§17.1Brock-Mirman 1972 — Stochastic NGM
Brock-Mirman 19729 eq
(17.2) ★
★ Stochastic NGM utility
★ Stochastic NGM Utility
DRAFT
vc-passed
$$\max \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t\, u(c(t))$$
Intuition
DRAFT — ★ Brock-Mirman stochastic NGM 평생효용.
3세 비유
DRAFT — ★ Brock-Mirman stochastic NGM 평생효용.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ Brock-Mirman stochastic NGM 평생효용.
(17.3)
Stochastic capital constraint
Stochastic Capital Constraint
DRAFT
vc-passed
$$k(t+1) = f(k(t), z(t)) + (1-\delta)\, k(t) - c(t),\ k(t) \geq 0$$
Intuition
DRAFT — stochastic capital dyn (16.2 동일).
3세 비유
DRAFT — stochastic capital dyn (16.2 동일).
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — stochastic capital dyn (16.2 동일).
(17.4) ★
★ ★ Brock-Mirman Bellman
★ ★ Brock-Mirman Bellman
DRAFT
vc-passed
$$V(k, z) = \max_{k' \in [0, f(k,z) + (1-\delta)k]} u(\cdot) + \beta\, \mathbb{E}[V(k', z') \mid z]$$
Intuition
DRAFT — ★ ★ ★ stochastic NGM Bellman.
3세 비유
DRAFT — ★ ★ ★ stochastic NGM Bellman.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ ★ stochastic NGM Bellman.
(17.5) ★
★ ★ Stochastic Euler (NGM)
★ ★ Stochastic Euler (NGM)
DRAFT
vc-passed
$$u'(c) = \beta\, \mathbb{E}[(f'(\pi(k,z), z') + (1-\delta))\, u'(c')]$$
Intuition
DRAFT — ★ ★ Brock-Mirman Euler — 자본의 한계 가치 = 미래 한계 효용 + 한계 산출.
3세 비유
DRAFT — ★ ★ Brock-Mirman Euler — 자본의 한계 가치 = 미래 한계 효용 + 한계
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Brock-Mirman Euler — 자본의 한계 가치 = 미래 한계 효용 + 한계 산출.
(17.6) ★
★ Lucas asset pricing
★ Lucas Asset Pricing
DRAFT
vc-passed
$$u'(c(t)) = \beta\, \mathbb{E}_t[p(t+1)\, u'(c(t+1))]$$
Intuition
DRAFT — ★ ★ Lucas tree pricing — 자산가격의 SDF.
3세 비유
DRAFT — ★ ★ Lucas tree pricing — 자산가격의 SDF.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Lucas tree pricing — 자산가격의 SDF.
(17.7) ★
★ Stochastic NGM TVC
★ Stochastic NGM TVC
DRAFT
vc-passed
$$\lim_{t\to\infty} \mathbb{E}[\beta^t (f'(k(t), z(t)) + (1-\delta))\, u'(c(t))\, k(t) \mid z(0)] = 0$$
Intuition
DRAFT — ★ stochastic transversality.
3세 비유
DRAFT — ★ stochastic transversality.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ stochastic transversality.
(17.8)
Markov policy
Markov Policy
DRAFT
vc-passed
$$k(t+1) = \pi(k(t), z(t))$$
Intuition
DRAFT — ★ Markov policy.
3세 비유
DRAFT — ★ Markov policy.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ Markov policy.
(17.9)
Example 17.1 — Cobb-Douglas
Example 17.1 — Cobb-Douglas
DRAFT
vc-passed
$$\frac{1}{z\, k^{\alpha} - \pi(k, z)} = \beta\, \mathbb{E}\!\left[\frac{\alpha\, z'\, \pi(k, z)^{\alpha-1}}{z'\, \pi(k, z)^{\alpha} - \pi(\pi(k, z), z')}\right]$$
Intuition
DRAFT — Cobb-Douglas case Euler.
3세 비유
DRAFT — Cobb-Douglas case Euler.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — Cobb-Douglas case Euler.
(17.10) ★
★ Cobb-Douglas closed form
★ Cobb-Douglas Closed Form
DRAFT
vc-passed
$$\pi(k, z) = B_0 + B_1\, z\, k^{\alpha},\ \text{with } B_0, B_1\ \text{determined}$$
Intuition
DRAFT — ★ Brock-Mirman 의 closed-form policy.
3세 비유
DRAFT — ★ Brock-Mirman 의 closed-form policy.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ Brock-Mirman 의 closed-form policy.
§17.2Arrow-Debreu state-contingent claims
Arrow-Debreu State-Contingent13 eq
(17.11) ★
★ Arrow-Debreu sequence budget
★ Arrow-Debreu Sequence Budget
DRAFT
vc-passed
$$\sum_{t=0}^{\infty} \sum_{z^t \in Z^{\infty}} p_0[z^t]\, c[z^t] \leq \sum_{t=0}^{\infty} \sum_{z^t \in Z^{\infty}} w[z^t]$$
Intuition
DRAFT — ★ AD state-contingent budget.
3세 비유
DRAFT — ★ AD state-contingent budget.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ AD state-contingent budget.
(17.12)
AD utility (state-contingent)
AD Utility (State-Contingent)
DRAFT
vc-passed
$$\sum_{t=0}^{\infty} \beta^t \sum_{z^t \in Z^{\infty}} q[z^t \mid z_0]\, u(c[z^t])$$
Intuition
DRAFT — AD 평생효용.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AD 평생효용.
(17.13)
★ AD FOC
AD FOC
DRAFT
vc-passed
$$\beta^t\, q[z^t \mid z_0]\, u'(c[z^t]) = \lambda\, p_0[z^t]$$
Intuition
DRAFT — ★ AD 의 FOC.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AD 의 FOC.
(17.14)
Conditional expansion
Conditional Expansion
DRAFT
vc-passed
$$z^{t+1} = (z^t, z(t+1)),\ \beta^{t+1}\, q[z^{t+1} \mid z_0]\, u'(c[z^{t+1}]) = \lambda\, p_0[z^{t+1}]$$
Intuition
DRAFT — ★ recursive update.
3세 비유
DRAFT — ★ recursive update.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ recursive update.
(17.15)
AD factor pricing
AD Factor Pricing
DRAFT
vc-passed
$$p_0[z^t]\, [\partial F(K_e[z^t], L[z^t], z(t))/\partial K_e + (1-\delta)] = R_0[z^t]$$
Intuition
DRAFT — ★ AD factor pricing.
3세 비유
DRAFT — ★ AD factor pricing.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AD factor pricing.
(17.16)
Labor normalization
Labor Normalization
DRAFT
vc-passed
$$L[z^t] = 1\quad \forall z^t$$
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — L = 1.
(17.17)
Capital identity
Capital Identity
DRAFT
vc-passed
$$k_e[z^{t+1}] = k[z^t]$$
Intuition
DRAFT — ★ contingent vs spot capital.
3세 비유
DRAFT — ★ contingent vs spot capital.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ contingent vs spot capital.
(17.18)
Resource (state-contingent)
Resource (State-Contingent)
DRAFT
vc-passed
$$c[z^t] + k[z^t] \leq f(k[z^{t-1}], z(t)) + (1-\delta)\, k[z^{t-1}]$$
Intuition
DRAFT — AD resource.
3세 비유
DRAFT — AD resource.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AD resource.
(17.19)
Spot price = sum AD
Spot Price = Sum AD
DRAFT
vc-passed
$$p_0[z^t] = \sum_{z(t+1) \in Z} R_0[(z^t, z(t+1))]$$
Intuition
DRAFT — ★ AD spot price recursion.
3세 비유
DRAFT — ★ AD spot price recursion.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AD spot price recursion.
(17.20)
FOC ratio (AD)
FOC Ratio (AD)
DRAFT
vc-passed
$$u'(c[z^t]) = \sum_{z(t+1) \in Z} \lambda\, p_0[z^{t+1}]$$
Intuition
DRAFT — AD FOC sum.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AD FOC sum.
(17.21)
FOC ratio compact
FOC Ratio Compact
DRAFT
vc-passed
$$\beta\, u'(c[z^{t+1}]) = \lambda\, p_0[z^{t+1}]\, \beta^t\, q[z^{t+1} \mid z_0]$$
Intuition
DRAFT — ★ AD ratio.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AD ratio.
(17.22)
Bayes factorization
Bayes Factorization
DRAFT
vc-passed
$$q[z^{t+1} \mid z_0] = q[z^{t+1} \mid z_t]\, q[z^t \mid z_0]$$
Intuition
DRAFT — ★ Bayes 분해.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ Bayes 분해.
(17.23)
AD No-Ponzi
AD No-Ponzi
DRAFT
vc-passed
$$\lim_{t\to\infty} \sum_{z^{t-1} \in Z^{t-1}} p_0[z^{t-1}]\, k[z^{t-1}] \geq 0$$
Intuition
DRAFT — AD No-Ponzi.
3세 비유
DRAFT — AD No-Ponzi.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AD No-Ponzi.
§17.3Recursive Arrow security equilibrium
Recursive Arrow Security4 eq
(17.24) ★
★ Recursive Bellman Arrow
★ Recursive Bellman Arrow
DRAFT
vc-passed
$$V(a, z) = \max_{\{a'[z'\mid z]\}_{z' \in Z}} u(a + w - \sum_{z'} \bar p[z'\mid z]\, a'[z'\mid z]) + \beta\, \sum_{z'} q[z'\mid z]\, V(a'[z'\mid z], z')$$
Intuition
DRAFT — ★ ★ Arrow security 의 recursive Bellman.
3세 비유
DRAFT — ★ ★ Arrow security 의 recursive Bellman.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Arrow security 의 recursive Bellman.
(17.25)
★ Arrow Euler
★ Arrow Euler
DRAFT
vc-passed
$$\bar p[z'\mid z]\, u'(c[a, z]) = \beta\, q[z'\mid z]\, \frac{\partial V(a'[z'\mid z], z')}{\partial a}$$
Intuition
DRAFT — ★ Arrow security FOC.
3세 비유
DRAFT — ★ Arrow security FOC.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ Arrow security FOC.
(17.26)
Market clearing (Arrow)
Market Clearing (Arrow)
DRAFT
vc-passed
$$a'[z'\mid z] = R[z'\mid z]\, k$$
Intuition
DRAFT — Arrow market clearing.
3세 비유
DRAFT — Arrow market clearing.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — Arrow market clearing.
(17.27) ★
★ Arrow no-arb
★ Arrow No-Arb
DRAFT
vc-passed
$$\sum_{z' \in Z} \bar p[z'\mid z]\, R[z'\mid z] = 1$$
Intuition
DRAFT — ★ ★ Arrow security no-arbitrage.
3세 비유
DRAFT — ★ ★ Arrow security no-arbitrage.
Genealogy
| 인물 | 저작 | note |
|---|
| Arrow & Debreu (1954) | Existence of an Equilibrium for a Competitive Economy | DRAFT — state-contingent claims framework. |
| Roy Radner (1972) | Existence of Equilibrium of Plans, Prices, Price Expectations | DRAFT — sequential trading equivalent. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Arrow security no-arbitrage.
§17.4Heterogeneous agents (Bewley-Aiyagari)
Heterogeneous Agents7 eq
(17.28)
Utility u(C, L)
Utility u(C, L)
DRAFT
vc-passed
$$u(C, L)\ \text{convex compact set } [0, \bar L]$$
Intuition
DRAFT — labor + consumption joint.
3세 비유
DRAFT — labor + consumption joint.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — labor + consumption joint.
(17.29) ★
★ Stochastic Euler with labor
★ Stochastic Euler with Labor
DRAFT
vc-passed
$$u_c(\pi_c, \pi_l) = \beta\, \mathbb{E}[R(\pi_k(K, z), z')\, u_c(\pi_c', \pi_l') \mid z]$$
Intuition
DRAFT — ★ labor-extended Euler.
3세 비유
DRAFT — ★ labor-extended Euler.
Genealogy
| 인물 | 저작 | note |
|---|
| William Brock & Leonard Mirman (1972) | Optimal Economic Growth and Uncertainty (JET) | DRAFT — stochastic NGM 정전. |
| Robert Lucas (1978) | Asset Prices in an Exchange Economy (Econometrica) | DRAFT — Lucas tree asset pricing. |
| Stokey & Lucas with Prescott (1989) | Recursive Methods in Economic Dynamics | DRAFT — Bellman-style stochastic growth. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ labor-extended Euler.
(17.30) ★
★ ★ Borrowing constraint
★ ★ Borrowing Constraint
DRAFT
vc-passed
$$a_h(t) \geq -\frac{z_{\min}}{R - 1} \equiv -b$$
Intuition
DRAFT — ★ ★ Aiyagari natural debt limit.
3세 비유
DRAFT — ★ ★ Aiyagari natural debt limit.
Genealogy
| 인물 | 저작 | note |
|---|
| Mark Huggett (1993) | The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies (JEDC) | DRAFT — Bewley-Huggett incomplete-markets. |
| Truman Bewley (1986) | Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers | DRAFT — heterogeneous-agent incomplete markets pioneer. |
| S. Rao Aiyagari (1994) | Uninsured Idiosyncratic Risk and Aggregate Saving (QJE) | DRAFT — Aiyagari precautionary savings. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Aiyagari natural debt limit.
(17.31) ★
★ ★ Bewley-Aiyagari Bellman
★ ★ Bewley-Aiyagari Bellman
DRAFT
vc-passed
$$V_h(a, z) = \max_{a' \in [-b, R\, a + w\, z]} u(R\, a + w\, z - a') + \beta\, \mathbb{E}[V_h(a', z') \mid z]$$
Intuition
DRAFT — ★ ★ ★ heterogeneous-agent Bellman 정전.
3세 비유
DRAFT — ★ ★ ★ heterogeneous-agent Bellman 정전.
Genealogy
| 인물 | 저작 | note |
|---|
| Mark Huggett (1993) | The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies (JEDC) | DRAFT — Bewley-Huggett incomplete-markets. |
| Truman Bewley (1986) | Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers | DRAFT — heterogeneous-agent incomplete markets pioneer. |
| S. Rao Aiyagari (1994) | Uninsured Idiosyncratic Risk and Aggregate Saving (QJE) | DRAFT — Aiyagari precautionary savings. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ ★ heterogeneous-agent Bellman 정전.
(17.32) ★
★ Aiyagari k**
★ Aiyagari k**
DRAFT
vc-passed
$$f'(k^{**}) = \beta^{-1} - (1-\delta)$$
Intuition
DRAFT — ★ ★ Aiyagari modified golden rule (deterministic).
3세 비유
DRAFT — ★ ★ Aiyagari modified golden rule (deterministic).
Genealogy
| 인물 | 저작 | note |
|---|
| Mark Huggett (1993) | The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies (JEDC) | DRAFT — Bewley-Huggett incomplete-markets. |
| Truman Bewley (1986) | Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers | DRAFT — heterogeneous-agent incomplete markets pioneer. |
| S. Rao Aiyagari (1994) | Uninsured Idiosyncratic Risk and Aggregate Saving (QJE) | DRAFT — Aiyagari precautionary savings. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ Aiyagari modified golden rule (deterministic).
(17.33)
★ Aiyagari k* < k**
★ Aiyagari k* < k**
DRAFT
vc-passed
$$f'(k^*) < \beta^{-1} - (1-\delta)$$
Intuition
DRAFT — ★ Aiyagari stochastic capital > deterministic — precautionary saving.
3세 비유
DRAFT — ★ Aiyagari stochastic capital > deterministic — pr
Genealogy
| 인물 | 저작 | note |
|---|
| Mark Huggett (1993) | The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies (JEDC) | DRAFT — Bewley-Huggett incomplete-markets. |
| Truman Bewley (1986) | Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers | DRAFT — heterogeneous-agent incomplete markets pioneer. |
| S. Rao Aiyagari (1994) | Uninsured Idiosyncratic Risk and Aggregate Saving (QJE) | DRAFT — Aiyagari precautionary savings. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ Aiyagari stochastic capital > deterministic — precautionary saving.
(17.34) ★
★ ★ Precautionary k*
★ ★ Precautionary k*
DRAFT
vc-passed
$$k^* > k^{**}$$
Intuition
DRAFT — ★ ★ ★ Aiyagari 의 핵심 — 불확실성 → 저축 ↑ → 자본 ↑ (낮은 r).
3세 비유
DRAFT — ★ ★ ★ Aiyagari 의 핵심 — 불확실성 → 저축 ↑ → 자본 ↑ (낮은 r).
Genealogy
| 인물 | 저작 | note |
|---|
| Mark Huggett (1993) | The Risk-Free Rate in Heterogeneous-Agent Incomplete-Insurance Economies (JEDC) | DRAFT — Bewley-Huggett incomplete-markets. |
| Truman Bewley (1986) | Stationary Monetary Equilibrium with a Continuum of Independently Fluctuating Consumers | DRAFT — heterogeneous-agent incomplete markets pioneer. |
| S. Rao Aiyagari (1994) | Uninsured Idiosyncratic Risk and Aggregate Saving (QJE) | DRAFT — Aiyagari precautionary savings. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ ★ Aiyagari 의 핵심 — 불확실성 → 저축 ↑ → 자본 ↑ (낮은 r).
§17.5Stochastic OLG (Diamond + shock)
Stochastic OLG6 eq
(17.35)
OLG utility (stochastic)
OLG Utility (Stochastic)
DRAFT
vc-passed
$$U_t(c_1(t), c_2(t+1)) = \log c_1(t) + \beta\, \log c_2(t+1)$$
Intuition
DRAFT — ★ stochastic OLG.
3세 비유
DRAFT — ★ stochastic OLG.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ stochastic OLG.
(17.36)
Population n
Population n
DRAFT
vc-passed
$$L(t) = (1+n)^t\, L(0)$$
Intuition
DRAFT — OLG population.
3세 비유
DRAFT — OLG population.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — OLG population.
(17.37)
Stochastic Cobb-Douglas Y
Stochastic Cobb-Douglas Y
DRAFT
vc-passed
$$Y(t) = z(t)\, K(t)^{\alpha}\, L(t)^{1-\alpha},\ R(k, z) = \alpha\, z\, k^{\alpha-1},\ w(k, z) = (1-\alpha)\, z\, k^{\alpha}$$
Intuition
DRAFT — Cobb-Douglas + shock.
3세 비유
DRAFT — Cobb-Douglas + shock.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — Cobb-Douglas + shock.
(17.38) ★
★ OLG saving = β/(1+β) w
★ OLG Saving = β/(1+β) w
DRAFT
vc-passed
$$s(k(t), z(t)) = \frac{\beta}{1+\beta}\, w(k(t), z(t))$$
Intuition
DRAFT — ★ stochastic OLG 의 저축 — log utility 결과.
3세 비유
DRAFT — ★ stochastic OLG 의 저축 — log utility 결과.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ stochastic OLG 의 저축 — log utility 결과.
(17.39)
OLG capital dynamic
OLG Capital Dynamic
DRAFT
vc-passed
$$k(t+1) = \pi(k, z) = s(k, z)/(1+n)$$
Intuition
DRAFT — ★ Diamond OLG dynamic.
3세 비유
DRAFT — ★ Diamond OLG dynamic.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ Diamond OLG dynamic.
(17.40) ★
★ OLG steady-state k*
★ OLG Steady-State k*
DRAFT
vc-passed
$$k^* = \left(\frac{\beta(1-\alpha)\, \bar z}{(1+n)(1+\beta)}\right)^{1/(1-\alpha)}$$
Intuition
DRAFT — ★ ★ stochastic OLG steady-state.
3세 비유
DRAFT — ★ ★ stochastic OLG steady-state.
Genealogy
| 인물 | 저작 | note |
|---|
| Peter Diamond (1965) | National Debt in a Neoclassical Growth Model | DRAFT — OLG. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ stochastic OLG steady-state.
§17.6Acemoglu-Zilibotti 1997 finance-growth
Acemoglu-Zilibotti 199724 eq
(17.41)
AZ 생산함수
AZ Production
DRAFT
vc-passed
$$Y(t) = K(t)^{\alpha}\, L(t)^{1-\alpha}$$
Intuition
DRAFT — AZ 1997 setup.
3세 비유
DRAFT — AZ 1997 setup.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ 1997 setup.
(17.42)
Risky vs safe
Risky vs Safe
DRAFT
vc-passed
$$q < Q$$
Intuition
DRAFT — ★ q (safe), Q (risky) returns.
3세 비유
DRAFT — ★ q (safe), Q (risky) returns.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ q (safe), Q (risky) returns.
(17.43)
Threshold M(j)
Threshold M(j)
DRAFT
vc-passed
$$M(j) = \max\!\{0,\ D(1-\gamma)(j-\gamma)\}$$
Intuition
DRAFT — ★ AZ minimum scale.
3세 비유
DRAFT — ★ AZ minimum scale.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ minimum scale.
(17.44)
AZ utility
AZ Utility
DRAFT
vc-passed
$$\mathbb{E}_t U_t = \log c_1(t) + \beta \int_0^1 \log c_2(j, t+1)\, dj$$
Intuition
DRAFT — ★ AZ 평생효용 — risky payoff.
3세 비유
DRAFT — ★ AZ 평생효용 — risky payoff.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ 평생효용 — risky payoff.
(17.45)
AZ wage
AZ Wage
DRAFT
vc-passed
$$w(j, t+1) = (1-\alpha)\, K(j, t+1)^{\alpha}$$
Intuition
DRAFT — AZ wage.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ wage.
(17.46)
AZ R(j, t+1)
AZ R(j, t+1)
DRAFT
vc-passed
$$R(j, t+1) = \alpha\, K(j, t+1)^{\alpha-1}$$
Intuition
DRAFT — AZ interest rate per project.
3세 비유
DRAFT — AZ interest rate per project.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ interest rate per project.
(17.47)
AZ portfolio max
AZ Portfolio Max
DRAFT
vc-passed
$$\max_{s, X, [I(j)]} \log c(t) + \beta \int_0^1 \log c(j, t+1)\, dj$$
Intuition
DRAFT — AZ portfolio choice.
3세 비유
DRAFT — AZ portfolio choice.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ portfolio choice.
(17.48)
Investment constraint
Investment Constraint
DRAFT
vc-passed
$$X(t) + \int_0^1 I(j, t)\, dj = s(t)$$
Intuition
DRAFT — 총 투자 = 저축.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — 총 투자 = 저축.
(17.49)
c(j, t+1)
c(j, t+1)
DRAFT
vc-passed
$$c(j, t+1) = R(j, t+1)\, [q\, X(t) + Q\, I(j, t)]$$
Intuition
DRAFT — ★ AZ 노년 소비.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ 노년 소비.
(17.50)
Sparse investment
Sparse Investment
DRAFT
vc-passed
$$I(j, t) = 0\quad \forall j \notin J(t)$$
Intuition
DRAFT — ★ J(t) 이외 0.
3세 비유
DRAFT — ★ J(t) 이외 0.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ J(t) 이외 0.
(17.51)
Income constraint
Income Constraint
DRAFT
vc-passed
$$c(t) + s(t) \leq w(t)$$
Intuition
DRAFT — 젊은 시기 budget.
3세 비유
DRAFT — 젊은 시기 budget.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — 젊은 시기 budget.
(17.52)
★ AZ saving
★ AZ Saving
DRAFT
vc-passed
$$s^*(t) = \frac{\beta}{1+\beta}\, w(t)$$
Intuition
DRAFT — ★ AZ saving = log utility.
3세 비유
DRAFT — ★ AZ saving = log utility.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ saving = log utility.
(17.53)
AZ simpler
AZ Simpler
DRAFT
vc-passed
$$\max_{X, I} n^*(t)\, \log R_G(t+1)(q\, X + Q\, I) + (1-n^*(t))\, \log R_B(\cdots)$$
Intuition
DRAFT — ★ G/B (good/bad) outcome.
3세 비유
DRAFT — ★ G/B (good/bad) outcome.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ G/B (good/bad) outcome.
(17.54)
Sparse constraint
Sparse Constraint
DRAFT
vc-passed
$$X(t) + n^*(t)\, I(t) \leq s^*(t)$$
Intuition
DRAFT — AZ portfolio.
3세 비유
DRAFT — AZ portfolio.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ portfolio.
(17.55)
★ X* solution
★ X* Solution
DRAFT
vc-passed
$$X^*(t) = \frac{(1-n^*(t))\, Q}{Q - q\, n^*(t)}\, s^*(t)$$
Intuition
DRAFT — ★ AZ safe investment.
3세 비유
DRAFT — ★ AZ safe investment.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ safe investment.
(17.56)
★ I* solution
★ I* Solution
DRAFT
vc-passed
$$I^*(j, t) = \frac{Q-q}{Q - q\, n(t)}\, s^*(t)\quad \text{for } j \leq n^*(t),\ 0\ \text{else}$$
Intuition
DRAFT — ★ AZ risky investment.
3세 비유
DRAFT — ★ AZ risky investment.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ risky investment.
(17.57) ★
★ ★ AZ portfolio threshold
★ ★ AZ Portfolio Threshold
DRAFT
vc-passed
$$n^*[K(t)] = \frac{(Q + q\gamma) - \sqrt{(Q+q)^2 - 4q[D^{-1}(Q-q)(1-\gamma)\, K^{\alpha} + \gamma]}}{\cdots}$$
Intuition
DRAFT — ★ ★ AZ diversification threshold.
3세 비유
DRAFT — ★ ★ AZ diversification threshold.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ AZ diversification threshold.
(17.58) ★
★ ★ K(t+1) random walk
★ ★ K(t+1) Random Walk
DRAFT
vc-passed
$$K(t+1) = \begin{cases} \frac{q(1-n^*[K(t)])}{Q - q\, n^*[K(t)]}\, Q\, K(t)^{\alpha} & \text{w/p } 1-n^*[K(t)] \\ Q\, K(t)^{\alpha} & \text{w/p } n^*[K(t)] \end{cases}$$
Intuition
DRAFT — ★ ★ AZ K transition — random.
3세 비유
DRAFT — ★ ★ AZ K transition — random.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ AZ K transition — random.
(17.59)
★ Volatility of growth
★ Volatility of Growth
DRAFT
vc-passed
$$\sigma_e(n^*[K(t)]) = (1-n^*[K(t)])\, \frac{q(1-n^*[K(t)])}{Q - q\, n^*[K(t)]}\, Q + n^*[K(t)]\, Q$$
Intuition
DRAFT — ★ AZ growth volatility.
3세 비유
DRAFT — ★ AZ growth volatility.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ growth volatility.
(17.60) ★
★ K_QSSB and K_QSSG
★ K_QSSB and K_QSSG
DRAFT
vc-passed
$$K_{QSSB} = \cdots,\ K_{QSSG} = Q^{1/(1-\alpha)}$$
Intuition
DRAFT — ★ ★ AZ two steady states (poverty trap + advanced).
3세 비유
DRAFT — ★ ★ AZ two steady states (poverty trap + advanced)
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ canonical bridge — ★ ★ AZ two steady states (poverty trap + advanced).
(17.61)
Poverty trap condition
Poverty Trap Condition
DRAFT
vc-passed
$$D < \frac{1}{1-\alpha}\, Q^{\alpha/(1-\alpha)}$$
Intuition
DRAFT — ★ AZ poverty trap 존재 조건.
3세 비유
DRAFT — ★ AZ poverty trap 존재 조건.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ poverty trap 존재 조건.
(17.62)
★ K growth log linear
★ K Growth Log Linear
DRAFT
vc-passed
$$\Delta \log K(t+1) = \log[\sigma(n^*[K(t)])] - (1-\alpha)\, \log K(t)$$
Intuition
DRAFT — ★ AZ log dynamics.
3세 비유
DRAFT — ★ AZ log dynamics.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ log dynamics.
(17.63)
★ AZ social planner
★ AZ Social Planner
DRAFT
vc-passed
$$\max_{n, X, I} n(t)\, \int_0^{n(t)} \log(qX + QI(j, t))\, dj + (1-n(t)) \log(\cdots)$$
Intuition
DRAFT — AZ social planner problem.
3세 비유
DRAFT — AZ social planner problem.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — AZ social planner problem.
(17.64)
Proposition 17.11 (closing)
Proposition 17.11 (Closing)
DRAFT
vc-passed
$$I^S(j, t) = M(j^*) > M(j)\ \text{for } j < j^*(t),\ I^S(j) = M(j)\ \text{for } j \geq j^*(t)$$
Intuition
DRAFT — ★ AZ social planner solution.
3세 비유
DRAFT — ★ AZ social planner solution.
Genealogy
| 인물 | 저작 | note |
|---|
| Daron Acemoglu & Fabrizio Zilibotti (1997) | Was Prometheus Unbound by Chance? Risk, Diversification, and Growth (JPE) | DRAFT — finance-growth poverty trap. |
| Greenwood-Jovanovic (1990) | Financial Development, Growth, and the Distribution of Income | DRAFT — finance-growth interaction. |
Applications
- Brock-Mirman 1972 stochastic NGM
- Lucas 1978 asset pricing
- Aiyagari 1994 incomplete markets
- Acemoglu-Zilibotti 1997 finance-growth
- Krusell-Smith 1998 hetero agent macro
Civilizational Relevance
DRAFT — ★ AZ social planner solution.