★ ★ Stochastic Euler (NGM)
$$u'(c) = \beta\, \mathbb{E}[(f'(\pi(k,z), z') + (1-\delta))\, u'(c')]$$
★ Lucas asset pricing
$$u'(c(t)) = \beta\, \mathbb{E}_t[p(t+1)\, u'(c(t+1))]$$
★ Stochastic NGM TVC
$$\lim_{t\to\infty} \mathbb{E}[\beta^t (f'(k(t), z(t)) + (1-\delta))\, u'(c(t))\, k(t) \mid z(0)] = 0$$
Markov policy
$$k(t+1) = \pi(k(t), z(t))$$
Example 17.1 — Cobb-Douglas
$$\frac{1}{z\, k^{\alpha} - \pi(k, z)} = \beta\, \mathbb{E}\!\left[\frac{\alpha\, z'\, \pi(k, z)^{\alpha-1}}{z'\, \pi(k, z)^{\alpha} - \pi(\pi(k, z), z')}\right]$$
★ Cobb-Douglas closed form
$$\pi(k, z) = B_0 + B_1\, z\, k^{\alpha},\ \text{with } B_0, B_1\ \text{determined}$$
★ Arrow-Debreu sequence budget
$$\sum_{t=0}^{\infty} \sum_{z^t \in Z^{\infty}} p_0[z^t]\, c[z^t] \leq \sum_{t=0}^{\infty} \sum_{z^t \in Z^{\infty}} w[z^t]$$
AD utility (state-contingent)
$$\sum_{t=0}^{\infty} \beta^t \sum_{z^t \in Z^{\infty}} q[z^t \mid z_0]\, u(c[z^t])$$
★ AD FOC
$$\beta^t\, q[z^t \mid z_0]\, u'(c[z^t]) = \lambda\, p_0[z^t]$$
Conditional expansion
$$z^{t+1} = (z^t, z(t+1)),\ \beta^{t+1}\, q[z^{t+1} \mid z_0]\, u'(c[z^{t+1}]) = \lambda\, p_0[z^{t+1}]$$
AD factor pricing
$$p_0[z^t]\, [\partial F(K_e[z^t], L[z^t], z(t))/\partial K_e + (1-\delta)] = R_0[z^t]$$
Labor normalization
$$L[z^t] = 1\quad \forall z^t$$
Capital identity
$$k_e[z^{t+1}] = k[z^t]$$
Resource (state-contingent)
$$c[z^t] + k[z^t] \leq f(k[z^{t-1}], z(t)) + (1-\delta)\, k[z^{t-1}]$$
Spot price = sum AD
$$p_0[z^t] = \sum_{z(t+1) \in Z} R_0[(z^t, z(t+1))]$$
FOC ratio (AD)
$$u'(c[z^t]) = \sum_{z(t+1) \in Z} \lambda\, p_0[z^{t+1}]$$
FOC ratio compact
$$\beta\, u'(c[z^{t+1}]) = \lambda\, p_0[z^{t+1}]\, \beta^t\, q[z^{t+1} \mid z_0]$$
Bayes factorization
$$q[z^{t+1} \mid z_0] = q[z^{t+1} \mid z_t]\, q[z^t \mid z_0]$$
AD No-Ponzi
$$\lim_{t\to\infty} \sum_{z^{t-1} \in Z^{t-1}} p_0[z^{t-1}]\, k[z^{t-1}] \geq 0$$
★ Recursive Bellman Arrow
$$V(a, z) = \max_{\{a'[z'\mid z]\}_{z' \in Z}} u(a + w - \sum_{z'} \bar p[z'\mid z]\, a'[z'\mid z]) + \beta\, \sum_{z'} q[z'\mid z]\, V(a'[z'\mid z], z')$$
★ Arrow Euler
$$\bar p[z'\mid z]\, u'(c[a, z]) = \beta\, q[z'\mid z]\, \frac{\partial V(a'[z'\mid z], z')}{\partial a}$$
Market clearing (Arrow)
$$a'[z'\mid z] = R[z'\mid z]\, k$$
★ Arrow no-arb
$$\sum_{z' \in Z} \bar p[z'\mid z]\, R[z'\mid z] = 1$$
Utility u(C, L)
$$u(C, L)\ \text{convex compact set } [0, \bar L]$$
★ Stochastic Euler with labor
$$u_c(\pi_c, \pi_l) = \beta\, \mathbb{E}[R(\pi_k(K, z), z')\, u_c(\pi_c', \pi_l') \mid z]$$
★ ★ Borrowing constraint
$$a_h(t) \geq -\frac{z_{\min}}{R - 1} \equiv -b$$
★ ★ Bewley-Aiyagari Bellman
$$V_h(a, z) = \max_{a' \in [-b, R\, a + w\, z]} u(R\, a + w\, z - a') + \beta\, \mathbb{E}[V_h(a', z') \mid z]$$
★ Aiyagari k**
$$f'(k^{**}) = \beta^{-1} - (1-\delta)$$
★ Aiyagari k* < k**
$$f'(k^*) < \beta^{-1} - (1-\delta)$$
★ ★ Precautionary k*
$$k^* > k^{**}$$
OLG utility (stochastic)
$$U_t(c_1(t), c_2(t+1)) = \log c_1(t) + \beta\, \log c_2(t+1)$$
Population n
$$L(t) = (1+n)^t\, L(0)$$
Stochastic Cobb-Douglas Y
$$Y(t) = z(t)\, K(t)^{\alpha}\, L(t)^{1-\alpha},\ R(k, z) = \alpha\, z\, k^{\alpha-1},\ w(k, z) = (1-\alpha)\, z\, k^{\alpha}$$
★ OLG saving = β/(1+β) w
$$s(k(t), z(t)) = \frac{\beta}{1+\beta}\, w(k(t), z(t))$$
OLG capital dynamic
$$k(t+1) = \pi(k, z) = s(k, z)/(1+n)$$
★ OLG steady-state k*
$$k^* = \left(\frac{\beta(1-\alpha)\, \bar z}{(1+n)(1+\beta)}\right)^{1/(1-\alpha)}$$
AZ 생산함수
$$Y(t) = K(t)^{\alpha}\, L(t)^{1-\alpha}$$
Threshold M(j)
$$M(j) = \max\!\{0,\ D(1-\gamma)(j-\gamma)\}$$
AZ utility
$$\mathbb{E}_t U_t = \log c_1(t) + \beta \int_0^1 \log c_2(j, t+1)\, dj$$
AZ wage
$$w(j, t+1) = (1-\alpha)\, K(j, t+1)^{\alpha}$$
AZ R(j, t+1)
$$R(j, t+1) = \alpha\, K(j, t+1)^{\alpha-1}$$
AZ portfolio max
$$\max_{s, X, [I(j)]} \log c(t) + \beta \int_0^1 \log c(j, t+1)\, dj$$
Investment constraint
$$X(t) + \int_0^1 I(j, t)\, dj = s(t)$$
c(j, t+1)
$$c(j, t+1) = R(j, t+1)\, [q\, X(t) + Q\, I(j, t)]$$
Sparse investment
$$I(j, t) = 0\quad \forall j \notin J(t)$$
Income constraint
$$c(t) + s(t) \leq w(t)$$
★ AZ saving
$$s^*(t) = \frac{\beta}{1+\beta}\, w(t)$$
AZ simpler
$$\max_{X, I} n^*(t)\, \log R_G(t+1)(q\, X + Q\, I) + (1-n^*(t))\, \log R_B(\cdots)$$
Sparse constraint
$$X(t) + n^*(t)\, I(t) \leq s^*(t)$$
★ X* solution
$$X^*(t) = \frac{(1-n^*(t))\, Q}{Q - q\, n^*(t)}\, s^*(t)$$
★ I* solution
$$I^*(j, t) = \frac{Q-q}{Q - q\, n(t)}\, s^*(t)\quad \text{for } j \leq n^*(t),\ 0\ \text{else}$$
★ ★ AZ portfolio threshold
$$n^*[K(t)] = \frac{(Q + q\gamma) - \sqrt{(Q+q)^2 - 4q[D^{-1}(Q-q)(1-\gamma)\, K^{\alpha} + \gamma]}}{\cdots}$$
★ ★ K(t+1) random walk
$$K(t+1) = \begin{cases} \frac{q(1-n^*[K(t)])}{Q - q\, n^*[K(t)]}\, Q\, K(t)^{\alpha} & \text{w/p } 1-n^*[K(t)] \\ Q\, K(t)^{\alpha} & \text{w/p } n^*[K(t)] \end{cases}$$
★ Volatility of growth
$$\sigma_e(n^*[K(t)]) = (1-n^*[K(t)])\, \frac{q(1-n^*[K(t)])}{Q - q\, n^*[K(t)]}\, Q + n^*[K(t)]\, Q$$
★ K_QSSB and K_QSSG
$$K_{QSSB} = \cdots,\ K_{QSSG} = Q^{1/(1-\alpha)}$$
Poverty trap condition
$$D < \frac{1}{1-\alpha}\, Q^{\alpha/(1-\alpha)}$$
★ K growth log linear
$$\Delta \log K(t+1) = \log[\sigma(n^*[K(t)])] - (1-\alpha)\, \log K(t)$$
★ AZ social planner
$$\max_{n, X, I} n(t)\, \int_0^{n(t)} \log(qX + QI(j, t))\, dj + (1-n(t)) \log(\cdots)$$
Proposition 17.11 (closing)
$$I^S(j, t) = M(j^*) > M(j)\ \text{for } j < j^*(t),\ I^S(j) = M(j)\ \text{for } j \geq j^*(t)$$
1인당 산출 (effective labor 정규화)
$$y_j(t) \equiv \frac{Y_j(t)}{L_j(t)} = A_j(t) f(k_j(t))$$
국가별 기술 성장률 정의
$$g_j(t) \equiv \frac{\dot A_j(t)}{A_j(t)}$$
기술 확산 ODE — OU process / Brownian motion 동형 ★★★
$$\dot A_j(t) = \sigma_j\bigl(A(t) - A_j(t)\bigr) + \lambda_j A_j(t)$$
정규화 비율 ODE (linear)
$$\dot a_j(t) = \sigma_j - (\sigma_j + g - \lambda_j)\, a_j(t)$$
Diffusion ODE solution
$$a_j(t) = a_j^* + (a_j(0) - a_j^*)\, e^{-(\sigma_j + g - \lambda_j) t}$$
Lifetime utility (diffusion)
$$U_j = \int_0^{\infty} e^{-(\rho - n_j) t}\, \frac{c_j(t)^{1-\theta} - 1}{1-\theta}\, dt$$
Expanding-variety 생산함수 (Romer 1990)
$$Y_j(t) = \frac{1}{1-\beta} \int_0^{N_j(t)} x_j(\nu, t)^{1-\beta}\, d\nu \cdot L_j^{\beta}$$
Resource constraint
$$C_j(t) + X_j(t) + \zeta_j\, Z_j(t) \leq Y_j(t)$$
변종 수 증식 ODE (frontier diffusion)
$$\dot N_j(t) = \eta_j \left(\frac{N(t)}{N_j(t)}\right)^{\varphi} Z_j(t)$$
Aggregate g
$$\dot N(t) = g\, N(t)$$
BGP free-entry
$$(N_j)^{-\varphi}\, V^*_j\quad \text{vs}\quad \mu^*_j = \frac{\eta_j\, \beta\, L_j}{\zeta_j\, r^*}$$
★ Aggregate N
$$N(t) = \frac{1}{J} \sum_{j=1}^{J} N_j(t)$$
★ ★ Cross-country μ_j*
$$\frac{1}{J} \sum_{j=1}^{J} \mu^*_j = \frac{1}{J} \sum_{j=1}^{J} \frac{\eta_j\, \beta\, L_j}{\zeta_j\, \rho}$$
Sustained growth condition
$$\frac{1}{J} \sum_{j=1}^{J} \left(\frac{\eta_j\, \beta\, L_j}{\zeta_j\, \rho}\right)^{1/\varphi} > 1$$
Appropriate technology 함수
$$A(k \mid k') = A \min\!\left\{1,\, \left(\frac{k}{k'}\right)^{\gamma}\right\}$$
Cobb-Douglas Y_j
$$Y_j(t) = \exp\!\left(\int_0^1 \log y_j(i, t)\, di\right)$$
Sector production
$$y_j(i, t) = \frac{1}{1-\beta}\, \int_0^{N_L(t)} x_{L,j}(i, \nu, t)^{1-\beta}\, d\nu \cdot ((1-i)\, l_j(i, t))^{\beta}$$
Sector i = closing
$$\text{output sector } i = \cdots$$
Relative price (Trade-DTC)
$$\frac{P_{H,j}(t)}{P_{L,j}(t)} = \frac{N_H(t)}{N_L(t)}\, \omega_{Hj} L_j$$
★ Price ratio (β/2 power)
$$\frac{P_{H,j}(t)}{P_{L,j}(t)} = \left(\frac{N_H(t)}{N_L(t)}\right)\, \left(\frac{\omega_{Hj}}{L_j}\right)^{-\beta/2}$$
Trade share I_j
$$\frac{I_j(t)}{1 - I_j(t)} = \left(\frac{N_H(t)}{N_L(t)}\right)\, \left(\frac{\omega_{Hj}}{L_j}\right)^{-1/2}$$
★ Aggregate Y_j
$$Y_j(t) = e^{-\beta}\, [(N_L(t)\, L_j)^{1/2} + (N_H(t)\, \omega_{Hj})^{1/2}]^2$$
★ Skill premium (trade)
$$\frac{w_{H,j}(t)}{w_{L,j}(t)} = \omega\, \left(\frac{N_H(t)}{N_L(t)}\right)^{1/2}\, \left(\frac{\omega_{Hj}}{L_j}\right)^{-1/2}$$
R&D dynamics (NL, NH)
$$\dot N_L(t) = \eta\, Z_L(t),\ \dot N_H(t) = \eta\, Z_H(t)$$
★ N_H/N_L closed
$$\left(\frac{N_H}{N_L}\right)^* = \omega_H^n\, L^n$$
Contracting utility
$$u = \left(\int_0^M q(\nu)^{\beta}\, d\nu\right)^{1/\beta} - \psi\, e,\quad 0 < \beta < 1$$
★ Revenue R
$$R = A^{1-\beta}\, q^{\beta}$$
★ Quality q (CES of inputs)
$$q = N^{\kappa + 1 - 1/\alpha}\, \int_0^N X(j)^{\alpha}\, dj$$
X(j) sub-aggregator
$$X(j) = \exp\!\left(\int_0^1 \log x(i, j)\, di\right)$$
Concavity condition
$$\forall N > 0,\ \frac{\Gamma''(N)}{\Gamma'(N) + w_0} > \frac{\beta(\kappa+1) - 1}{1-\beta}$$
★ Profit π
$$\pi = R - \int_0^N [\tau(j) + s(j)]\, dj - \Gamma(N)$$
R closed
$$R = A^{1-\beta}\, N^{\beta(\kappa+1)}\, x^{\beta\, \cdots}$$
★ Optimal contract problem
$$\max_{N, \{x(i,j)\}, \{s(j), \tau(j)\}} R - \int_0^N [\tau(j) + s(j)]\, dj - \Gamma(N)$$
★ Supplier IR constraint
$$s(j) + \tau(j) - \psi\, \int_0^1 x(i, j)\, di \geq w_0\quad \forall j \in [0, N]$$
Reduced max
$$\max_{N, x} A^{1-\beta}\, N^{\beta(\kappa+1)}\, x^{\beta} - N\, w_0 - N\, \psi\, x - \Gamma(N)$$
★ ★ N* equation
$$(N^*)^{[\beta(\kappa+1)-1]/(1-\beta)}\, A^{\kappa\beta/(1-\beta)}\, \psi^{-\beta/(1-\beta)} = \Gamma'(N^*)$$
x* solution
$$x^* = \frac{\Gamma'(N^*) + w_0}{\kappa\, \psi}$$
x_n max
$$x_n \in \arg \max_{x_n(j)} \bar s_x(N, x_c, x_n, x_n(j)) - (1-\mu)\, \psi\, x_n(j)$$
★ Bargaining IR
$$\bar s_x(N, x_c, x_n, x_n) + \tau \geq \mu\, \psi\, x_c + (1-\mu)\, \psi\, x_n + w_0$$
★ Reduced contract max
$$\max_{N, x_c, x_n} s_q(N, x_c, x_n) + N\, \bar s_x(N, x_c, x_n)$$
Tilde τ
$$\tilde \tau = \mu\, \psi\, \tilde x_c + (1-\mu)\, \psi\, \tilde x_n + w_0 - \bar s_x(\tilde N, \tilde x_c, \tilde x_n)$$
Proposition 18.10
$$\bar s_x(N, x_c, x_n(-j), \cdots) = \cdots$$
★ γ definition
$$\gamma \equiv \frac{\alpha}{\alpha + \beta}$$
★ s_x symmetric
$$x_n(j) = x_n(-j) = x_n,\ s_x(N, x_c, x_n) = (1-\gamma)\, A^{1-\beta}\, x_c^{\beta\mu}\, x_n^{\beta(1-\mu)}\, N^{\beta(\kappa+1)}$$
★ s_q symmetric
$$s_q(N, x_c, x_n) = \gamma\, A^{1-\beta}\, x_c^{\beta\mu}\, x_n^{\beta(1-\mu)}\, N^{\beta(\kappa+1)}$$
★ x_n optimal
$$x_n = \bar x_n(N, x_c) = [\alpha(1-\gamma)\, \psi^{-1}\, x_c^{\beta\mu}\, A^{1-\beta}\, N^{\beta(\kappa+1)}]^{1/(1-\beta(1-\mu))}$$
★ Reduced max (with constraint)
$$\max_{N, x_c} A^{1-\beta}\, x_c^{\beta\mu}\, [\alpha(1-\gamma)/\psi]^{\beta(1-\mu)/(1-\beta(1-\mu))} \cdots$$
★ ★ Tilde N — endogenous adoption
$$(\tilde N)^{[\beta(\kappa+1)-1]/(1-\beta)}\, A^{\kappa\beta/(1-\beta)} \cdots = \Gamma'(\tilde N)$$
Tilde x_c
$$\tilde x_c = \frac{\Gamma'(\tilde N) + w_0}{\kappa\, \psi}$$
★ Tilde x_n closed
$$\tilde x_n = \frac{\alpha(1-\gamma)\, [1-\beta(1-\mu)]}{\beta\, [1-\alpha(1-\gamma)(1-\mu)]}\, \frac{\Gamma'(\tilde N) + w_0}{\psi}$$
★ ★ Distortion ratio
$$\frac{\tilde x_n}{\tilde x_c} = \frac{\alpha(1-\gamma)\, [1-\beta(1-\mu)]}{\beta\, [1-\alpha(1-\gamma)(1-\mu)]} < 1$$
★ Shapley share
$$s_j = \frac{1}{(K+1)!} \sum_{g \in G} [v(z^j_g \cup j) - v(z^j_g)]$$
두 국가 무역 — Heckscher-Ohlin baseline
$$Y_j(t) = F_j(K_j(t), L_j(t), A_j(t))$$
Open NGM ODE
$$\dot k_j(t) = f(k_j(t)) - \tilde c_j(t) + b_j(t) - (n+g+\delta)\, k_j(t)$$
★ Asset evolution
$$\dot A_j(t) = r(t)\, A_j(t) - B_j(t)$$
Per-capita asset
$$\dot a_j(t) = (r(t) - g - n)\, a_j(t) - b_j(t)$$
Free trade equilibrium
$$p_H(t)\, MP_{L,H}(t) = p_F(t)\, MP_{L,F}(t)$$
Trade balance closure
$$\sum_{j=1}^{J} A_j\, b_j(t) = 0$$
Trade in factors Y_j
$$Y_j(t) = F(X^K_j(t), X^L_j(t))$$
Y^L_j
$$Y^L_j(t) = A_j\, L_j(t)$$
Y^K_j
$$Y^K_j(t) = K_j(t)$$
Specialization — comparative advantage
$$\frac{a_{Lj}}{a_{Hj}} > \frac{a_{Lk}}{a_{Hk}} \Rightarrow j \text{ specializes in } L\text{-good}$$
Trade balance (KL)
$$p_K(t)\, [X^K_j - Y^K_j] + p_L(t)\, [X^L_j - Y^L_j] = 0$$
K dynamics
$$\dot K_j(t) = F(X^K_j, X^L_j) - \delta\, K_j(t) - C_j(t)$$
★ World market clearing
$$\sum_j X^L_j(t) = \sum_j Y^L_j(t),\ \sum_j X^K_j(t) = \sum_j Y^K_j(t)$$
Trade NGM utility
$$U_j = \int_0^{\infty} e^{-(\rho-n) t}\, \frac{c_j(t)^{1-\theta} - 1}{1-\theta}\, dt$$
Trade-induced growth ODE
$$\dot A_j(t) = \sigma_j\, T_j(t) + \lambda_j A_j(t)$$
x_j definition
$$x_j(t) \equiv \frac{X^K_j(t)}{X^L_j(t)},\ Y_j = X^L_j\, f(x_j)$$
★ ★ World convergence
$$f'(x^*_j) = f'(k^*_A) = \rho + \delta\quad \forall j$$
★ x* definition
$$x^*_j = x^* = \frac{\sum_j K_j(t)}{L(t) \sum_j A_j},\ k^* = \frac{\sum_j K_j(t)}{J\, L(t)}$$
p_K* = ρ+δ
$$p_K^* = \rho + \delta$$
Multi-country log utility
$$\int_0^{\infty} e^{-\rho_j t}\, \log C_j(t)\, dt$$
Multi-country budget
$$p_I^j(t)\, \dot K_j(t) + p_C^j(t)\, C_j(t) = Y_j(t)$$
Variety conservation
$$\sum_{j=1}^{J} \mu_j = N$$
Variety price = rate
$$p_j(t) = r_j(t)$$
★ ★ Multi-country C_j (CES)
$$C_j(t) = \chi\, K^C_j(t)^{1-\tau}\, \int_0^N x^C_j(t, \nu)^{(\varepsilon-1)/\varepsilon}\, d\nu$$
Krugman gravity equation
$$T_{ij} = G\, \frac{Y_i\, Y_j}{D_{ij}^{\theta}}$$
Capital allocation
$$K^C_j(t) + K^I_j(t) + K^{\mu}_j(t) \leq K_j(t)$$
B_C demand
$$B^C_j(r_j, [p(\nu)]) = r_j^{1-\tau}\, \int_0^N p(t, \nu)^{1-\varepsilon}\, d\nu$$
B_I demand
$$B^I_j(r_j, [p(\nu)]) = \zeta_j\, r_j^{1-\tau}\, \int_0^N p(t, \nu)^{1-\varepsilon}\, d\nu$$
★ Inflation arbitrage
$$r_j(t) + \frac{\dot p^I_j}{p^I_j} - \frac{\dot p^C_j}{p^C_j} = \rho_j + \frac{\dot C_j}{C_j}$$
Trade TVC
$$\lim_{t\to\infty} e^{-\rho_j t}\, \frac{p^I_j(t)\, K_j(t)}{p^C_j(t)\, C_j(t)} = 0$$
★ p_C C = ρ p_I K
$$p^C_j(t)\, C_j(t) = \rho_j\, p^I_j(t)\, K_j(t)$$
Price index normalize
$$\left(\int_0^N p(t, \nu)^{1-\varepsilon}\, d\nu\right)^{1/(1-\varepsilon)} = 1$$
★ Price formulas
$$p^C_j(t) = r_j(t)^{1-\tau},\ p^I_j(t) = \zeta_j\, r_j(t)^{1-\tau}$$
Y_j formula
$$Y_j(t) = \mu_j\, r_j(t)^{1-\varepsilon}\, Y(t)$$
★ K growth
$$\frac{\dot K_j}{K_j} = \frac{r_j(t)^{\tau}}{\zeta_j} - \rho_j$$
Income identity
$$r_j(t)\, K_j(t) = \mu_j\, r_j(t)^{1-\varepsilon}\, \sum_{i=1}^{J} r_i(t)\, K_i(t)$$
★ K and Y BGP equal
$$\frac{\dot K_j}{K_j} = \frac{\dot Y_j}{Y_j} = g^*$$
★ ★ BGP equilibrium
$$\sum_{j=1}^{J} \mu_j\, [\zeta_j(\rho_j + g^*)]^{(1-\varepsilon)/\tau} = 1$$
★ r* and p*
$$r^*_j = p^*_j = [\zeta_j(\rho_j + g^*)]^{1/\tau}$$
World income distribution (Pareto)
$$\Pr(y > y_0) = \left(\frac{y_0}{y_{\min}}\right)^{-\alpha}$$
Y_j formula closed
$$y^*_j = A_j\, s_j\, (g^*)^{\alpha/(1-\alpha)}$$
Trade B with labor
$$B^C_j(w_j, r_j, [p(\nu)]) = w_j^{(1-\tau)(1-\gamma)}\, \cdots$$
★ p_C C as ρ p_I K
$$p^C_j(t)\, C_j(t) = \rho_j\, p^I_j(t)\, K_j(t) + \int_t^{\infty} \exp\!\left(-\int_t^z (r_j + \dot p^I_j/p^I_j - \rho_j)\right)\, dz \cdot w_j(t)$$
Wage demand share
$$(1-\gamma)(1-\tau)\ \text{of consumption expenditure on goods}$$
★ p_C C = ρ/(1-(1-γ)(1-τ)) p_I K
$$p^C_j(t)\, C_j(t) = \frac{\rho_j}{1 - (1-\gamma)(1-\tau)}\, p^I_j(t)\, K_j(t)$$
★ K + W income
$$r_j(t)\, K_j(t) + w_j(t) = \mu_j\, r_j(t)^{1-\varepsilon}\, \sum_i [r_i(t) K_i(t) + w_i(t)]$$
★ Wage share
$$\frac{w_j(t)}{r_j(t) K_j(t) + w_j(t)} = \frac{(1-\gamma)(1-\tau) \rho_j}{[\gamma + (1-\gamma)\tau]\, \zeta_j}$$
★ N-S consumption aggregator
$$C_j(t) = \left(\int_0^{N(t)} c_j(t, \nu)^{(\varepsilon-1)/\varepsilon}\, dz\right)^{\varepsilon/(\varepsilon-1)}$$
N-S prices = wages
$$p_n(t) = w_n(t),\ p_o(t) = w_s(t)$$
Demand ratio
$$\frac{c_n(t)}{c_o(t)} = \left(\frac{p_n}{p_o}\right)^{-\varepsilon}$$
Quantity per variety
$$c_n(t) = \frac{L_n}{N_n(t)},\ c_o(t) = \frac{L_s}{N_o(t)}$$
★ Wage gap (north/south)
$$\frac{w_n(t)}{w_s(t)} \equiv \omega(t) = \left(\frac{N_n(t)}{N_o(t)}\right)\, \left(\frac{L_s}{L_n}\right)^{1/\varepsilon}$$
★ ★ N_n/N_o BGP
$$\frac{N_n(t)}{N_o(t)} = \frac{\eta}{\iota}$$
★ Wage gap closed
$$\frac{w_n(t)}{w_s(t)} = \max\!\left\{\frac{\eta}{\iota}\, \left(\frac{L_s}{L_n}\right)^{1/\varepsilon},\, 1\right\}$$
Wellposed conditions
$$\eta\, \beta > \rho,\ 2(1-\theta)\, \eta\, \beta < \rho$$
★ N-S innovation g_A
$$g_A = \frac{1}{\theta}\, (\eta\, \beta - \rho)$$
★ Imitation rate
$$\frac{\dot A_j(t)}{A_j(t)} = \eta\, L^1_j(t)$$
Initial conditions
$$A_n(0) = 1,\ A_s(0) = 1 - \delta$$
Aggregate p1, p2 link
$$p^1_j(t)\, A_j(t) = p^2_j(t)$$
Producer price ratio
$$\frac{p^1_j(t)}{p^2_j(t)} = \frac{X^1_j(t)}{X^2_j(t)}$$
★ Trade balance condition
$$(1-\delta)^{-\varepsilon} < \frac{L_S}{L_N} < \varepsilon^{-1} + (1-\delta)^{-\varepsilon}$$
Two-sector structural change
$$Y(t) = \bigl[\eta_A Y_A(t)^{\sigma} + \eta_M Y_M(t)^{\sigma}\bigr]^{1/\sigma}$$
★ KRX consumption Stone-Geary
$$c(t) = (c_A(t) - \gamma_A)^{\eta_A}\, c_M(t)^{\eta_M}\, (c_S(t) + \gamma_S)^{\eta_S}$$
3-sector production
$$Y_A = B_A\, F(K_A, X L_A),\ Y_M = B_M\, F(K_M, X L_M),\ Y_S = B_S\, F(K_S, X L_S)$$
X (productivity) growth
$$\frac{\dot X(t)}{X(t)} = g$$
Engel's law
$$\frac{\partial \log c_A}{\partial \log Y} < 1$$
Labor allocation
$$L_A(t) + L_M(t) + L_S(t) = L(t)$$
M sector resource
$$\dot K(t) + c_M(t)\, L(t) = Y_M(t)$$
A & S resource
$$c_A(t)\, L(t) = Y_A(t),\ c_S(t)\, L(t) = Y_S(t)$$
★ Relative price A
$$p_A(t)\, \frac{c_A(t) - \gamma_A}{\eta_A} = \frac{c_M(t)}{\eta_M}$$
★ Relative price S
$$p_S(t)\, \frac{c_S(t) + \gamma_S}{\eta_S} = \frac{c_M(t)}{\eta_M}$$
Wage = MP_L (M sector)
$$w(t) = \frac{\partial B_M F(K_M, X L_M)}{\partial L_M}$$
Interest = MP_K (M sector)
$$r(t) = \frac{\partial B_M F(K_M, X L_M)}{\partial K_M}$$
Equilibrium tuple
$$\{[K_A(t), K_M(t), K_S(t), L_A(t), L_M(t), L_S(t)]\}, \{p_A(t), p_M(t), p_S(t)\}$$
★ ★ KRX equal capital ratio
$$\frac{K_A}{X L_A} = \frac{K_M}{X L_M} = \frac{K_S}{X L_S} = \frac{K(t)}{X L(t)}\quad \text{(Prop 20.1)}$$
Sectoral labor allocation
$$L_A(t) + L_M(t) = L(t)$$
★ M consumption Euler
$$\frac{\dot c_M(t)}{c_M(t)} = \frac{1}{\theta}\, (r(t) - \rho)\quad \text{(Prop 20.2)}$$
B-ratio identity
$$\frac{B_M(c_A - \gamma_A)}{B_A\, \eta_A} = \frac{c_M}{\eta_M},\ \frac{B_M(c_S + \gamma_S)}{B_S\, \eta_S} = \frac{c_M}{\eta_M}$$
★ ★ KRX CGP existence
$$\gamma_A B_A = \gamma_S B_S\quad \text{(Prop 20.4)}$$
★ KRX growth differentials
$$\frac{\dot c_A}{c_A} = g\, \frac{c_A - \gamma_A}{c_A},\ \frac{\dot c_M}{c_M} = g,\ \frac{\dot c_S}{c_S} = g\, \frac{c_S}{c_S + \gamma_S}$$
2-sector aggregator
$$Y(t) = F(Y_1(t), Y_2(t))$$
Y1 production
$$Y_1(t) = A_1(t)\, G_1(K_1(t), L_1(t))$$
Y2 + resource
$$Y_2(t) = A_2(t)\, G_2(K_2(t), L_2(t)),\ K_1 + K_2 = K,\ L_1 + L_2 = L$$
Price ratio = MP ratio
$$\frac{p_1(t)}{p_2(t)} = \frac{\partial F/\partial Y_1}{\partial F/\partial Y_2}$$
★ FPE
$$w = p_1\, A_1\, \frac{\partial G_1}{\partial L_1} = p_2\, A_2\, \frac{\partial G_2}{\partial L_2}$$
Equilibrium tuple
$$\{[p_1(t), p_2(t), w(t), r(t)]\}_{t=0}^{\infty}$$
Per-unit productivity
$$g_1 k_1 \equiv \frac{G_1(K_1, L_1)}{L_1},\ g_2 k_2 \equiv \frac{G_2(K_2, L_2)}{L_2}$$
Constant relative price
$$\frac{\dot p_1}{p_1} = \frac{\dot p_2}{p_2} = 0$$
★ Capital arbitrage
$$r = p_1\, A_1\, g'_1(k_1) = p_2\, A_2\, g'_2(k_2)$$
★ Labor arbitrage
$$w = p_1 A_1\, [g_1(k_1) - g'_1(k_1) k_1] = p_2 A_2\, [g_2(k_2) - g'_2(k_2) k_2]$$
★ Growth implication
$$\frac{\varepsilon\, g'_1\, \dot k_1}{k_1} = \frac{\varepsilon\, g'_2\, \dot k_2}{k_2}$$
★ AG CES aggregator
$$Y(t) = \left[\gamma\, Y_1(t)^{(\varepsilon-1)/\varepsilon} + (1-\gamma)\, Y_2(t)^{(\varepsilon-1)/\varepsilon}\right]^{\varepsilon/(\varepsilon-1)}$$
Resource constraint
$$\dot K(t) + L(t)\, c(t) \leq Y(t)$$
★ AG sector CD
$$Y_1 = A_1\, K_1^{\alpha_1}\, L_1^{1-\alpha_1},\ Y_2 = A_2\, K_2^{\alpha_2}\, L_2^{1-\alpha_2}$$
★ ★ Capital intensity asymmetry
$$\alpha_1 < \alpha_2$$
★ Productivity growth diff
$$\frac{\dot A_1}{A_1} = a_1 > 0,\ \frac{\dot A_2}{A_2} = a_2 > 0$$
Labor balance
$$L_1(t) + L_2(t) = L(t)$$
Capital balance
$$K_1(t) + K_2(t) = K(t)$$
Price index condition
$$1 = \gamma^{\varepsilon}\, p_1(t)^{1-\varepsilon} + (1-\gamma)^{\varepsilon}\, p_2(t)^{1-\varepsilon}$$
Wage = MP_L (sector 1)
$$\gamma\, (1-\alpha_1)\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{L_1(t)} = (1-\gamma)\, (1-\alpha_2)\, \left(\frac{Y(t)}{Y_2(t)}\right)^{1/\varepsilon}\, \frac{Y_2(t)}{L_2(t)}$$
Interest = MP_K (sector 1)
$$\gamma\, \alpha_1\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{K_1(t)} = (1-\gamma)\, \alpha_2\, \left(\frac{Y(t)}{Y_2(t)}\right)^{1/\varepsilon}\, \frac{Y_2(t)}{K_2(t)}$$
Wage formula
$$w(t) = \gamma\, (1-\alpha_1)\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{L_1(t)}$$
Rate formula
$$r(t) = \gamma\, \alpha_1\, \left(\frac{Y(t)}{Y_1(t)}\right)^{1/\varepsilon}\, \frac{Y_1(t)}{K_1(t)}$$
★ κ definition
$$\kappa(t) = 1 + \frac{\alpha_1}{\alpha_2}\, \frac{1-\alpha_2}{1-\alpha_1} \cdots$$
★ λ definition
$$\lambda(t) = 1 + \frac{\alpha_1}{\alpha_2}\, \frac{1-\alpha_2}{1-\alpha_1}\, \frac{1-\kappa}{\kappa} \cdots$$
Proposition 20.6 (κ elasticity)
$$\frac{d \log \kappa}{d \log K} = -\frac{d \log \kappa}{d \log L} = \cdots$$
Prop 20.6 (A elasticity)
$$\frac{d \log \kappa}{d \log A_2} = -\frac{d \log \kappa}{d \log A_1} = (1-\varepsilon)\, \cdots$$
★ w/r ratio
$$\frac{w(t)}{r(t)} = \frac{1-\alpha_1}{\alpha_1}\, \frac{\kappa(t)\, K(t)}{\lambda(t)\, L(t)}$$
★ Capital share σ_K
$$\sigma_K(t) \equiv \frac{r(t)\, K(t)}{Y(t)} = \gamma\, \alpha_1\, \left(\frac{Y_1(t)}{Y(t)}\right)^{(\varepsilon-1)/\varepsilon}$$
Prop 20.7 (w/r elasticity)
$$\frac{d \log(w/r)}{d \log L} = -\frac{d \log(w/r)}{d \log L} = \frac{1}{1 + (\alpha_2 - \alpha_1)/\cdots}$$
w/r vs A elasticity
$$\frac{d \log(w/r)}{d \log A_2} = -\frac{d \log(w/r)}{d \log A_1} = -(1-\varepsilon)\, \cdots$$
★ σ_K dynamics (K)
$$\frac{d \log \sigma_K}{d \log K}\ \text{positive iff } (\alpha_2 - \alpha_1)(1-\varepsilon) > 0$$
σ_K dynamics (A)
$$\frac{d \log \sigma_K}{d \log A_2} = -\frac{d \log \sigma_K}{d \log A_1} < 0$$
Y1/Y formula
$$\left(\frac{Y_1}{Y}\right)^{(\varepsilon-1)/\varepsilon} = \gamma + (1-\gamma)\, \cdots$$
★ σ_K growth (A)
$$\frac{d \log \sigma_K}{d \log A_2} = -\frac{d \log \sigma_K}{d \log A_1} = \frac{1-\sigma_K}{\sigma_K}\, \alpha_1\, \cdots$$
★ Consumption Euler
$$\frac{\dot c}{c} = \frac{1}{\theta}\, (r(t) - \rho)$$
TVC
$$\lim_{t\to\infty} K(t)\, e^{-\int_0^t r(\tau) d\tau} = 0$$
Prop 20.8 — n_1, n_2 ordering
$$\varepsilon < 1: n_1 \gtrless n_2 \Leftrightarrow \cdots$$
Growth equality
$$\frac{1}{\varepsilon}\, g + \frac{\varepsilon-1}{\varepsilon}\, g_1 - z_1 = \frac{1}{\varepsilon}\, g + \frac{\varepsilon-1}{\varepsilon}\, g_2 - z_2$$
Prop 20.9 g* existence
$$g^*_1, g^*_2\ \text{exist},\ \varepsilon < 1 \Rightarrow \min,\ \varepsilon > 1 \Rightarrow \max$$
Prop 20.9 (cont)
$$\lim_{t\to\infty} \dot K/K = \min\{g^*_1, g^*_2\}\ \text{or}\ \max$$
TVC condition
$$a_1/(1-\alpha_1) < a_2/(1-\alpha_2)\ \text{and}\ \varepsilon < 1\ \text{(or symm)}$$
★ Prop 20.10 — sector 1 dominant
$$g^* = g^*_C = g^*_1 = z^*_1 = n + g^*_c = n + \frac{a_1}{1-\alpha_1}$$
z*_2
$$z^*_2 = n - (1-\varepsilon)\, a_2 + (1 + (1-\varepsilon)(1-\alpha_2))\, \frac{a_1}{1-\alpha_1}$$
g*_2 > g*
$$g^*_2 = n + \varepsilon\, a_2 + (1-\varepsilon(1-\alpha_2))\, \frac{a_1}{1-\alpha_1} > g^*$$
Employment balance n*
$$n^*_1 = n,\ n^*_2 = n - \frac{(1-\varepsilon)(1-\alpha_2)\, a_2}{1-\alpha_2} - \frac{a_1}{1-\alpha_1}$$
TVC checks
$$z^*_1, z^*_2, m^*_1, m^*_2, g^*_1, g^*_2\ \text{satisfy TVC}$$
★ Reverse Prop
$$g^* = g^*_C = g^*_2 = z^*_2 = n + \frac{a_2}{1-\alpha_2},\ z^*_1 = n - (1-\varepsilon)\, a_1 + \cdots$$
★ HP utility (CRRA)
$$\int_0^{\infty} e^{-\rho t}\, (c_A(t) - \gamma_A)^{\eta}\, c_M(t)^{1-\eta}\, dt$$
★ M, A productions
$$Y_M(t) = X(t)\, F(L_M(t)),\ Y_A(t) = B_A\, G(L_A(t))$$
L balance + X dyn
$$L_M(t) + L_A(t) \leq 1,\ \dot X(t) = \kappa\, \cdots$$
★ Wage equality
$$B_A\, G'(1 - n(t)) = p(t)\, X(t)\, F'(n(t))$$
Subsistence condition
$$B_A\, G(1) > \gamma_A > 0$$
★ ★ HP relative price
$$c_A(t) = \gamma_A + \eta\, p(t)\, c_M(t) / (1-\eta)$$
★ ★ Malthus → Solow transition
$$\varphi(n(t)) = \gamma_A / B_A$$
n* solution
$$n^* = \varphi^{-1}(\gamma_A / B_A)$$
★ φ growth
$$\frac{\dot \varphi(t)}{\varphi(t)} = \frac{1}{\theta}\, [\alpha_1\, \gamma\, \eta(t)^{1/\varepsilon}\, \lambda(t)^{1-\alpha_2}\, \cdots]$$
η definition
$$\eta(t) \equiv \gamma^{\varepsilon}\, \frac{\varepsilon-1}{\varepsilon}\, [1 + \alpha_2\, \cdots]$$
★ HP boundary
$$X(0)/B_A\to X^F(0)/B_F:\quad n^*(0)\ \text{satisfies threshold}$$
★ GZ utility (log)
$$\mathbb{E}_t U_t(c(t), c(t+1)) = \log c(t) + \beta\, \mathbb{E}_t \log c(t+1)$$
GZ initial wealth
$$W_i(0) = w(0)\, l_i,\ w(t) = (1-\alpha)\, K(t)^{\alpha}$$
★ ★ Threshold W*
$$W^* \equiv \frac{\xi}{1 - (q/Q)^{\beta/(1+\beta)}} > 0$$
★ Fraction below g_F
$$g_F(t) \equiv 1 - G(W^*/w(t)) = 1 - G(W^*/[(1-\alpha)\, K(t)^{\alpha}])$$
★ K(t+1) dynamics
$$K(t+1) = \frac{\beta}{1+\beta}\, [q\, \int_{\chi(t)} l\, dG(l) + Q\, \int^{\chi(t)} l\, dG(l)]$$
★ ★ GW Becker quality-quantity
$$c(t)^{\beta}\, [y(t+1)\, n(t+1) - \frac{1}{2}\, \eta_0\, n(t+1)^2]$$
GW Y
$$Y(t) = Z^{\alpha}\, L(t)^{1-\alpha}$$
L dynamics
$$L(t+1) = n(t+1)\, L(t)$$
Wage = (1-α) L^(-α)
$$w(t+1) = (1-\alpha)\, L(t+1)^{-\alpha}$$
★ Fertility n
$$n(t+1) = \frac{(1-\alpha)\, \eta_0^{-1}\, L(t+1)^{-\alpha}}{1+\alpha}$$
★ Steady-state L*
$$L^* \equiv (1-\alpha)^{1/\alpha}\, \eta_0^{-1/\alpha}$$
★ M-sector Y_M
$$Y_M(t) = X(t)\, S(t)$$
★ Q-Q with education
$$c(t)^{\beta}\, [y(t+1)\, n(t+1) - \frac{1}{2}\, \eta_0\, (1-e(t)) + \eta_1\, X(t+1)\, e(t)]$$
★ X dynamics
$$X(t+1) - X(t) = \kappa\, S(t)$$
Unskilled wage
$$w_U(t) = (1-\alpha)\, U(t)^{-\alpha}$$
Skilled wage
$$w_S(t) = X(t)$$
Unskilled fertility
$$n_U(t+1) = w_U(t+1)\, \eta_0^{-1} = (1-\alpha)\, \eta_0^{-1}\, U(t+1)^{-\alpha}$$
Skilled fertility
$$n_S(t+1) = \eta_1^{-1}\, w_S(t+1)\, X(t+1)^{-1} = \eta_1^{-1}$$
★ Value functions V_U, V_S
$$V_U(t) = \frac{1}{2}\, (1-\alpha)^2\, \eta_0^{-1}\, U(t+1)^{-2\alpha},\ V_S(t) = \frac{1}{2}\, X(t+1)\, \eta_1^{-1}$$
★ Pre-take-off condition
$$\eta_1^{-1}\, X(0) < (1-\alpha)^2\, \eta_0^{-1}\, (L^*)^{-2\alpha}$$
U dynamics
$$U(t+1) = (1-\alpha)^{2/(1+\alpha)}\, \eta_0^{-1/(1+\alpha)}\, \cdots$$
S dynamics
$$S(t+1) = (1 - u(t+1))\, n_S(t+1)\, L(t) = \eta_1^{-1}\, (1 - u(t+1))\, L(t)$$
★ ★ X dynamics (post-takeoff)
$$X(t+1) = (1-\alpha)^{2/(1+\alpha)}\, \eta_0^{-(1-\alpha)/(1+\alpha)}\, \eta_1\, u(t+1)\, \cdots$$
★ K dynamics + migration
$$\dot K(t) = s\, F(K(t), L_U(t)) - \delta\, K(t)$$
Wage = MP_L
$$w_U(t) = \frac{\partial F(K(t), L_U(t))}{\partial L_U}$$
★ Migration ODE
$$\dot L_R(t) = \begin{cases} -\mu\, L_R(t) & \text{if } w_U > w_R \\ \in [-\mu\, L_R, 0] & \text{if } w_U = w_R \\ 0 & \text{else} \end{cases}$$
★ k(t) ODE (per capita)
$$\dot k(t) = s\, f(k(t)) - (\delta + \mu)\, k(t)$$
k condition no migration
$$k(t) f'(k(t)) \leq B_A:\quad \dot k(t) = s\, f(k(t)) - \delta\, k(t)$$
★ Threshold k-bar
$$f(\bar k) - \bar k\, f'(\bar k) = B_A$$
Stable hat-k
$$\frac{s\, f(\hat k)}{\hat k} = \delta$$
★ Roy max problem
$$\max_{[L(h)]} \int_0^{\bar h} A_h\, L(h)\, dh$$
★ Roy constraints + ability
$$\int_0^{\bar h} L(h)\, dh = L,\ \int_0^{\bar h} h\, L(h)\, dh = H,\ A_h \leq \lambda_L + h\, \lambda_H$$
★ ★ AC Y
$$Y(t) = \int_0^1 A(\nu, t)^{\beta}\, x(\nu, t)^{1-\beta}\, d\nu$$
Markup price
$$p(\nu, t) = \chi > 1$$
★ Profit
$$\pi(\nu, t) = \delta\, A(\nu, t)$$
★ Aggregate A growth
$$\bar A(t) = (1+g)\, \bar A(t-1)$$
★ g definition
$$g \equiv \eta + \bar \gamma - 1$$
★ A dynamics (entry + incumb)
$$A(\nu, t) = \eta\, \bar A(t-1) + \gamma\, A(t-1) + \varepsilon(\nu, t)$$
Difference equation a
$$a(t) = \frac{1}{1+g}\, [\eta + \gamma\, a(t-1)]$$
★ Stagnation regime switch
$$a(t) = \begin{cases} \frac{1}{1+g}\, [\bar \eta + \gamma\, a(t-1)] & \text{if } R(t) = 1 \\ \frac{1}{1+g}\, [\eta + \gamma\, a(t-1)] & \text{else} \end{cases}$$
Conditions
$$\bar \eta > \eta,\ \gamma < \bar \gamma < 1+g$$
★ a-hat threshold
$$\hat a \equiv \frac{\bar \eta - \eta}{\bar \gamma - \gamma} \in (0, 1)$$
Lock-in resource
$$C(1) + I(1) \leq Y(1),\ C(2) \leq Y(2)$$
Two production techs
$$y(\nu, 1) = l(\nu, 1),\ y(\nu, 2) = l(\nu, 2)\ \text{(old)},\ \alpha\, l(\nu, 2)\ \text{(new)}$$
Labor constraint
$$\int_0^1 l(\nu, t)\, d\nu \leq L$$
★ ★ Discount factor
$$\hat r = \beta^{-1} - 1$$
Variety demand
$$y(\nu, 2) = p(\nu, 2)^{-\varepsilon}\, Y(2)$$
★ Markup condition for new tech
$$\frac{\varepsilon}{\varepsilon - 1}\, \frac{1}{\alpha} > 1$$
Profit (period 2)
$$\pi(\nu, 2) = \frac{\alpha - 1}{\alpha}\, w(2)^{1-\varepsilon}\, Y(2)$$
★ ★ Lock-in profit signs
$$\pi^N < 0,\ \pi^I > 0$$
★ Lock-in condition
$$\frac{\beta\, \alpha\, L}{L - F} - \theta\, (\alpha - 1)\, L > F > \beta\, \frac{\alpha - 1}{\alpha}\, L$$
★ h(t+1) = e^γ Becker
$$h_i(t+1) = \begin{cases} e_i(t)^{\gamma} & \text{if } e_i(t) \geq 1 \\ \bar h & \text{if } e_i(t) < 1 \end{cases}$$
★ Education = δ × wage
$$e_i(t) = \delta\, w_i(t) = \delta\, A\, h_i(t)$$
★ ★ Two-regime threshold
$$\delta\, A > 1 > \delta\, A\, \bar h$$
Constant saving rate
$$\delta\ \text{constant saving rate (capital investment)}$$
★ ★ x(t+1) bistable
$$x(t+1) = \begin{cases} b_u(x) = \delta((1+r)(w_u + x) + w_u) & \text{if } x < f \\ \cdots & \text{else} \end{cases}$$
Continuous y
$$y(t) = f(k(t), x(t))$$
x ODE
$$\dot x(t) = g(k(t), x(t))$$
k ODE
$$\dot k(t) = s\, f(k(t), x(t)) - \delta\, k(t)$$
★ Closing utility
$$u = (1-\delta)^{-(1-\delta)}\, \delta^{-\delta}\, c^{1-\delta}\, b^{\delta}$$
★ Elite preferences
$$\mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t\, C^i(t)$$
Production Y_i
$$Y_i(t) = F(K_i(t), L_i(t))$$
Labor constraint
$$\int_{S_m} L_i(t)\, di \leq 1$$
First-best k*
$$k_i(t) = k^* \equiv (f')^{-1}(\beta^{-1} + \delta - 1)$$
Labor allocation
$$L_i(t) = L^* \equiv \min\!\left\{\bar L,\ \frac{1}{\theta_m}\right\}$$
Capacity condition
$$\theta_m\, \bar L > 1$$
Wage = MP_L
$$w(t) = w^* \equiv f(k^*) - k^*\, f'(k^*)$$
★ ★ Tax-transfer constraint
$$T_w(t) + \theta_m\, T_m(t) + \theta_e\, T_e(t) \leq \tau(t)\, \int_{S_m} F(K_i, L_i)\, di$$
★ Equilibrium definition
$$\{[K_i(s+1), L_i(s)]_{i \in S_m}\}_s\ \text{taking } p_t \text{ as given}$$
★ ★ Distorted Euler
$$\beta\, [(1-\tau(t+1))\, f'(k_i(t+1)) + (1-\delta)] = 1$$
★ Distorted k(τ)
$$\hat k(\tau) \equiv (f')^{-1}\!\left(\frac{\beta^{-1} + \delta - 1}{1 - \tau}\right)$$
★ k'(τ) < 0
$$\hat k'(\tau) = \frac{f'(\hat k)}{(1-\tau)\, f''(\hat k)} < 0$$
Distorted wage
$$\hat w(\tau(t)) = (1-\tau(t))\, [f(\hat k(\tau)) - \hat k(\tau)\, f'(\hat k(\tau))]$$
★ Elite transfer
$$T^e(t) = \frac{1}{\theta_e}\, \tau(t)\, \cdots$$
★ ★ Elite Bellman
$$V^e(\tau(t), [K_i]) = \max_{\tau(t+1) \in [0,1]} \{T^e(t) + \beta\, V^e(\tau(t+1), \cdots)\}$$
★ Optimal tau hat
$$f(\hat k(\hat \tau)) + \frac{\hat \tau}{1 - \hat \tau}\, \frac{[f'(\hat k(\hat \tau))]^2}{f''(\hat k(\hat \tau))} = 0$$
★ Cobb-Douglas
$$Y_i(t) = \frac{1}{\alpha}\, K_i(t)^{\alpha}\, [A_i(t)\, L_i(t)]^{1-\alpha}$$
Per capita CD
$$f(k_i) = \frac{1}{\alpha}\, (A_m)^{1-\alpha}\, k_i^{\alpha}$$
Tax with entry
$$T_w + \theta_m\, T_m + \theta_e\, T_e \leq \varphi\, \int_{S_m \cup S_e} \tau_i(t)\, F(K_i, L_i)\, di$$
★ k(τ) closed-form (CD)
$$k_i(t+1) = \hat k_i(\tau(t+1)) = (\beta(1-\tau(t+1)))^{1/(1-\alpha)}\, A_i$$
★ Net marginal product
$$(1-\alpha)\, \beta^{\alpha/(1-\alpha)}\, (1-\tau)^{\alpha/(1-\alpha)}\, A$$
★ ★ Wage equilibrium (Prop 22.3)
$$w(t) = \min\!\left\{\frac{1-\alpha}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, (1-\tau_e)^{\alpha/(1-\alpha)}\, A_e,\ \cdots\right\}$$
★ τ_RE definition
$$\tau_m(t) = \tau_{RE} \equiv \cdots$$
Elite max τ_m problem
$$\max_{\tau_m(t)} \frac{1-\alpha}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, A_e - w(t) \cdots$$
Constraint Le, Lm
$$\theta_e\, L_e(t) + \theta_m\, L_m(t) = 1$$
★ Lm condition
$$L_m(t) = \bar L\ \text{if } (1-\tau_m)^{1/(1-\alpha)}\, A_m \geq A_e$$
Prop 22.6 conditions
$$A_e \geq \varphi\, \alpha^{\alpha/(1-\alpha)}\, A_m\, \frac{\theta_m}{\theta_e}$$
★ ★ τ_COM
$$\tau_m(t) = \tau_{COM} \equiv \frac{\kappa(\bar L, \theta_e, \alpha, \varphi)}{1 + \kappa}$$
★ κ definition
$$\kappa(\bar L, \theta_e, \alpha, \varphi) \equiv \frac{1-\alpha}{\alpha}\, [1 + \theta_e\, \bar L]$$
★ η endogenous (entry)
$$\eta(t) = \eta(\theta_m\, C_m(t)) \in [0, 1]$$
Tax revenue equation
$$\varphi\, (1-\beta)(1-\alpha)\, \alpha^{-(1-2\alpha)/(1-\alpha)}\, \beta^{\alpha/(1-\alpha)}\, \cdots$$
Tax revenue alt
$$\varphi\, \alpha^{-(1-2\alpha)/(1-\alpha)}\, \beta^{\alpha/(1-\alpha)}\, \cdots$$
★ A_m FOC
$$\Gamma'(A_m) = \frac{1-\alpha}{\alpha\, (1-\beta)}\, \beta^{\alpha/(1-\alpha)}\, (1-\tau_{RE})^{1/(1-\alpha)} \cdots$$
A_m envelope
$$A_m - \frac{\alpha}{1-\alpha}\, \frac{\tau_m}{1-\tau_m}\, A_m + \tau_m\, \frac{dA_m}{d\tau_m} = 0$$
Welfare function J
$$J\ \text{be the social welfare function}$$
★ Voter sigma
$$\tilde \sigma^g_i(A) = 0\ \text{normalization}$$
★ Voting probability
$$v^g_i(p_A, p_B) = \begin{cases} 1 & U^g(p_A) - U^g(p_B) > \tilde \sigma^g_i \\ 1/2 & = \\ 0 & \text{else} \end{cases}$$
★ ★ Vote share π_A
$$\pi_A = \sum_g \lambda_g\, H^g(U^g(p_A) - U^g(p_B))$$
★ Equilibrium FOC
$$\sum_g \lambda_g\, h^g(0)\, DU^g(p^*) = 0$$
★ Pareto weights
$$\sum_g \chi^g\, \lambda_g\, U^g(p)$$
★ Heterogeneous Bellman
$$V_i(k_i \mid p_t) = \max_{k_i' \geq 0} \{(1-\tau(t))\, A_i\, f(k_i) - A_i\, k_i' + T(t+1) + \beta\, V_i(k_i' \mid p_{t+1})\}$$
★ Distorted Euler
$$\beta(1-\tau(t+1))\, f'(k_i(t+1)) = 1$$
k(τ)
$$\hat k(\tau) = (f')^{-1}((\beta(1-\tau))^{-1})$$
★ Tax revenue T(t+1)
$$T(t+1) = \int_0^1 \tau(t+1)\, A_i\, f(\hat k(\tau(t+1)))\, di = \tau(t+1)\, \bar A\, f(\hat k(\tau(t+1)))$$
★ ★ Single-crossing
$$\tau' \in [0, 1]\ \text{satisfy single-crossing condition} \Rightarrow \text{median voter决定}$$
★ Endogenous tech Y_i
$$Y_i(t) = \frac{1}{\alpha}\, K_i(t)^{\alpha}\, (A(t)\, L_i(t))^{1-\alpha}$$
Tax revenue
$$\text{Tax}(t) = \tau(t)\, \int_0^1 Y_i(t)\, di$$
★ Innovation A(t)
$$A(t) = \alpha\, \zeta^{1-\alpha}\, G(t)^{1/\zeta}$$
★ Sticky tax
$$\tau(t) = \bar \tau\quad \forall t$$
★ k_i closed
$$k_i(t) = (\beta(1-\bar \tau))^{1/(1-\alpha)}\, A(t)$$
★ Tax revenue function
$$T(A(t)) = (\beta(1-\bar \tau))^{\alpha/(1-\alpha)}\, \bar \tau\, A(t)^{\alpha}$$
★ ★ Elite Bellman (V^e)
$$V^e(A(t)) = \max_{A(t+1)} T(A(t)) - \frac{1-\alpha}{\alpha\, \zeta}\, A(t+1)^{\zeta} + \beta\, V^e(A(t+1))$$
FOC (V^e)
$$\frac{1-\alpha}{\alpha}\, A(t+1)^{\zeta - 1} = \beta\, (V^e)'(A(t+1))$$
Envelope
$$(V^e)'(A(t+1)) = T'(A(t)) = (\beta(1-\bar \tau))^{\alpha/(1-\alpha)}\, \bar \tau\, \alpha\, A(t)^{\alpha-1}$$
★ A(t+1) closed
$$A(t+1) = A[\bar \tau] \equiv \beta^{1/(1-\alpha)}\, (1-\alpha)^{-1}\, \cdots$$
★ V^e(A) closed
$$V^e(A(t)) = (\beta(1-\bar \tau))^{\alpha/(1-\alpha)}\, \bar \tau\, A(t)^{\alpha} + \beta^{1/(1-\alpha)}\, \cdots$$
★ ★ Y(τ-bar)
$$Y(t) = Y[\bar \tau] \equiv \frac{1}{\alpha}\, (\beta(1-\bar \tau))^{\alpha/(1-\alpha)}\, A^{\alpha}$$
★ ★ τ* optimal extraction
$$\bar \tau^* = \frac{1-\alpha}{1-\alpha + \alpha\, \zeta}$$
★ Closing argument
$$g = 0\ \text{(elite prefers no growth)}$$
★ Log Bellman
$$V^e(k) = \max_{\tau \in [0,1]} \log(\tau\, A\, k^{\alpha}) + \beta\, V^e(\alpha\, \beta(1-\tau)\, A\, k^{\alpha})$$
★ FOC log
$$\frac{1}{\tau} = \beta^2\, \alpha\, A\, k^{\alpha}\, (V^e)'(k') = \frac{\beta\, k'\, (V^e)'(k')}{1-\tau}$$
G(t) capital identity
$$G(t) = \tau(t)\, \bar K(t)$$
★ ★ Final condition (Prop 23.1)
$$A_m \geq \varphi\, \alpha^{\alpha/(1-\alpha)}\, A_e\, \theta_e/\theta_m$$
★ Heterogeneous y_i
$$y_i(t) = \frac{1}{\alpha}\, k_i(t)^{\alpha}\, (a_i(t)\, l_i(t))^{1-\alpha}$$
★ Profit
$$\pi(k_i \mid a_i, w, \tau) = \frac{1}{\alpha}\, (1-\tau)\, k_i^{\alpha}\, (a_i\, \bar L)^{1-\alpha} - w\, \bar L - \frac{1}{\beta}\, k_i$$
★ Z function
$$Z(\tau, w) = \max_{k_i} \pi(k_i \mid a_i = A_z, w, \tau)$$
★ Labor constraint
$$\int_0^1 e_i(t)\, l_i(t)\, di = \int_{i \in S^E_t} \bar L\, di \leq 1$$
★ ★ Markov productivity
$$a_i(t+1) = \begin{cases} A_H & \text{w/p } \sigma_H\ \text{if } a_i(t) = A_H \\ A_H & \text{w/p } \sigma_L\ \text{if } a_i(t) = A_L \\ A_L & \text{else} \end{cases}$$
★ ★ M stationary distribution
$$M \equiv \frac{\sigma_L}{1 - \sigma_H + \sigma_L} \in (0, 1)$$
★ k_i closed form
$$k_i(t) = (\beta(1-\tau(t)))^{1/(1-\alpha)}\, a_i(t)\, \bar L$$
★ Z closed
$$Z(\tau, w) = \frac{1-\alpha}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, (1-\tau)^{1/(1-\alpha)}\, A_z\, \bar L - w\, \bar L$$
★ Tax revenue
$$\sum_{i \in S^E_t} T(t) = \frac{1}{\alpha}\, \tau(t)\, (\beta(1-\tau(t)))^{\alpha/(1-\alpha)}\, \bar L\, \int_{i \in S^E_t} a_i(t)\, di$$
★ Worker value W^z
$$W^z(q_t) = w(t) + T(t) + \beta\, CW^z(q_{t+1})$$
★ Continuation value (worker)
$$CW^z(q_{t+1}) = \sigma_z\, \max\{W^H(q_{t+1});\ V^H(q_{t+1}) - b(t+1)\, \bar L\}$$
★ Entrepreneur value V^z
$$V^z(q_t) = w(t) + T(t) + Z(\tau, w) + \beta\, CV^z(q_{t+1})$$
★ Continuation (entrepreneur)
$$CV^z(q_{t+1}) = \sigma_z\, \max\{W^H; V^H\} + (1-\sigma_z)\, \max\{W^L; V^L\}$$
★ ★ wH wage equation
$$w_H(t) \equiv \max\!\left\{\frac{1-\alpha}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, (1-\tau)^{1/(1-\alpha)}\, A_H - b(t) + \beta(CV^H - CW^H)\, \bar L\right\}$$
★ wL wage equation
$$w_L(t) \equiv \max\!\left\{\frac{1-\alpha}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, (1-\tau)^{1/(1-\alpha)}\, A_L\, \cdots\right\}$$
★ Wage gap
$$w_H(t) \geq w_L(t)$$
★ Equilibrium wage
$$w_E(t) = w_H(t)$$
★ ★ μ(t) Markov dynamics
$$\mu(t) = \begin{cases} \sigma_H\, \mu(t-1) + \sigma_L(1 - \mu(t-1)) & \text{if (23.17) does not hold} \\ 1 & \text{if (23.17) holds} \end{cases}$$
★ T^E (entrepreneur tax revenue)
$$T^E(t) = \frac{1}{\alpha}\, \bar \tau\, (\beta(1-\bar \tau))^{\alpha/(1-\alpha)}\, \bar L\, \int_{i \in S^E_t} a_i(t)\, di$$
★ ★ Equilibrium b^E
$$b(t) = b^E \equiv \frac{1}{1-\beta}\, \frac{(1-\beta(1-\sigma_L))\, A_H + \beta(1-\sigma_H)\, A_L}{1 - \beta(\sigma_H - \sigma_L)}\, \cdots$$
★ Y^E (entrepreneur output)
$$Y^E(t) = \frac{1}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, [\mu(t)\, A_H + (1 - \mu(t))\, A_L]\, \bar L$$
★ ★ Y^E∞ limit
$$\lim_{t\to\infty} Y^E(t) = Y^E_{\infty} \equiv \frac{1}{\alpha}\, \beta^{\alpha/(1-\alpha)}\, [A_L + M(A_H - A_L)]$$
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