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Re-based and cherry picked from 1505 #1568

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74 changes: 44 additions & 30 deletions examples/ConsNewKeynesianModel/SSJ_explanation.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -5,21 +5,35 @@
"id": "5d86d99a",
"metadata": {},
"source": [
"# Sequence Space Jacobian method to Solve HANK models"
"# The Sequence Space Jacobian (SSJ) method"
]
},
{
"cell_type": "markdown",
"id": "d54e4a66",
"metadata": {},
"source": [
"- The sequence space jacobian method (SSJ) is a computational method to solve general equilibrium models, both with and without rich micro heterogeneity\n",
"\n",
"- Method linearizes the model to first order to solve for solutions to small MIT shocks.\n",
"\n",
"- The algorithm assumes that agents have perfect foresight on macro aggregates (i.e. model is deterministic with respect to aggregate states).\n",
"\n",
"- Method allows for the fast computation of Jacobian matrices that summarize the aggregate behavior of agents with rich micro heterogeneity (heterogenous households, firms, etc.)"
"- [Krusell and Smith (1997)](https://www.journals.uchicago.edu/doi/abs/10.1086/250034) solved their HA macro model with a \"state space\" method\n",
" - They approximate the model solution for every conceivable combination of states\n",
" - This is very expensive computationally\n",
"\n",
"- The [sequence space Jacobian method (SSJ)](https://www.econometricsociety.org/publications/econometrica/2021/09/01/using-sequence-space-jacobian-solve-and-estimate-heterogeneous) (Auclert et al., 2021) is an alternative\n",
" - It computes a solution to the model for an expected path of shocks\n",
" - This allows computation of the solution for only the interesting shocks\n",
"\n",
"- The SSJ method solves for solutions to 'MIT shocks'\n",
" - The key requirement is that the MIT shocks be small, _relative_ to the distribution of microeconomic state variables\n",
" - This permits -- in fact, requires -- rich micro heterogeneity\n",
" - SSJ is usable when there is an aggregate equilibrium and micro heterogeneity\n",
" - So long as the \"aggregate shocks\" are _relatively_ small\n",
" - So it can be used to study industry or market equilibrium\n",
"\n",
"- The algorithm assumes that agents have perfect foresight wrt aggregates\n",
" - The model is deterministic wrt the evolution of aggregate \"shocks\"\n",
"\n",
"- As a result, easily calculable Jacobian matrices are a \"sufficient statistic\"\n",
" - They provide information sufficient to compute the macro dynamics\n",
" - Even when the microfoundations have rich complexity and realism\n"
]
},
{
Expand All @@ -29,17 +43,20 @@
"source": [
"## Advantages of SSJ\n",
"\n",
"Can solve general equilibrium models with rich microeconomic heterogeneity quickly. (Basic HANK models can take 3 seconds, previous methods take at least 15 minutes)\n",
"- Can solve equilibrium models with rich microeconomic heterogeneity\n",
" - Basic HANK models can take 3 seconds, previous methods take at least 15 minutes\n",
"\n",
"Can add additional exogenous shocks at virtually no cost."
"- Can add additional exogenous aggregate shocks at virtually no cost."
]
},
{
"cell_type": "markdown",
"id": "095bae35",
"metadata": {},
"source": [
"## Krusell Smith Model in Sequence Space"
"## Krusell Smith Model in Sequence Space\n",
"\n",
"Here we solve a slightly simplified version of the Krusell-Smith model to clarify the exposition. In particular, we assume each household inelastically supplies $\\ell$ units of labor.\n"
]
},
{
Expand All @@ -49,7 +66,7 @@
"source": [
"### Households\n",
"\n",
"Assume a continuum of atomistic households on the unit interval $[0,1]$ indexed by i. \n",
"Assume a continuum of atomistic households on the unit interval $[0,1]$ indexed by $i$. \n",
"\n",
"Assume households have perfect foresight over the real interest rate $r_{t}$ and the real wage $w_{t}$.\n",
"\n"
Expand All @@ -74,18 +91,12 @@
"\n",
"$$k_{it} \\geq 0,$$\n",
"\n",
"where \n",
"\n",
"$m_{it}$ is cash on hand,\n",
"\n",
"$k_{it}$ is capital holdings,\n",
"\n",
"$y_{it}$ is labor income.\n",
"where $m_{it}$ is market resources (`cash on hand`), $k_{it}$ is capital holdings and $y_{it}$ is labor income.\n",
"\n",
"\n",
"#### Labor Income\n",
"\n",
"Labor income is the product of an idiosyncratic transitory shock $\\theta$, the wage rate $w$ (determined in aggregate), and the exogenously fixed individual labor supply $\\ell$. \n",
"Labor income is the product of an idiosyncratic transitory shock $\\theta$, the wage rate $w$ (determined in aggregate), and the exogenously fixed individual labor supply $\\ell$:\n",
"\n",
"$$y_{t} = \\theta_{it} w_{t} \\ell,$$\n",
"\n",
Expand Down Expand Up @@ -147,15 +158,16 @@
"id": "12cc8957",
"metadata": {},
"source": [
"# The model as a system of difference equations in sequence space"
"# Model as defined by diff eqns in sequence space"
]
},
{
"cell_type": "markdown",
"id": "01ee68b8",
"metadata": {},
"source": [
"The equilibrium of this model can be expressed as a root of a system of difference equations on the sequence of current and future prices and productivity shocks. For notational convenience, the sequence of aggregate outcomes will be represented by $\\textbf{U}$, and the sequence of productivity shocks by $\\textbf{Z}$. For period $t$, the equilibrium conditions are:\n",
"\n",
"The sequence-space equilibrium can be expressed as a root of a system of difference equations on the sequence of current and future prices and productivity shocks. For notational convenience, the sequence of aggregate outcomes will be represented by $\\textbf{U}$, and the sequence of productivity shocks by $\\textbf{Z}$. For period $t$, the equilibrium conditions are:\n",
"\n",
"$$ H_{t}(\\mathbf{U},\\mathbf{Z}) = \\begin{pmatrix} \n",
" \\mathcal{K}_{t}(\\{r_{s} ,w_{s} \\}_{s=t}^{s=T}) - K_{t} \\\\ \\\\\n",
Expand Down Expand Up @@ -233,9 +245,9 @@
"id": "1488f183",
"metadata": {},
"source": [
"These Jacobian matrices are the most computationally complex object to compute in the model. Direct methods can take up to 20 minutes for each matrix.\n",
"These Jacobian matrices are the most computationally complex object to compute in the model. On a 2024 laptop used to produce the results here, such methods can take up to 20 minutes for each matrix.\n",
"\n",
"The sequence space Jacobian methodology proposes a 'fake news' algorithm to solve these matrices in under 3 seconds!"
"But the sequence space Jacobian methodology uses a 'fake news' algorithm to solve these matrices in under 3 seconds!"
]
},
{
Expand Down Expand Up @@ -276,7 +288,7 @@
"id": "c66c3791",
"metadata": {},
"source": [
"Because labor is exogenously supplied, aggregate capital $K_t$ is a perfect instrument for factor prices $r_t$ and $w_t$. The system above can thus be reduced to simply:\n",
"Here, because labor is exogenously supplied, aggregate capital $K_t$ is a sufficient statistic for factor prices $r_t$ and $w_t$. The system above can thus be reduced to simply:\n",
"\n",
"$$ H_{t}(\\mathbf{U},\\mathbf{Z}) \\equiv \\begin{pmatrix} \n",
" \\mathcal{K}_{t}\\left(\\{\\alpha Z_{t} K_{t}^{\\alpha - 1} \\bar{L}^{1-\\alpha} ,(1-\\alpha) Z_{s} K_{s}^{\\alpha} \\bar{L}^{-\\alpha} \\}_{s=0}^{s=T}\\right) - K_{t} \\\\\n",
Expand Down Expand Up @@ -318,12 +330,14 @@
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3ee1c897-4a63-4437-a596-0c8af34ada66",
"cell_type": "markdown",
"id": "be891e16-d276-49c7-8374-2478de587664",
"metadata": {},
"outputs": [],
"source": []
"source": [
"## Solving Using the HANK-SSJ Link\n",
"\n",
"A companion notebook, `KS-HARK-presentation` presents the solution to the model described above using the HARK toolkit in combination with the SSJ toolkit."
]
}
],
"metadata": {
Expand Down