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1D Problem

The full solution can be found at problem1.ipynb.

Problem Setup

Parameter and Variable Definitions

Parameter Definition Value
\(\gamma^j\) relative risk aversion \(\gamma^i=2, \gamma^h=5\)
\(\rho^j\) discount rate \(\rho^i=\rho^h=0.05\)
\(\zeta^j\) intertemporal elasticity of substitution \(\zeta^i=\zeta^h=1.00005\)
\(\mu^a\) growth rate of capital \(\mu^a=0.04\)
\(\sigma^a\) volatility of capital \(\sigma^{a}=0.2\)
\(\mu^O\) sentiment factor \(\mu^O=0.04\)
\(\alpha^a\) productivity \(\alpha^a=0.1\)
\(\kappa\) investment cost \(\kappa=10000\)
Type Definition
State Variables \(\eta_t\in[0,1]\) (\(\eta\))
Agents \(\xi_t^i\), \(\xi_t^h\)
Endogenous Variables \(\mu_t^{\eta}\), \(\sigma_t^{\eta a}\), \(q_t^a\), \(w_t^{ia}\), \(w_t^{ha}\)

Equations

\[\iota_t^a = \frac{q_t^a - 1}{\kappa}\]
\[\Phi(\iota_t^a) = \frac{1}{\kappa}\log(1+\kappa\iota_t^a)\]
\[c_t^j = (\rho^j)^{\zeta^j}(\xi_t^j)^{1-\zeta^j}\]
\[\sigma_t^{qa} = \frac{1}{q_t^a} \frac{\partial q_t^a}{\partial \eta_t} \sigma_t^{\eta a} \eta_t\]
\[\sigma_t^{nja} = w_t^{ja}(\sigma^a + \sigma_t^{qa})\]
\[\sigma_t^{\xi ja} = \frac{1}{\xi_t^j}\frac{\partial \xi_t^j}{\partial \eta_t}\sigma_t^{\eta a}\eta_t \]
\[\sigma_t^{na} = \eta_t\sigma_t^{nia} + (1-\eta_t)\sigma_t^{nha}\]
\[\mu_t^{qa} = \frac{1}{q_t^a} \left(\frac{\partial q_t^a}{\partial \eta_t} \mu_t^{\eta} \eta_t + \frac{1}{2} \frac{\partial^2 q_t^a}{\partial \eta_t^2} (\sigma_t^{\eta a} \eta_t)^2\right)\]
\[r_t^{ka} = \mu_t^{qa} + \mu^a + \Phi_t^a + \sigma^a\sigma^{qa} + \frac{\alpha^a - \iota_t^a}{q_t^a}\]
\[r_t = r_t^{ka} - \gamma^hw_t^{ha}(\sigma^a + \sigma_t^{qa})^2 + (1-\gamma^h)\sigma_t^{\xi ha}(\sigma^a + \sigma_t^{qa})\]
\[\mu_t^{nj} = r_t - c_t^j + w_t^{ja}(r_t^{ka} - r_t)\]
\[\mu_t^{\xi j} = \frac{1}{\xi_t^j}\left(\frac{\partial \xi_t^j}{\partial \eta_t}\mu_t^{\eta}\eta_t + \frac{1}{2}\frac{\partial^2 \xi_t^j}{\partial \eta_t^2}(\sigma_t^{\eta a}\eta_t)^2\right)\]
\[\hat{r_t^{ka}} = r_t^{ka} + \frac{\mu^O - \mu^a}{\sigma^a}(\sigma^a + \sigma_t^{qa})\]

Endogenous Equations

\[\mu_t^{\eta} = (1-\eta_t)(\mu_t^{ni} - \mu_t^{nh}) +(\sigma_t^{na})^2 - \sigma_t^{nia}\sigma_t^{na}\]
\[\sigma_t^{\eta a} = (1-\eta_t)(\sigma_t^{nia} - \sigma_t^{nha})\]
\[\hat{r_t^{ka}} - r_t = \gamma^iw_t^{ia}(\sigma^a + \sigma_t^{qa})^2 - (1-\gamma^i)\sigma_t^{\xi ia}(\sigma^{a} + \sigma_t^{qa})\]
\[1 = w_t^{ia}\eta_t + w_t^{ha}(1-\eta_t)\]
\[\alpha^a - \iota_t^a = (c_t^i\eta_t + c_t^h(1 - \eta_t))q_t^a\]

HJB Equations

\[ \frac{\rho^i}{1-\frac{1}{\zeta^i}}\left( \left(\frac{c_t^i}{\xi_t^i} \right)^{1-1/\zeta^i}-1 \right) + \mu_t^{\xi i} + \mu_t^{ni} - \frac{\gamma^i}{2}(\sigma_t^{nia})^2 - \frac{\gamma^i}{2}(\sigma_t^{\xi ia})^2 + (1-\gamma^i)\sigma_t^{\xi ia}\sigma_t^{nia}\]
\[ \frac{\rho^h}{1-\frac{1}{\zeta^h}}\left( \left(\frac{c_t^h}{\xi_t^h} \right)^{1-1/\zeta^h}-1 \right) + \mu_t^{\xi h} + \mu_t^{nh} - \frac{\gamma^h}{2}(\sigma_t^{nha})^2 - \frac{\gamma^h}{2}(\sigma_t^{\xi ha})^2 + (1-\gamma^h)\sigma_t^{\xi ha}\sigma_t^{nha}\]

Implementation

  1. Import necessary packages 

    import os
import numpy as np
import torch
import matplotlib.pyplot as plt
from deep_macrofin import PDEModel
from deep_macrofin import OptimizerType, plot_loss_df, set_seeds

  2. Define LaTex-variable mapping so that we can parse LaTex equations 

    latex_var_mapping = {
    # constants
    r"\gamma^i": "gammai",
    r"\gamma^h": "gammah",
    r"\rho^i": "rhoi",
    r"\rho^h": "rhoh",
    r"\sigma^a": "siga",
    r"\sigma^{a}": "siga",
    r"\mu^a": "mua",
    r"\mu^O": "muO",
    r"\zeta^i": "zetai",
    r"\zeta^h": "zetah",
    r"\alpha^a": "aa",
    r"\kappa": "kappa",

    # state variables
    r"\eta_t": "eta",

    # learnable variables (agents + endogenous variables)
    r"\xi_t^i": "xii",
    r"\xi_t^h": "xih",
    r"\mu_t^{\eta}": "mue",
    r"\sigma_t^{\eta a}": "sigea",
    r"q_t^a": "qa",
    r"w_t^{ia}": "wia",
    r"w_t^{ha}": "wha",

    # variables defined by equations
    r"\iota_t^a": "iota_a",
    r"\Phi_t^a": "phi_a",
    r"c_t^i": "ci",
    r"c_t^h": "ch",
    r"\sigma^{qa}": "sigqa",
    r"\sigma_t^{qa}": "sigqa",
    r"\sigma_t^{nia}": "signia",
    r"\sigma_t^{nha}": "signha",
    r"\sigma_t^{\xi ia}": "sigxia",
    r"\sigma_t^{\xi ha}": "sigxha",
    r"\sigma_t^{na}": "signa",
    r"\mu_t^{qa}": "muqa",
    r"\hat{r_t^{ka}}": "rka_hat", # define hat first, and replace hat first
    r"r_t^{ka}": "rka",
    r"\mu_t^{ni}": "muni",
    r"\mu_t^{nh}": "munh",
    r"\mu_t^{\xi i}": "muxi",
    r"\mu_t^{\xi h}": "muxh",
}

  3. Define the problem 

    set_seeds(0)
pde_model = PDEModel("problem1", {"num_epochs": 2000, "loss_log_interval": 100, "optimizer_type": OptimizerType.Adam}, latex_var_mapping)
pde_model.set_state(["eta"], {"eta": [0.01, 0.99]})
pde_model.add_agents(["xii", "xih"], {"xii": {"positive": True}, "xih": {"positive": True}})
pde_model.add_endogs(["mue", "qa", "wia", "wha", "sigea"], {"qa": {"positive": True}})
pde_model.add_params({
    "gammai": 2.0,
    "gammah": 5.0,
    "rhoi": 0.05,
    "rhoh": 0.05,
    "siga": 0.2, #\sigma^{a}
    "mua": 0.04,
    "muO": 0.04,
    "zetai":1.00005,
    "zetah":1.00005,
    "aa":0.1,
    "kappa":10000
})
pde_model.add_equation(r"$\iota_t^a = \frac{q_t^a - 1}{\kappa}$")
pde_model.add_equation(r"$\Phi_t^a = \frac{1}{\kappa} * \log(1+\kappa * \iota_t^a)$")
pde_model.add_equation(r"$c_t^i = (\rho^i)^{\zeta^i} * (\xi_t^i)^{1-\zeta^i}$")
pde_model.add_equation(r"$c_t^h = (\rho^h)^{\zeta^h} * (\xi_t^h)^{1-\zeta^h}$")
pde_model.add_equation(r"$\sigma_t^{qa} = \frac{1}{q_t^a}  * \frac{\partial q_t^a}{\partial \eta_t}  * \sigma_t^{\eta a}  * \eta_t$")
pde_model.add_equation(r"$\sigma_t^{nia} = w_t^{ia}*(\sigma^a + \sigma_t^{qa})$")
pde_model.add_equation(r"$\sigma_t^{nha} = w_t^{ha}*(\sigma^a + \sigma_t^{qa})$")
pde_model.add_equation(r"$\sigma_t^{\xi ia}  = \frac{1}{\xi_t^i} * \frac{\partial \xi_t^i}{\partial \eta_t} * \sigma_t^{\eta a} * \eta_t$")
pde_model.add_equation(r"$\sigma_t^{\xi ha}  = \frac{1}{\xi_t^h} * \frac{\partial \xi_t^h}{\partial \eta_t} * \sigma_t^{\eta a} * \eta_t$")
pde_model.add_equation(r"$\sigma_t^{na} = \eta_t * \sigma_t^{nia} + (1-\eta_t) * \sigma_t^{nha}$")
pde_model.add_equation(r"$\mu_t^{qa} = \frac{1}{q_t^a}  * \left(\frac{\partial q_t^a}{\partial \eta_t}  * \mu_t^{\eta}  * \eta_t + \frac{1}{2}  * \frac{\partial^2 q_t^a}{\partial \eta_t^2}  * (\sigma_t^{\eta a}  * \eta_t)^2\right)$")
pde_model.add_equation(r"$r_t^{ka} = \mu_t^{qa} + \mu^a + \Phi_t^a + \sigma^a * \sigma^{qa} + \frac{\alpha^a - \iota_t^a}{q_t^a}$")
pde_model.add_equation(r"$r_t = r_t^{ka} - \gamma^h * w_t^{ha} * (\sigma^a + \sigma_t^{qa})^2 + (1-\gamma^h) * \sigma_t^{\xi ha} * (\sigma^a + \sigma_t^{qa})$")
pde_model.add_equation(r"$\mu_t^{ni} = r_t - c_t^i + w_t^{ia} * (r_t^{ka} - r_t)$")
pde_model.add_equation(r"$\mu_t^{nh} = r_t - c_t^h + w_t^{ha} * (r_t^{ka} - r_t)$")
pde_model.add_equation(r"$\mu_t^{\xi i} = \frac{1}{\xi_t^i} * \left(\frac{\partial \xi_t^i}{\partial \eta_t} * \mu_t^{\eta} * \eta_t + \frac{1}{2} * \frac{\partial^2 \xi_t^i}{\partial \eta_t^2} * (\sigma_t^{\eta a} * \eta_t)^2\right)$")
pde_model.add_equation(r"$\mu_t^{\xi h} = \frac{1}{\xi_t^h} * \left(\frac{\partial \xi_t^h}{\partial \eta_t} * \mu_t^{\eta} * \eta_t + \frac{1}{2} * \frac{\partial^2 \xi_t^h}{\partial \eta_t^2} * (\sigma_t^{\eta a} * \eta_t)^2\right)$")
pde_model.add_equation(r"$\hat{r_t^{ka}} = r_t^{ka} + \frac{\mu^O - \mu^a}{\sigma^a} * (\sigma^a + \sigma_t^{qa})$")

pde_model.add_endog_equation(r"$\mu_t^{\eta} = (1-\eta_t) * (\mu_t^{ni} - \mu_t^{nh}) +(\sigma_t^{na})^2  - \sigma_t^{nia} * \sigma_t^{na}$")
pde_model.add_endog_equation(r"$\sigma_t^{\eta a} = (1-\eta_t) * (\sigma_t^{nia} - \sigma_t^{nha})$")
pde_model.add_endog_equation(r"$\hat{r_t^{ka}} - r_t = \gamma^i * w_t^{ia} * (\sigma^a  + \sigma_t^{qa})^2 - (1-\gamma^i) * \sigma_t^{\xi ia} * (\sigma^{a}  + \sigma_t^{qa})$")
pde_model.add_endog_equation(r"$1 = w_t^{ia} * \eta_t + w_t^{ha} * (1-\eta_t)$")
pde_model.add_endog_equation(r"$\alpha^a - \iota_t^a = (c_t^i*\eta_t + c_t^h * (1 - \eta_t)) * q_t^a$")

pde_model.add_hjb_equation(r"$\frac{\rho^i}{1-\frac{1}{\zeta^i}} * \left( \left(\frac{c_t^i}{\xi_t^i} \right)^{1-1/\zeta^i}-1 \right) + \mu_t^{\xi i} +  \mu_t^{ni} - \frac{\gamma^i}{2} * (\sigma_t^{nia})^2  - \frac{\gamma^i}{2} * (\sigma_t^{\xi ia})^2 + (1-\gamma^i) * \sigma_t^{\xi ia} * \sigma_t^{nia}$")
pde_model.add_hjb_equation(r"$\frac{\rho^h}{1-\frac{1}{\zeta^h}} * \left( \left(\frac{c_t^h}{\xi_t^h} \right)^{1-1/\zeta^h}-1 \right) + \mu_t^{\xi h} +  \mu_t^{nh} - \frac{\gamma^h}{2} * (\sigma_t^{nha})^2  - \frac{\gamma^h}{2} * (\sigma_t^{\xi ha})^2 + (1-\gamma^h) * \sigma_t^{\xi ha} * \sigma_t^{nha}$")

print(pde_model)

  4. Train and evaluate the best model with min loss 

    pde_model.train_model("./models/1d_prob", "1d_prob.pt", True)
pde_model.load_model(torch.load("./models/1d_prob/1d_prob_best.pt"))
pde_model.eval_model(True)

  5. Plot the solutions, with additional variables defined by equations. 

    pde_model.plot_vars([r"$q_t^a$", r"$\sigma_t^{qa}$", r"$w_t^{ia}$", r"$w_t^{ha}$", "rp_a = rka-r_t"])