From d0457bc4da7a7d705c71be314aadbc935c53d1bb Mon Sep 17 00:00:00 2001 From: jdebacker Date: Thu, 19 Sep 2024 02:44:24 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20EAPD-DRB?= =?UTF-8?q?/OG-ZAF@44397a135968d7df73598a07d89d1b2a54960b99=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- _modules/index.html | 2 +- _modules/ogzaf/calibrate.html | 18 +- _modules/ogzaf/income.html | 4 +- _modules/ogzaf/input_output.html | 43 ++- _modules/ogzaf/macro_params.html | 226 +++++++------ _modules/ogzaf/utils.html | 15 +- _sources/content/calibration/demographics.md | 4 +- .../calibration/exogenous_parameters.md | 103 ++++-- _sources/content/calibration/firms.md | 27 +- _sources/content/calibration/macro.md | 13 + _sources/content/calibration/taxes.md | 2 +- _sphinx_design_static/sphinx-design.min.css | 2 +- _static/sphinx-design.min.css | 2 +- content/OGZAF_references.html | 10 +- content/api/calibrate.html | 4 +- content/api/income.html | 6 +- content/api/input_output.html | 2 +- content/api/macro_params.html | 21 +- content/api/public_api.html | 2 +- content/api/utils.html | 2 +- content/calibration/UBI.html | 10 +- content/calibration/demographics.html | 38 +-- content/calibration/earnings.html | 4 +- content/calibration/exogenous_parameters.html | 298 ++++++++++++++++-- content/calibration/firms.html | 33 +- content/calibration/households.html | 2 +- content/calibration/macro.html | 19 +- content/calibration/matching_lwi.html | 2 +- content/calibration/taxes.html | 6 +- content/citations.html | 2 +- content/contributing/contributor_guide.html | 2 +- content/intro/intro.html | 2 +- genindex.html | 2 +- objects.inv | Bin 1795 -> 1756 bytes py-modindex.html | 2 +- search.html | 2 +- searchindex.js | 2 +- 37 files changed, 678 insertions(+), 256 deletions(-) diff --git a/_modules/index.html b/_modules/index.html index db98771..bc30dfe 100644 --- a/_modules/index.html +++ b/_modules/index.html @@ -33,7 +33,7 @@ - + diff --git a/_modules/ogzaf/calibrate.html b/_modules/ogzaf/calibrate.html index 65f0ab0..bd30501 100644 --- a/_modules/ogzaf/calibrate.html +++ b/_modules/ogzaf/calibrate.html @@ -33,7 +33,7 @@ - + @@ -366,6 +366,7 @@

Source code for ogzaf.calibrate

 from ogzaf import input_output as io
 import os
 import numpy as np
+import datetime
 from ogcore import demographics
 
 
@@ -377,8 +378,8 @@ 
Source code for ogzaf.calibrate
     def __init__(
         self,
         p,
-        estimate_tax_functions=False,
-        estimate_chi_n=False,
+        macro_data_start_year=datetime.datetime(1947, 1, 1),
+        macro_data_end_year=datetime.date.today(),
         demographic_data_path=None,
         output_path=None,
     ):
@@ -387,9 +388,6 @@ 
Source code for ogzaf.calibrate
 
         Args:
             p (OG-Core Specifications object): model parameters
-            estimate_tax_functions (bool): whether to estimate tax
-                function parameters
-            estimate_chi_n (bool): whether to estimate chi_n
             demographic_data_path (str): path to save demographic data
             output_path (str): path to save output to
 
@@ -401,11 +399,11 @@ 
Source code for ogzaf.calibrate
         if output_path is not None:
             if not os.path.exists(output_path):
                 os.makedirs(output_path)
-        self.estimate_tax_functions = estimate_tax_functions
-        self.estimate_chi_n = estimate_chi_n
 
         # Macro estimation
-        self.macro_params = macro_params.get_macro_params()
+        self.macro_params = macro_params.get_macro_params(
+            macro_data_start_year, macro_data_end_year
+        )
 
         # io matrix and alpha_c
         if p.I > 1:  # no need if just one consumption good
@@ -462,8 +460,6 @@ 
Source code for ogzaf.calibrate
     # method to return all newly calibrated parameters in a dictionary
     def get_dict(self):
         dict = {}
-        # if self.estimate_chi_n:
-        #     dict["chi_n"] = self.chi_n
         dict.update(self.macro_params)
         dict["e"] = self.e
         dict["alpha_c"] = self.alpha_c
diff --git a/_modules/ogzaf/income.html b/_modules/ogzaf/income.html
index 5c42068..133cea2 100644
--- a/_modules/ogzaf/income.html
+++ b/_modules/ogzaf/income.html
@@ -33,7 +33,7 @@
     
     
     
-    
+    
   
   
   
@@ -694,7 +694,7 @@ 
Source code for ogzaf.income
 

 [docs]
 def get_e_orig(age_wgts, abil_wgts, plot_path=None):
-    """
+    r"""
     This function generates the 80 x 7 matrix of lifetime earnings
     ability profiles, corresponding to annual ages from 21 to 100 and to
     paths based on income percentiles 0-25, 25-50, 50-70, 70-80, 80-90,
diff --git a/_modules/ogzaf/input_output.html b/_modules/ogzaf/input_output.html
index 2ca916d..ef2ac03 100644
--- a/_modules/ogzaf/input_output.html
+++ b/_modules/ogzaf/input_output.html
@@ -33,7 +33,7 @@
     
     
     
-    
+    
   
   
   
@@ -364,6 +364,7 @@ 

   
Source code for ogzaf.input_output
 import pandas as pd
 import numpy as np
+from ogzaf.utils import is_connected
 from ogzaf.constants import CONS_DICT, PROD_DICT
 
 """
@@ -372,13 +373,39 @@ 
Source code for ogzaf.input_output
 # Read in SAM file
 storage_options = {"User-Agent": "Mozilla/5.0"}
 SAM_path = "https://www.wider.unu.edu/sites/default/files/Data/SASAM-2015-Data-Resource.xlsx"
-SAM = pd.read_excel(
-    SAM_path,
-    sheet_name="Micro SAM 2015",
-    skiprows=6,
-    index_col=0,
-    storage_options=storage_options,
-)
+SAM_path_alt = "https://raw.githubusercontent.com/EAPD-DRB/SAM-files/main/Data/ZAF/SASAM-2015-Data-Resource.xlsx"
+
+if is_connected():
+    try:
+        SAM = pd.read_excel(
+            SAM_path,
+            sheet_name="Micro SAM 2015",
+            skiprows=6,
+            index_col=0,
+            storage_options=storage_options,
+        )
+        print("Successfully read SAM from WIDER.")
+    except Exception as e:
+        print(f"Failed to read from WIDER: {e}")
+        try:
+            # Attempt to read from the GitHub repository
+            SAM = pd.read_excel(
+                SAM_path_alt,
+                sheet_name="Micro SAM 2015",
+                skiprows=6,
+                index_col=0,
+                storage_options=storage_options,
+            )
+            print("Successfully read SAM from GitHub repository.")
+        except Exception as e:
+            print(f"Failed to read from the GitHub repository: {e}")
+            SAM = None
+    # If both attempts fail, SAM will be None
+    if SAM is None:
+        print("Failed to read SAM from both sources.")
+else:
+    SAM = None
+    print("No internet connection. SAM cannot be read.")
 
 
 

diff --git a/_modules/ogzaf/macro_params.html b/_modules/ogzaf/macro_params.html
index 0593309..ea5638e 100644
--- a/_modules/ogzaf/macro_params.html
+++ b/_modules/ogzaf/macro_params.html
@@ -33,7 +33,7 @@
     
     
     
-    
+    
   
   
   
@@ -364,7 +364,7 @@ 

   
Source code for ogzaf.macro_params
 """
 This module uses data from World Bank WDI, World Bank Quarterly Public
-Sector Debt (QPSD) database, UN Data Portal, and FRED to find values for
+Sector Debt (QPSD) database, the IMF, and UN ILO to find values for
 parameters for the OG-ZAF model that rely on macro data for calibration.
 """
 
@@ -380,91 +380,127 @@ 
Source code for ogzaf.macro_params
 
 

 [docs]
-def get_macro_params():
+def get_macro_params(
+    data_start_date=datetime.datetime(1947, 1, 1),
+    data_end_date=datetime.date.today(),
+    country_iso="ZAF",
+):
     """
     Compute values of parameters that are derived from macro data
-    """
 
-    # set beginning and end dates for data
-    # format is year (1940),month (1),day (1)
-    country_iso = "ZAF"
-    start = datetime.datetime(1947, 1, 1)
-    end = datetime.date.today()  # go through today
-    baseline_date = datetime.datetime(2019, 3, 31)
-    baseline_yearquarter = (
-        "2024Q1"  # The WB QPSD database has the date in YYYYQQ format
+    Args:
+        data_start_date (datetime): start date for data
+        data_end_date (datetime): end date for data
+        country_iso (str): ISO code for country
+
+    Returns:
+        macro_parameters (dict): dictionary of parameter values
+    """
+    # initialize a dictionary of parameters
+    macro_parameters = {}
+    # baseline date formatted for World Bank data
+    baseline_YYYYQ = (
+        str(data_end_date.year)
+        + "Q"
+        + str(pd.Timestamp(data_end_date).quarter)
     )
 
     """
-    This retrieves annual data from the World Bank World Development Indicators.
+    Retrieve data from the World Bank World Development Indicators.
     """
-
+    # Dictionaries of variables and their corresponding World Bank codes
+    # Annual data
     wb_a_variable_dict = {
         "GDP per capita (constant 2015 US$)": "NY.GDP.PCAP.KD",
         "Real GDP (constant 2015 US$)": "NY.GDP.MKTP.KD",
         "Nominal GDP (current US$)": "NY.GDP.MKTP.CD",
         "General government final consumption expenditure (current US$)": "NE.CON.GOVT.CD",
-        # "General government debt (percentage of GDP)": "GC.DOD.TOTL.GD.ZS",
-        # "GDP per capita (current US$)": "NY.GDP.PCAP.CD",
-        # "GDP per person employed (constant 2017 PPP$)": "SL.GDP.PCAP.EM.KD",
+    }
+    # Quarterly data
+    wb_q_variable_dict = {
+        "Gross PSD USD - domestic creditors": "DP.DOD.DECD.CR.PS.CD",
+        "Gross PSD USD - external creditors": "DP.DOD.DECX.CR.PS.CD",
+        "Gross PSD Gen Gov - percentage of GDP": "DP.DOD.DECT.CR.GG.Z1",
     }
 
-    # pull series of interest using pandas_datareader
-    # wb_data_a = wb.download(
-    #     indicator=wb_a_variable_dict.values(),
-    #     country=country_iso,
-    #     start=start,
-    #     end=end,
-    # )
-    # wb_data_a.rename(
-    #     columns=dict((y, x) for x, y in wb_a_variable_dict.items()),
-    #     inplace=True,
-    # )
-
-    """
-    This retrieves quarterly data from the World Bank Quarterly Public Sector Debt database.
-    The command extracts all available data even when start and end dates are specified.
-    """
-
-    # wb_q_variable_dict = {
-    #     "Gross PSD USD - domestic creditors": "DP.DOD.DECD.CR.PS.CD",
-    #     "Gross PSD USD - external creditors": "DP.DOD.DECX.CR.PS.CD",
-    #     "Gross PSD Gen Gov - percentage of GDP": "DP.DOD.DECT.CR.GG.Z1",
-    # }
-
-    # pull series of interest using pandas_datareader
-    # wb_data_q = wb.download(
-    #     indicator=wb_q_variable_dict.values(),
-    #     country=country_iso,
-    #     start=start,
-    #     end=end,
-    # )
-    # wb_data_q.rename(
-    #     columns=dict((y, x) for x, y in wb_q_variable_dict.items()),
-    #     inplace=True,
-    # )
-
-    # # Remove the hierarchical index (country and year) of wb_data_q and create a single row index using year
-    # wb_data_q = wb_data_q.reset_index()
-    # wb_data_q = wb_data_q.set_index("year")
+    try:
+        # pull series of interest from the WB using pandas_datareader
+        # Annual data
+        wb_data_a = wb.download(
+            indicator=wb_a_variable_dict.values(),
+            country=country_iso,
+            start=data_start_date,
+            end=data_end_date,
+        )
+        wb_data_a.rename(
+            columns=dict((y, x) for x, y in wb_a_variable_dict.items()),
+            inplace=True,
+        )
+        # Quarterly data
+        wb_data_q = wb.download(
+            indicator=wb_q_variable_dict.values(),
+            country=country_iso,
+            start=data_start_date,
+            end=data_end_date,
+        )
+        wb_data_q.rename(
+            columns=dict((y, x) for x, y in wb_q_variable_dict.items()),
+            inplace=True,
+        )
+        # Remove the hierarchical index (country and year) of
+        # wb_data_q and create a single row index using year
+        wb_data_q = wb_data_q.reset_index()
+        wb_data_q = wb_data_q.set_index("year")
+
+        # Compute macro parameters from WB data
+        macro_parameters["initial_debt_ratio"] = (
+            pd.Series(wb_data_q["Gross PSD Gen Gov - percentage of GDP"]).loc[
+                baseline_YYYYQ
+            ]
+        ) / 100
+        macro_parameters["initial_foreign_debt_ratio"] = pd.Series(
+            wb_data_q["Gross PSD USD - external creditors"]
+            / (
+                wb_data_q["Gross PSD USD - domestic creditors"]
+                + wb_data_q["Gross PSD USD - external creditors"]
+            )
+        ).loc[baseline_YYYYQ]
+        # zeta_D = share of new debt issues from government that are
+        # purchased by foreigners
+        macro_parameters["zeta_D"] = [
+            pd.Series(
+                wb_data_q["Gross PSD USD - external creditors"]
+                / (
+                    wb_data_q["Gross PSD USD - domestic creditors"]
+                    + wb_data_q["Gross PSD USD - external creditors"]
+                )
+            ).loc[baseline_YYYYQ]
+        ]
+        macro_parameters["g_y_annual"] = (
+            wb_data_a["GDP per capita (constant 2015 US$)"]
+            .pct_change(-1)
+            .mean()
+        )
+    except:
+        print("Failed to retrieve data from World Bank")
+        print("Will not update the following parameters:")
+        print("[initial_debt_ratio, initial_foreign_debt_ratio, zeta_D, g_y]")
 
     """
-    This retrieves labour share data from the ILOSTAT Data API
+    Retrieve labour share data from the United Nations ILOSTAT Data API
     (see https://rshiny.ilo.org/dataexplorer9/?lang=en)
     """
-
     target = (
         "https://rplumber.ilo.org/data/indicator/"
         + "?id=LAP_2GDP_NOC_RT_A"
         + "&ref_area="
         + str(country_iso)
         + "&timefrom="
-        + str(start.year)
+        + str(data_start_date.year)
         + "&timeto="
-        + str(end.year)
+        + str(data_end_date.year)
         + "&type=both&format=.csv"
     )
-
     response = requests.get(target)
     if response.status_code == 200:
         csv_content = StringIO(response.text)
@@ -473,68 +509,34 @@ 
Source code for ogzaf.macro_params
         print(
             f"Failed to retrieve data. HTTP status code: {response.status_code}"
         )
-
     ilo_data = df_temp[["time", "obs_value"]]
-
-    # initialize a dictionary of parameters
-    macro_parameters = {}
-
-    # print(fred_data.loc(str(baseline_date)))
-    # find initial_debt_ratio
-    # macro_parameters["initial_debt_ratio"] = (
-    #     pd.Series(wb_data_q["Gross PSD Gen Gov - percentage of GDP"]).loc[
-    #         baseline_yearquarter
-    #     ]
-    # ) / 100
-
-    # find initial_foreign_debt_ratio
-    # macro_parameters["initial_foreign_debt_ratio"] = pd.Series(
-    #     wb_data_q["Gross PSD USD - external creditors"]
-    #     / (
-    #         wb_data_q["Gross PSD USD - domestic creditors"]
-    #         + wb_data_q["Gross PSD USD - external creditors"]
-    #     )
-    # ).loc[baseline_yearquarter]
-
-    # find zeta_D (Share of new debt issues from government that are purchased by foreigners)
-    # macro_parameters["zeta_D"] = [
-    #     pd.Series(
-    #         wb_data_q["Gross PSD USD - external creditors"]
-    #         / (
-    #             wb_data_q["Gross PSD USD - domestic creditors"]
-    #             + wb_data_q["Gross PSD USD - external creditors"]
-    #         )
-    #     ).loc[baseline_yearquarter]
-    # ]
-
-    # alpha_T, non-social security benefits as a fraction of GDP
-    # source: https://data.imf.org/?sk=b052f0f0-c166-43b6-84fa-47cccae3e219&hide_uv=1
-    macro_parameters["alpha_T"] = [0.041 - 0.0]
-    # alpha_G, gov't consumption expenditures as a fraction of GDP
-    # source: https://data.imf.org/?sk=edb0cd70-0af3-40e1-a9c3-bdef83ee4d1e&hide_uv=1
-    macro_parameters["alpha_G"] = [0.351 - 0.043 - 0.041]
-
     # find gamma, capital's share of income
     macro_parameters["gamma"] = [
         1
         - (
             (
                 ilo_data.loc[
-                    ilo_data["time"] == baseline_date.year, "obs_value"
+                    ilo_data["time"] == data_end_date.year, "obs_value"
                 ].squeeze()
             )
             / 100
         )
     ]
 
-    # find g_y
-    # macro_parameters["g_y_annual"] = (
-    #     wb_data_a["GDP per capita (constant 2015 US$)"].pct_change(-1).mean()
-    # )
+    """
+    Calibrate parameters from IMF data
+    """
+    # alpha_T, non-social security benefits as a fraction of GDP
+    # source: https://data.imf.org/?sk=b052f0f0-c166-43b6-84fa-47cccae3e219&hide_uv=1
+    macro_parameters["alpha_T"] = [0.041 - 0.0]
+    # alpha_G, gov't consumption expenditures as a fraction of GDP
+    # source: https://data.imf.org/?sk=edb0cd70-0af3-40e1-a9c3-bdef83ee4d1e&hide_uv=1
+    macro_parameters["alpha_G"] = [0.351 - 0.043 - 0.041]
 
     """"
-    We want to use the non linear relationship estimated by
-    Li, Magud, Werner, Witte (2021), available here:
+    Esimate the discount on sovereign yields relative to private debt
+    Follow the methodology in Li, Magud, Werner, Witte (2021)
+    available at:
     https://www.imf.org/en/Publications/WP/Issues/2021/06/04/The-Long-Run-Impact-of-Sovereign-Yields-on-Corporate-Yields-in-Emerging-Markets-50224
 
     Steps:
@@ -553,14 +555,10 @@ 
Source code for ogzaf.macro_params
         corp_yhat,
     )
     res = mod.fit()
-    # first term is the constant and needs to be divided by 100 to have
+    # First term is the constant and needs to be divided by 100 to have
     # the correct unit. Second term is the coefficient
-    macro_parameters["r_gov_shift"] = [
-        (-res.params[0] / 100)
-    ]  # constant = -0.0337662504
-    macro_parameters["r_gov_scale"] = [
-        res.params[1]
-    ]  # coefficient = 0.24484764
+    macro_parameters["r_gov_shift"] = [(-res.params[0] / 100)]
+    macro_parameters["r_gov_scale"] = [res.params[1]]
 
     return macro_parameters

 
diff --git a/_modules/ogzaf/utils.html b/_modules/ogzaf/utils.html
index a760f59..e218f30 100644
--- a/_modules/ogzaf/utils.html
+++ b/_modules/ogzaf/utils.html
@@ -33,7 +33,7 @@
     
     
     
-    
+    
   
   
   
@@ -365,6 +365,7 @@ 
Source code for ogzaf.utils
 import requests
 import urllib3
 import ssl
+import socket
 
 
 

@@ -399,6 +400,18 @@ 
Source code for ogzaf.utils
     session = requests.session()
     session.mount("https://", CustomHttpAdapter(ctx))
     return session
+
+
+# Function to check if connected to internet
+def is_connected():
+    try:
+        # connect to the host -- tells us if the host is actually
+        # reachable
+        socket.create_connection(("1.1.1.1", 53))
+        return True
+    except OSError:
+        pass
+    return False
 

 
                 
diff --git a/_sources/content/calibration/demographics.md b/_sources/content/calibration/demographics.md
index ce09f62..9f09f24 100644
--- a/_sources/content/calibration/demographics.md
+++ b/_sources/content/calibration/demographics.md
@@ -293,7 +293,7 @@ We discuss the approach to estimating fertility rates $f_{s,t}$, mortality rates
   Theoretical steady-state population distribution vs. population distribution at period $t=120$
   ```
 
-  Further, we find that the maximum absolute difference between the population levels $\hat{\omega}_{s,t}$ and $\hat{\omega}_{s,t+1}$ was $1.3852\times 10^{-5}$ after 160 periods. That is to say, that after 160 periods, given the estimated mortality, fertility, and immigration rates, the population has not achieved its steady state. For convergence in our solution method over a reasonable time horizon, we want the population to reach a stationary distribution after $T$ periods. To do this, we artificially impose that the population distribution in period $t=120$ is the steady-state. As can be seen from {numref}`Figure %s `, this assumption is not very restrictive. {numref}`Figure %s ` shows the change in immigration rates that would make the period $t=120$ population distribution equal be the steady-state. The maximum absolute difference between any two corresponding immigration rates in {numref}`Figure %s ` is 0.0028.
+  Further, we find that the maximum absolute difference between the population levels $\hat{\omega}_{s,t}$ and $\hat{\omega}_{s,t+1}$ was less than $1\times 10^{-4}$ after 160 periods. That is to say, that after 160 periods, given the estimated mortality, fertility, and immigration rates, the population has not achieved its steady state. For convergence in our solution method over a reasonable time horizon, we want the population to reach a stationary distribution after $T$ periods. To do this, we artificially impose that the population distribution in period $t=120$ is the steady-state. As can be seen from {numref}`Figure %s `, this assumption is not very restrictive. {numref}`Figure %s ` shows the change in immigration rates that would make the period $t=120$ population distribution equal be the steady-state. The maximum absolute difference between any two corresponding immigration rates in {numref}`Figure %s ` is very small.
 
   ```{figure} ./images/OrigVsAdjImm.png
   ---
@@ -303,7 +303,7 @@ We discuss the approach to estimating fertility rates $f_{s,t}$, mortality rates
   Original immigration rates vs. adjusted immigration rates to make fixed steady-state population distribution
   ```
 
-  The most recent year of population data come from {cite}`Census:2015` population estimates for both sexes for 2013. We those data and use the population transition matrix {eq}`EqPopLOMstatmat2` to age it to the current model year of 2015. We then use {eq}`EqPopLOMstatmat2` to generate the transition path of the population distribution over the time period of the model. {numref}`Figure %s ` shows the progression from the 2013 population data to the fixed steady-state at period $t=120$. The time path of the growth rate of the economically active population $\tilde{g}_{n,t}$ is shown in {numref}`Figure %s `.
+  We begin with 2023 population data and use the population transition matrix {eq}`EqPopLOMstatmat2` to age it to the start year of the model (e.g., 2024 or 2025). We then use {eq}`EqPopLOMstatmat2` to generate the transition path of the population distribution over the time period of the model. {numref}`Figure %s ` shows the progression from the 2023 population data to the fixed steady-state at period $t=120$. The time path of the growth rate of the economically active population $\tilde{g}_{n,t}$ is shown in {numref}`Figure %s `.
 
   ```{figure} ./images/pop_distribution.png
   ---
diff --git a/_sources/content/calibration/exogenous_parameters.md b/_sources/content/calibration/exogenous_parameters.md
index 2ec415d..b6163a7 100644
--- a/_sources/content/calibration/exogenous_parameters.md
+++ b/_sources/content/calibration/exogenous_parameters.md
@@ -21,33 +21,86 @@ kernelspec:
   ```{code-cell} ogzaf-dev
   :tags: [hide-cell]
   from myst_nb import glue
-  import ogzaf.parameter_tables as pt
-  from ogzaf import Specifications
+  import ogcore.parameter_tables as pt
+  from ogcore import Specifications
+  import ogzaf
+  import importlib
+  import json
   p = Specifications()
-  table = pt.param_table(p, table_format=None, path=None)
+  with importlib.resources.open_text(
+      "ogzaf", "ogzaf_default_parameters.json"
+      ) as file:
+          defaults = json.load(file)
+  table = pt.param_table(p, table_format="md", path=None)
   glue("param_table", table, display=False)
   ```
   -->
 
-  ```{list-table} **List of exogenous parameters and baseline calibration values.**
-  :header-rows: 1
-  :name: TabExogVars
-  * - **Symbol**
-    - **Description**
-    - **Value**
-  * - $S$
-    - Maximum periods in economically active household life
-    - 80
-  * - $E$
-    - Number of periods of youth economically outside the model
-    - $\text{round} \frac{S}{4}$=20
-  * - $T_1$
-    - Number of periods to steady state for initial time path guesses
-    - 160
-  * - $T_2$
-    - Maximum number of periods to steady state for nonsteady-state equilibrium
-    - 160
-  * - $\nu$
-    - Dampening parameter for TPI
-    - 0.4
-  ```
+ | Symbol                           | Description                                                             | Value                                                 |
+|:---------------------------------|:------------------------------------------------------------------------|:------------------------------------------------------|
+| $\texttt{start_year}$           | Initial year                                                            | 2025                                                  |
+| $\omega_{s,t}$                   | Population by age over time                                             | Too large to report here, see default parameters JSON |
+| $i_{s,t}$                        | Immigration rates by age                                                | Too large to report here, see default parameters JSON |
+| $\rho_{s,t}$                     | Mortality rates by age                                                  | Too large to report here, see default parameters JSON |
+| $e_{j,s,t}$                      | Deterministic ability process                                           | Too large to report here, see default parameters JSON |
+| $\lambda_{j}$                    | Lifetime income group percentages                                       | Too large to report here, see default parameters JSON |
+| $J$                              | Number of lifetime income groups                                        | 7                                                     |
+| $S$                              | Maximum periods in economically active individual life                  | 80                                                    |
+| $E$                              | Number of periods of youth economically outside the model               | 20                                                    |
+| $T$                              | Number of periods to steady-state                                       | 320                                                   |
+| $R$                              | Retirement age                                                          | [65.000...65.000]                                     |
+| $\tilde{l}$                      | Maximum hours of labor supply                                           | 1.000                                                 |
+| $\beta$                          | Discount factor                                                         | [0.960...0.960]                                       |
+| $\sigma$                         | Coefficient of constant relative risk aversion                          | 1.500                                                 |
+| $\nu$                            | Frisch elasticity of labor supply                                       | 0.400                                                 |
+| $b$                              | Scale parameter in utility of leisure                                   | 0.573                                                 |
+| $\upsilon$                       | Shape parameter in utility of leisure                                   | 2.856                                                 |
+| $\chi^{n}_{s}$                   | Disutility of labor level parameters                                    | Too large to report here, see default parameters JSON |
+| $\chi^{b}_{j}$                   | Utility of bequests level parameters                                    | [80.000...80.000]                                     |
+| $\texttt{use_zeta}$             | Whether to distribute bequests between lifetime income groups           | 0.00E+00                                              |
+| $\zeta$                          | Distribution of bequests                                                | Too large to report here, see default parameters JSON |
+| $Z_{t}$                          | Total factor productivity                                               | Too large to report here, see default parameters JSON |
+| $\gamma$                         | Capital share of income                                                 | [0.401...0.401]                                       |
+| $\varepsilon$                    | Elasticity of substitution between capital and labor                    | [1.000...1.000]                                       |
+| $\delta$                         | Capital depreciation rate                                               | 0.050                                                 |
+| $g_{y}$                          | Growth rate of labor augmenting technological progress                  | 0.00E+00                                              |
+| $\texttt{tax_func_type}$       | Functional form used for income tax functions                           | linear                                                |
+| $\texttt{analytical_mtrs}$      | Whether use analytical MTRs or estimate MTRs                            | 0.00E+00                                              |
+| $\texttt{age_specific}$         | Whether use age-specific tax functions                                  | 0.00E+00                                              |
+| $\tau^{p}_{t}$                   | Payroll tax rate                                                        | [0.000...0.000]                                       |
+| $\tau^{BQ}_{t}$                  | Bequest (estate) tax rate                                               | [0.200...0.200]                                       |
+| $\tau^{b}_{t}$                   | Entity-level business income tax rate                                   | Too large to report here, see default parameters JSON |
+| $\delta^{\tau}_{t}$              | Rate of depreciation for tax purposes                                   | Too large to report here, see default parameters JSON |
+| $\tau^{c}_{t,s,j}$               | Consumption tax rates                                                   | Too large to report here, see default parameters JSON |
+| $H$                              | Coefficient on linear term in wealth tax function                       | [0.100...0.100]                                       |
+| $M$                              | Constant in wealth tax function                                         | [1.000...1.000]                                       |
+| $P$                              | Coefficient on level term in wealth tax function                        | [0.000...0.000]                                       |
+| $\texttt{budget_balance}$       | Whether have a balanced budget in each period                           | 0.00E+00                                              |
+| $\texttt{baseline_spending}$    | Whether level of spending constant between the baseline and reform runs | 0.00E+00                                              |
+| $\alpha^{T}_{t}$                 | Transfers as a share of GDP                                             | [0.041...0.041]                                       |
+| $\eta_{j,s,t}$                   | Distribution of transfers                                               | Too large to report here, see default parameters JSON |
+| $\alpha^{G}_{t}$                 | Government spending as a share of GDP                                   | [0.267...0.267]                                       |
+| $t_{G1}$                         | Model period in which budget closure rule starts                        | 20                                                    |
+| $t_{G2}$                         | Model period in which budget closure rule ends                          | 256                                                   |
+| $\rho_{G}$                       | Budget closure rule smoothing parameter                                 | 0.100                                                 |
+| $\bar{\alpha}_{D}$               | Steady-state Debt-to-GDP ratio                                          | 1.200                                                 |
+| $\alpha_{D,0}$                   | Initial period Debt-to-GDP ratio                                        | 0.740                                                 |
+| $\tau_{d,t}$                     | Scale parameter in government interest rate wedge                       | [0.245...0.245]                                       |
+| $\mu_{d,t}$                      | Shift parameter in government interest rate wedge                       | [-0.034...-0.034]                                     |
+| $\texttt{avg_earn_num_years}$ | Number of years over which compute average earnings for pension benefit | 35                                                    |
+| $\texttt{AIME_bkt_1}$          | First AIME bracket threshold                                            | 749.000                                               |
+| $\texttt{AIME_bkt_2}$          | Second AIME bracket threshold                                           | 4517.000                                              |
+| $\texttt{PIA_rate_bkt_1}$     | First AIME bracket PIA rate                                             | 0.00E+00                                              |
+| $\texttt{PIA_rate_bkt_2}$     | Second AIME bracket PIA rate                                            | 0.00E+00                                              |
+| $\texttt{PIA_rate_bkt_3}$     | Third AIME bracket PIA rate                                             | 0.00E+00                                              |
+| $\texttt{PIA_maxpayment}$       | Maximum PIA payment                                                     | 0.00E+00                                              |
+| $\texttt{PIA_minpayment}$       | Minimum PIA payment                                                     | 0.00E+00                                              |
+| $\theta_{adj,t}$                 | Adjustment to replacement rate                                          | [1.000...1.000]                                       |
+| $r^{*}_{t}$                      | World interest rate                                                     | [0.040...0.040]                                       |
+| $D_{f,0}$                        | Share of government debt held by foreigners in initial period           | 0.237                                                 |
+| $\zeta_{D, t}$                   | Share of new debt issues purchased by foreigners                        | [0.237...0.237]                                       |
+| $\zeta_{K, t}$                   | Share of excess capital demand satisfied by foreigners                  | [0.900...0.900]                                       |
+| $\xi$                            | Dampening parameter for TPI                                             | 0.400                                                 |
+| $\texttt{maxiter}$               | Maximum number of iterations for TPI                                    | 250                                                   |
+| $\texttt{mindist_SS}$           | SS solution tolerance                                                   | 1.00E-09                                              |
+| $\texttt{mindist_TPI}$          | TPI solution tolerance                                                  | 1.00E-05                                              |
diff --git a/_sources/content/calibration/firms.md b/_sources/content/calibration/firms.md
index 61a23e4..8cc5f55 100644
--- a/_sources/content/calibration/firms.md
+++ b/_sources/content/calibration/firms.md
@@ -5,10 +5,33 @@
 
 The [OG-Core firm theory documentation](https://pslmodels.github.io/OG-Core/content/theory/firms.html) outlines the constant returns to scale, constant elasticity of substitution production function of the representative firm.  This function has two parameters; the elasticity of substitution and capital's share of output.
 
+The production function is given as:
+
+```{math}
+:label: EqFirmsCESprodfun
+  \begin{split}
+    Y_{m,t} &= F(K_{m,t}, K_{g,m,t}, L_{m,t}) \\
+    &\equiv Z_{m,t}\biggl[(\gamma_m)^\frac{1}{\varepsilon_m}(K_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + (\gamma_{g,m})^\frac{1}{\varepsilon_m}(K_{g,m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + \\
+    &\quad\quad\quad\quad\quad(1-\gamma_m-\gamma_{g,m})^\frac{1}{\varepsilon_m}(e^{g_y t}L_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m}\biggr]^\frac{\varepsilon_m}{\varepsilon_m-1} \quad\forall m,t
+  \end{split}
+```
+
+  This production function has the following parameters:
+  * $\varepsilon_m$ is the elasticity of substitution between capital, labor, and infrastructure in sector $m$.
+  * $\gamma_m$ is the share of capital in sector $m$.
+  * $\gamma_{g,m}$ is the share of government capital in sector $m$.
+  * $Z_{m,t}$ is the total factor productivity in sector $m$ at time $t$.
+
 ### Elasticity of substitution
 
 `OG-ZAF`'s default parameterization has an elasticity of substitution of $\varepsilon=1.0$, which implies a Cobb-Douglas production function.
 
-### Capital's share of output
+### Factor shares of output
+
+In the default calibration, we set infrastructure's share of output to $\gamma_{g,m}=0.0$ for all sectors.  This parameter is hard to identify from national accounts data and would entail an empirical study to tease out the relationship between infrastructure and output.
+
+We use a default value of $\gamma =0.40$, which corresponds to one minus labor's share of output, where labor's share of output is found as 0.60 in in the [UN ILOSTAT database](https://rshiny.ilo.org/dataexplorer9/?lang=en).
+
+### Total factor productivity
 
-Here, we use a default value of $\gamma =0.40$.
\ No newline at end of file
+In the case of the single prodcution sector, we can normalize $Z_{m,t}=1.0$.  In the case of multiple production sectors, we use {cite}`PRS2020 who identify TFP for various sectors in South Africa.
\ No newline at end of file
diff --git a/_sources/content/calibration/macro.md b/_sources/content/calibration/macro.md
index 64bb348..3898148 100644
--- a/_sources/content/calibration/macro.md
+++ b/_sources/content/calibration/macro.md
@@ -25,6 +25,18 @@ We set $\zeta_K = 0.9$. Note, this parameter is harder to pin down from the data
 
 The path of government debt is endogenous.  But the initial value is exogenous.  To avoid converting between model units and dollars, we calibrate the initial debt to GDP ratio, rather than the dollar value of the debt.  This is the model parameter $\alpha_D$.  We compute this from the ratio of publicly held debt outstanding to GDP.  Based on 2023 values, this gives us a ratio of 0.59.
 
+
+#### Interest rates on government debt
+
+We assume that there is a wedge between the real rate of return on private capital and the real interest rate on government debt.  We model this wedge a scale and level shift.  Specifically, we assume that the real interest rate on government debt, $r_{gov,t}$, is related to the real rate of return on private capital, $r_{t}$, by the following equation:
+
+```{math}
+:label: eqn:r_gov
+    r_{gov,t} = (1-\tau_{d,t})r_t + \mu_d
+```
+
+where $\tau_d$ is the scale parameter and $\mu_d$ is the level shift parameter.  We set the values of these two parameters to 0.245 and -0.034, respectively.  These are found by using the estimated relationship between corporate and sovereign yields in {cite}`LMW2023` (Table 8, Column 2) and simulating a series of corporate yields given a series of sovereign yields between 2% and 12%.  We then estimate the scale and level shift parameters that best fit these simulated data using ordinary least squares.
+
 ### Aggregate transfers
 
 Aggregate (non-Social Security) transfers to households are set as a share of GDP with the parameter $\alpha_T$. We exclude Social Security from transfers since it is modeled specifically. With this definition, the share of transfers to GDP in 2015 is 0.04 according to [IMF data](https://data.imf.org/?sk=b052f0f0-c166-43b6-84fa-47cccae3e219&hide_uv=1).
@@ -34,3 +46,4 @@ Aggregate (non-Social Security) transfers to households are set as a share of GD
 Government spending on goods and services are also set as a share of GDP with the parameter $\alpha_G$. We define government spending as:
     
Government Spending = Total Outlays - Transfers - Net Interest on Debt - Social Security

 With this definition, the share of government expenditure to GDP is 0.267 based on [data from the IMF](https://data.imf.org/?sk=b052f0f0-c166-43b6-84fa-47cccae3e219&hide_uv=1).
+
diff --git a/_sources/content/calibration/taxes.md b/_sources/content/calibration/taxes.md
index eab6ac0..1bdb2ec 100644
--- a/_sources/content/calibration/taxes.md
+++ b/_sources/content/calibration/taxes.md
@@ -5,7 +5,7 @@ The government is not an optimizing agent in `OG-ZAF`. The government levies tax
 
 
 ## Personal income taxes
-The government sector influences households through two terms in the household budget constraint {eq}`EqHHBC`---government transfers $TR_{t}$ and through the total tax liability function $T_{s,t}$, which can be decomposed into the effective tax rate times total income {eq}`EqTaxCalcLiabETR2`. In this chapter, we detail the household tax component of government activity $T_{s,t}$ in `OG-ZAF`.
+The government sector influences households through two terms in the household budget constraint {eq}`EqHHBC`---government transfers $TR_{t}$ and through the total tax liability function $T_{s,t}$, which can be decomposed into the effective tax rate times total income. In this chapter, we detail the household tax component of government activity $T_{s,t}$ in `OG-ZAF`.
 
 ```{math}
 :label: EqHHBC
diff --git a/_sphinx_design_static/sphinx-design.min.css b/_sphinx_design_static/sphinx-design.min.css
index a325746..860c36d 100644
--- a/_sphinx_design_static/sphinx-design.min.css
+++ b/_sphinx_design_static/sphinx-design.min.css
@@ -1 +1 @@
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diff --git a/_static/sphinx-design.min.css b/_static/sphinx-design.min.css
index a325746..860c36d 100644
--- a/_static/sphinx-design.min.css
+++ b/_static/sphinx-design.min.css
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diff --git a/content/OGZAF_references.html b/content/OGZAF_references.html
index 8267a1a..d77ba4b 100644
--- a/content/OGZAF_references.html
+++ b/content/OGZAF_references.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -438,6 +438,10 @@ 
References[EP17]
 
Richard W. Evans and Kerk L. Phillips. Advantages of an ellipse when modeling leisure utility. Computational Economics, 2017.

 

+
 

 [Nis15]
 
Shinichi Nishiyama. Fiscal policy effects in a heterogeneous-agent olg economy with an aging population. Journal of Economic Dynamics and Control, 61:114–132, December 2015.

@@ -454,10 +458,6 @@ 
References[BusinessTech24]
 
Business Tech. `reverse emigration' in south africa - this is how many people actually came home. Business Tech, April 2, 2024. URL: https://businesstech.co.za/news/lifestyle/764879/reverse-emigration-in-south-africa-this-is-how-many-people-actually-came-home/.

 

-

-[CensusBureau15]
-
Census Bureau. Annual estimates of the resident population by single year of age and sex: april 1, 2010 to july 1, 2013 (both sexes). National Characteristics Vintage 2013, U.S. Census Bureau, http://www.census.gov/popest/data/national/asrh/2013/index.html, 2015.

-

 

 

 
diff --git a/content/api/calibrate.html b/content/api/calibrate.html
index 87cb4cb..2c633ae 100644
--- a/content/api/calibrate.html
+++ b/content/api/calibrate.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -430,7 +430,7 @@ 
 Contents 

 
ogzaf.calibrate#

 

 

-class ogzaf.calibrate.Calibration(p, estimate_tax_functions=False, estimate_chi_n=False, demographic_data_path=None, output_path=None)[source]#

+class ogzaf.calibrate.Calibration(p, macro_data_start_year=datetime.datetime(1947, 1, 1), macro_data_end_year=datetime.date.today(), demographic_data_path=None, output_path=None)[source]#
 
OG-ZAF calibration class

 

 
diff --git a/content/api/income.html b/content/api/income.html
index b558f57..b2bbea5 100644
--- a/content/api/income.html
+++ b/content/api/income.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -670,8 +670,8 @@ 
percentiles, as it uses fitted polynomials to those percentiles.
-\[\ln(abil) = lpha + eta_{1}    ext{age} + eta_{2}     ext{age}^2
-    + eta_{3}  ext{age}^3\]

+\[\ln(abil) = \alpha + \beta_{1}\text{age} + \beta_{2}\text{age}^2
+    + \beta_{3}\text{age}^3\]

 
To calibrate for ZAF, the USA curves are adjusted in 2 ways (in this order)
 1) Adjustment by income (J): adjust the gaps between the J-income earning curves

 

diff --git a/content/api/input_output.html b/content/api/input_output.html
index dd05742..fb84c69 100644
--- a/content/api/input_output.html
+++ b/content/api/input_output.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
diff --git a/content/api/macro_params.html b/content/api/macro_params.html
index 72b98fd..dda0600 100644
--- a/content/api/macro_params.html
+++ b/content/api/macro_params.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -429,12 +429,27 @@ 
 Contents 

 

 
ogzaf.macro_params#

 
This module uses data from World Bank WDI, World Bank Quarterly Public
-Sector Debt (QPSD) database, UN Data Portal, and FRED to find values for
+Sector Debt (QPSD) database, the IMF, and UN ILO to find values for
 parameters for the OG-ZAF model that rely on macro data for calibration.

 

 

-ogzaf.macro_params.get_macro_params()[source]#

+ogzaf.macro_params.get_macro_params(data_start_date=datetime.datetime(1947, 1, 1), data_end_date=datetime.date.today(), country_iso='ZAF')[source]#
 
Compute values of parameters that are derived from macro data

+

+
Parameters:

+
    
+
  • data_start_date (datetime) – start date for data
    • 
+
  • data_end_date (datetime) – end date for data
    • 
+
  • country_iso (str) – ISO code for country
    • 
+

+

+
Returns:

+
dictionary of parameter values

+

+
Return type:

+
macro_parameters (dict)

+

+

 

 
 

diff --git a/content/api/public_api.html b/content/api/public_api.html
index a75ef40..3a6a3c0 100644
--- a/content/api/public_api.html
+++ b/content/api/public_api.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/api/utils.html b/content/api/utils.html
index de0d4c7..6d1eb3b 100644
--- a/content/api/utils.html
+++ b/content/api/utils.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/calibration/UBI.html b/content/calibration/UBI.html
index 2b18e96..f189394 100644
--- a/content/calibration/UBI.html
+++ b/content/calibration/UBI.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -428,14 +428,14 @@ 
 Contents 

   

 
Universal Basic Income (UBI)#

 
[TODO: This section is far along but needs to be updated.]

-
We have included the modeling of a universal basic income (UBI) policy directly in the theory and code for [OG-Core] on which dependency the OG-ZAF is based. UBI shows up in the household budget constraint (12), and is described in the Budget Constraint section of the Households chapter of the OG-Core documentation. We calculate the time series of a UBI matrix \(ubi_{j,s,t}\) representing the UBI transfer to every household with head of household age \(s\), lifetime income group \(j\), in period \(t\). We calculate the time series of this matrix from five parameters and some household composition data that we impose upon the existing demographics of OG-ZAF.

+
We have included the modeling of a universal basic income (UBI) policy directly in the theory and code for [OG-Core] on which dependency the OG-ZAF is based. UBI shows up in the household budget constraint (14), and is described in the Budget Constraint section of the Households chapter of the OG-Core documentation. We calculate the time series of a UBI matrix \(ubi_{j,s,t}\) representing the UBI transfer to every household with head of household age \(s\), lifetime income group \(j\), in period \(t\). We calculate the time series of this matrix from five parameters and some household composition data that we impose upon the existing demographics of OG-ZAF.

 

 
Calculating UBI#

 
We calculate the time series of UBI household transfers in model units \(ubi_{j,s,t)}\) and the time series of total UBI expenditures in model units \(UBI_t\) from five parameters described in the ogzaf_default_parameters.json file (ubi_growthadj, ubi_nom_017, ubi_nom_1864, ubi_nom_65p, and ubi_nom_max) interfaced with the OG-ZAF demographic dynamics over lifetime income groups \(j\) and ages \(s\), and multiplied by household composition matrices from the Calibrate class of the OG-ZAF/ogzaf/calibrate.py module in the repository.

 
From the OG-ZAF repository, we have four \(S\times J\) matrices ubi_num_017_mat\(_{j,s}\), ubi_num_1864_mat\(_{j,s}\), and ubi_num_65p_mat\(_{j,s}\) representing the number of children under age 0-17, number of adults ages 18-64, and the number of seniors age 65 and over, respectively, by lifetime ability group \(j\) and age \(s\) of head of household. Because our demographic age data match up well with head-of-household data from other datasets, we do not have to adjust the values in these matrices.[1]

 
Now we can solve for the dollar-valued (as opposed to model-unit-valued) UBI transfer to each household in the first period \(ubi^{\$}_{j,s,t=0}\) in the following way. Let the parameter ubi_nom_017 be the dollar value of the UBI transfer to each household per dependent child age 17 and under. Let the parameter ubi_nom_1864 be the dollar value of the UBI transfer to each household per adult between the ages of 18 and 64. Let ubi_nom_65p be the dollar value of UBI transfer to each household per senior 65 and over. And let ubi_nom_max be the maximum UBI benefit per household.

 

-(13)#\[\begin{split}  \begin{split}
+(15)#\[\begin{split}  \begin{split}
     ubi^{\$}_{j,s,t=0} = \min\Bigl(&\texttt{ubi_nom_max}, \\
     &\texttt{ubi_nom_017} * \texttt{ubi_num_017_mat}_{j,s} + \\
     &\texttt{ubi_nom_1864} * \texttt{ubi_num_1864_mat}_{j,s} + \\
@@ -447,10 +447,10 @@ 
 Contents 

 
UBI specification not adjusted for economic growth#

 
A non-growth adjusted UBI (ubi_growthadj = False) is one in which the initial nonstationary dollar-valued \(t=0\) UBI matrix \(ubi^{\$}_{j,s,t=0}\) does not grow, while the economy’s long-run growth rate is \(g_y\) for the most common parameterization where the long-run growth rate is positive \(g_y>0\).

 

-(14)#\[  ubi^{\$}_{j,s,t} = ubi^{\$}_{j,s,t=0} \quad\forall j,s,t\]

+(16)#\[  ubi^{\$}_{j,s,t} = ubi^{\$}_{j,s,t=0} \quad\forall j,s,t\]

 
As described in the OG-Core chapter on stationarization, the stationarized UBI transfer to each household \(\hat{ubi}_{j,s,t}\) is the nonstationary transfer divided by the growth rate since the initial period. When the long-run economic growth rate is positive \(g_y>0\) and the UBI specification is not growth-adjusted the steady-state stationary UBI household transfer is zero \(\overline{ubi}_{j,s}=0\) for all lifetime income groups \(j\) and ages \(s\) as time periods \(t\) go to infinity. However, to simplify, we assume in this case that the stationarized steady-state UBI transfer matrix to households is the stationarized value of that matrix in period \(T\).

 

-(15)#\[  \overline{ubi}_{j,s} = ubi_{j,s,t=T} \quad\forall j,s\]

+(17)#\[  \overline{ubi}_{j,s} = ubi_{j,s,t=T} \quad\forall j,s\]

 
Note that in non-growth-adjusted case, if \(g_y<0\), then the stationary value of \(\hat{ubi}_{j,s,t}\) is going to infinity as \(t\) goes to infinity. Therefore, a UBI specification must be growth adjusted for any assumed negative long run growth \(g_y<0\).[2]

 
 

diff --git a/content/calibration/demographics.html b/content/calibration/demographics.html
index ed4276a..a2a5f49 100644
--- a/content/calibration/demographics.html
+++ b/content/calibration/demographics.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -432,17 +432,17 @@ 
 Contents 

 
We define \(\omega_{s,t}\) as the number of households of age \(s\) alive at time \(t\). A measure \(\omega_{1,t}\) of households is born in each period \(t\) and live for up to \(E+S\) periods, with \(S\geq 4\).[1] Households are termed “youth”, and do not participate in market activity during ages \(1\leq s\leq E\). The households enter the workforce and economy in period \(E+1\) and remain in the workforce until they unexpectedly die or live until age \(s=E+S\). We model the population with households age \(s\leq E\) outside of the workforce and economy in order most closely match the empirical population dynamics.

 
The population of agents of each age in each period \(\omega_{s,t}\) evolves according to the following function,

 

-(1)#\[\begin{split}    \omega_{1,t+1} &= (1 - \rho_{0,t})\sum_{s=1}^{E+S} f_{s,t}\omega_{s,t} + i_1\omega_{1,t}\quad\forall t \\
+(3)#\[\begin{split}    \omega_{1,t+1} &= (1 - \rho_{0,t})\sum_{s=1}^{E+S} f_{s,t}\omega_{s,t} + i_1\omega_{1,t}\quad\forall t \\
     \omega_{s+1,t+1} &= (1 - \rho_{s,t})\omega_{s,t} + i_{s+1,t}\omega_{s+1,t}\quad\forall t\quad\text{and}\quad 1\leq s \leq E+S-1\end{split}\]

 
where \(f_{s,t}\geq 0\) is an age-specific fertility rate, \(i_{s,t}\) is an age-specific net immigration rate, \(\rho_{s,t}\) is an age-specific mortality hazard rate, and \(\rho_{0,t}\) is an infant mortality rate.[2] The total population in the economy \(N_t\) at any period is simply the sum of households in the economy, the population growth rate in any period \(t\) from the previous period \(t-1\) is \(g_{n,t}\), \(\tilde{N}_t\) is the working age population, and \(\tilde{g}_{n,t}\) is the working age population growth rate in any period \(t\) from the previous period \(t-1\).

 

-(2)#\[  N_t\equiv\sum_{s=1}^{E+S} \omega_{s,t} \quad\forall t\]

+(4)#\[  N_t\equiv\sum_{s=1}^{E+S} \omega_{s,t} \quad\forall t\]

 

-(3)#\[  g_{n,t+1} \equiv \frac{N_{t+1}}{N_t} - 1 \quad\forall t\]

+(5)#\[  g_{n,t+1} \equiv \frac{N_{t+1}}{N_t} - 1 \quad\forall t\]

 

-(4)#\[  \tilde{N}_t\equiv\sum_{s=E+1}^{E+S} \omega_{s,t} \quad\forall t\]

+(6)#\[  \tilde{N}_t\equiv\sum_{s=E+1}^{E+S} \omega_{s,t} \quad\forall t\]

 

-(5)#\[  \tilde{g}_{n,t+1} \equiv \frac{\tilde{N}_{t+1}}{\tilde{N}_t} - 1 \quad\forall t\]

+(7)#\[  \tilde{g}_{n,t+1} \equiv \frac{\tilde{N}_{t+1}}{\tilde{N}_t} - 1 \quad\forall t\]

 
We discuss the approach to estimating fertility rates \(f_{s,t}\), mortality rates \(\rho_{s,t}\), and immigration rates \(i_{s,t}\) in Sections Fertility rates, Mortality rates, and Immigration rates.

 

 
Fertility rates#

@@ -530,9 +530,9 @@ 
 Contents 

 

 

 
Immigration rates#

-
Because of the difficulty in getting accurate immigration rate data by age, we estimate the immigration rates by age in our model \(i_s\) as the average residual that reconciles the current-period population distribution with next period’s population distribution given fertility rates \(f_s\) and mortality rates \(\rho_{s,t}\). Solving equations (1) for the immigration rate \(i_s\) gives the following characterization of the immigration rates in given population levels in any two consecutive periods \(\omega_{s,t}\) and \(\omega_{s,t+1}\) and the fertility rates \(f_s\) and mortality rates \(\rho_{s,t}\).

+
Because of the difficulty in getting accurate immigration rate data by age, we estimate the immigration rates by age in our model \(i_s\) as the average residual that reconciles the current-period population distribution with next period’s population distribution given fertility rates \(f_s\) and mortality rates \(\rho_{s,t}\). Solving equations (3) for the immigration rate \(i_s\) gives the following characterization of the immigration rates in given population levels in any two consecutive periods \(\omega_{s,t}\) and \(\omega_{s,t+1}\) and the fertility rates \(f_s\) and mortality rates \(\rho_{s,t}\).

 

-(6)#\[\begin{split}    i_{1,t} &= \frac{\omega_{1,t+1} - (1 - \rho_{0,t})\sum_{s=1}^{E+S}f_{s,t}\omega_{s,t}}{\omega_{1,t}}\quad\forall t \\
+(8)#\[\begin{split}    i_{1,t} &= \frac{\omega_{1,t+1} - (1 - \rho_{0,t})\sum_{s=1}^{E+S}f_{s,t}\omega_{s,t}}{\omega_{1,t}}\quad\forall t \\
     i_{s+1,t+1} &= \frac{\omega_{s+1,t+1} - (1 - \rho_{s,t})\omega_{s,t}}{\omega_{s+1,t}}\qquad\qquad\forall t\quad\text{and}\quad 1\leq s \leq E+S-1\end{split}\]

 

 

@@ -574,17 +574,17 @@ 
 Contents 

 
Fig. 3 South Africa immigration rates by age \(\left(i_s\right)\) for \(E+S=100\): year 2023#

 
 
-
We calculate our immigration rates for the consecutive-year-periods of population distribution data 2022 and 2023. The immigration rates \(i_{s,t}\) that we use in our model are the the residuals described in (6) implied by these two consecutive periods. Figure 3 shows the estimated immigration rates for \(E+S=100\) and given the fertility rates from Section Fertility rates and the mortality rates from Section Mortality rates. These immigration rates show large out-migration from South Africa.[4]

+
We calculate our immigration rates for the consecutive-year-periods of population distribution data 2022 and 2023. The immigration rates \(i_{s,t}\) that we use in our model are the the residuals described in (8) implied by these two consecutive periods. Figure 3 shows the estimated immigration rates for \(E+S=100\) and given the fertility rates from Section Fertility rates and the mortality rates from Section Mortality rates. These immigration rates show large out-migration from South Africa.[4]

 
At the end of Section Population steady-state and transition path, we describe a small adjustment that we make to the immigration rates after a certain number of periods in order to make computation of the transition path equilibrium of the model compute more robustly.

 

 

 
Population steady-state and transition path#

-
This model requires information about mortality rates \(\rho_{s,t}\) in order to solve for the household’s problem each period. It also requires the steady-state stationary population distribution \(\bar{\omega}_{s}\) and population growth rate \(\bar{g}_n\) as well as the full transition path of the stationary population distribution \(\hat{\omega}_{s,t}\) and population grow rate \(\tilde{g}_{n,t}\) from the current state to the steady-state. To solve for the steady-state and the transition path of the stationary population distribution, we write the stationary population dynamic equations (7) and their matrix representation (8).

+
This model requires information about mortality rates \(\rho_{s,t}\) in order to solve for the household’s problem each period. It also requires the steady-state stationary population distribution \(\bar{\omega}_{s}\) and population growth rate \(\bar{g}_n\) as well as the full transition path of the stationary population distribution \(\hat{\omega}_{s,t}\) and population grow rate \(\tilde{g}_{n,t}\) from the current state to the steady-state. To solve for the steady-state and the transition path of the stationary population distribution, we write the stationary population dynamic equations (9) and their matrix representation (10).

 

-(7)#\[\begin{split}    \hat{\omega}_{1,t+1} &= \frac{(1-\rho_{0,t})\sum_{s=1}^{E+S} f_{s,t}\hat{\omega}_{s,t} + i_{1,t}\hat{\omega}_{1,t}}{1+\tilde{g}_{n,t+1}}\quad\forall t \\
+(9)#\[\begin{split}    \hat{\omega}_{1,t+1} &= \frac{(1-\rho_{0,t})\sum_{s=1}^{E+S} f_{s,t}\hat{\omega}_{s,t} + i_{1,t}\hat{\omega}_{1,t}}{1+\tilde{g}_{n,t+1}}\quad\forall t \\
     \hat{\omega}_{s+1,t+1} &= \frac{(1 - \rho_{s,t})\hat{\omega}_{s,t} + i_{s+1,t}\hat{\omega}_{s+1,t}}{1+\tilde{g}_{n,t+1}}\qquad\quad\:\forall t\quad\text{and}\quad 1\leq s \leq E+S-1\end{split}\]

 

-(8)#\[\begin{split}    & \begin{bmatrix}
+(10)#\[\begin{split}    & \begin{bmatrix}
       \hat{\omega}_{1,t+1} \\ \hat{\omega}_{2,t+1} \\ \hat{\omega}_{2,t+1} \\ \vdots \\ \hat{\omega}_{E+S-1,t+1} \\ \hat{\omega}_{E+S,t+1}
     \end{bmatrix}= \frac{1}{1 + g_{n,t+1}} \times ... \\
     & \begin{bmatrix}
@@ -598,12 +598,12 @@ 
 Contents 

     \begin{bmatrix}
       \hat{\omega}_{1,t} \\ \hat{\omega}_{2,t} \\ \hat{\omega}_{2,t} \\ \vdots \\ \hat{\omega}_{E+S-1,t} \\ \hat{\omega}_{E+S,t}
     \end{bmatrix}\end{split}\]

-
We can write system (8) more simply in the following way.

+
We can write system (10) more simply in the following way.

 

-(9)#\[  \boldsymbol{\hat{\omega}}_{t+1} = \frac{1}{1+g_{n,t+1}}\boldsymbol{\Omega}\boldsymbol{\hat{\omega}}_t \quad\forall t\]

-
The stationary steady-state population distribution \(\boldsymbol{\bar{\omega}}\) is the eigenvector \(\boldsymbol{\omega}\) with eigenvalue \((1+\bar{g}_n)\) of the matrix \(\boldsymbol{\Omega}\) that satisfies the following version of (9).

+(11)#\[  \boldsymbol{\hat{\omega}}_{t+1} = \frac{1}{1+g_{n,t+1}}\boldsymbol{\Omega}\boldsymbol{\hat{\omega}}_t \quad\forall t\]

+
The stationary steady-state population distribution \(\boldsymbol{\bar{\omega}}\) is the eigenvector \(\boldsymbol{\omega}\) with eigenvalue \((1+\bar{g}_n)\) of the matrix \(\boldsymbol{\Omega}\) that satisfies the following version of (11).

 

-(10)#\[  (1+\bar{g}_n)\boldsymbol{\bar{\omega}} = \boldsymbol{\Omega}\boldsymbol{\bar{\omega}}\]

+(12)#\[  (1+\bar{g}_n)\boldsymbol{\bar{\omega}} = \boldsymbol{\Omega}\boldsymbol{\bar{\omega}}\]

 

 
Proposition

 
If the age \(s=1\) immigration rate is \(i_1>-(1-\rho_0)f_1\) and the other immigration rates are strictly positive \(i_s>0\) for all \(s\geq 2\) such that all elements of \(\boldsymbol{\Omega}\) are nonnegative, then there exists a unique positive real eigenvector \(\boldsymbol{\bar{\omega}}\) of the matrix \(\boldsymbol{\Omega}\), and it is a stable equilibrium.

@@ -666,14 +666,14 @@ 
 Contents 

 
Fig. 4 Theoretical steady-state population distribution vs. population distribution at period \(t=120\)#

 
 
-
Further, we find that the maximum absolute difference between the population levels \(\hat{\omega}_{s,t}\) and \(\hat{\omega}_{s,t+1}\) was \(1.3852\times 10^{-5}\) after 160 periods. That is to say, that after 160 periods, given the estimated mortality, fertility, and immigration rates, the population has not achieved its steady state. For convergence in our solution method over a reasonable time horizon, we want the population to reach a stationary distribution after \(T\) periods. To do this, we artificially impose that the population distribution in period \(t=120\) is the steady-state. As can be seen from Figure 4, this assumption is not very restrictive. Figure 5 shows the change in immigration rates that would make the period \(t=120\) population distribution equal be the steady-state. The maximum absolute difference between any two corresponding immigration rates in Figure 5 is 0.0028.

+
Further, we find that the maximum absolute difference between the population levels \(\hat{\omega}_{s,t}\) and \(\hat{\omega}_{s,t+1}\) was less than \(1\times 10^{-4}\) after 160 periods. That is to say, that after 160 periods, given the estimated mortality, fertility, and immigration rates, the population has not achieved its steady state. For convergence in our solution method over a reasonable time horizon, we want the population to reach a stationary distribution after \(T\) periods. To do this, we artificially impose that the population distribution in period \(t=120\) is the steady-state. As can be seen from Figure 4, this assumption is not very restrictive. Figure 5 shows the change in immigration rates that would make the period \(t=120\) population distribution equal be the steady-state. The maximum absolute difference between any two corresponding immigration rates in Figure 5 is very small.

 

 ../../_images/OrigVsAdjImm.png
 

 
Fig. 5 Original immigration rates vs. adjusted immigration rates to make fixed steady-state population distribution#

 

 

-
The most recent year of population data come from [Census Bureau, 2015] population estimates for both sexes for 2013. We those data and use the population transition matrix (9) to age it to the current model year of 2015. We then use (9) to generate the transition path of the population distribution over the time period of the model. Figure 6 shows the progression from the 2013 population data to the fixed steady-state at period \(t=120\). The time path of the growth rate of the economically active population \(\tilde{g}_{n,t}\) is shown in Figure 6.

+
We begin with 2023 population data and use the population transition matrix (11) to age it to the start year of the model (e.g., 2024 or 2025). We then use (11) to generate the transition path of the population distribution over the time period of the model. Figure 6 shows the progression from the 2023 population data to the fixed steady-state at period \(t=120\). The time path of the growth rate of the economically active population \(\tilde{g}_{n,t}\) is shown in Figure 6.

 

 ../../_images/pop_distribution.png
 

@@ -738,7 +738,7 @@ 
 Contents 

 
 
 
 
diff --git a/content/calibration/earnings.html b/content/calibration/earnings.html
index 2703ae3..97cf298 100644
--- a/content/calibration/earnings.html
+++ b/content/calibration/earnings.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -417,7 +417,7 @@ 
Lifetime Earnings Profiles

 
Among households in OG-ZAF, we model variations in the labor productivity over the lifecycle and between households of different skill groups. Together, these variations in productivity generate a distribution of earnings that is calibrated to match the level of inequality in South Africa. This chapter describes the calibration of the lifecycle earnings profiles and the distribution of earnings in the model.

 
Differences among workers’ productivity is one of the key dimensions of heterogeneity to model in a micro-founded macroeconomy. In this chapter, we characterize this heterogeneity as deterministic lifetime productivity paths to which new cohorts of agents in the model are randomly assigned. In OG-ZAF, households’ labor income comes from the equilibrium wage and the agent’s endogenous quantity of labor supply. In this section, we augment the labor income expression with an individual productivity \(e_{j,s}\), where \(j\) is the index of the ability type or path of the individual and \(s\) is the age of the individual with that ability path.

 

-(11)#\[  \text{labor income:}\quad x_{j,s,t}\equiv w_t e_{j,s}n_{j,s,t} \quad\forall j,t \quad\text{and}\quad E+1\leq s\leq E+S\]

+(13)#\[  \text{labor income:}\quad x_{j,s,t}\equiv w_t e_{j,s}n_{j,s,t} \quad\forall j,t \quad\text{and}\quad E+1\leq s\leq E+S\]

 
In this specification, \(w_t\) is an equilibrium wage representing a portion of labor income that is common to all workers. Individual quantity of labor supply is \(n_{j,s,t}\), and \(e_{j,s}\) represents a labor productivity factor that augments or diminishes the productivity of a worker’s labor supply relative to average productivity.

 
We calibrate the model such that each lifetime income group has a different life-cycle profile of earnings. Since the distribution on income and wealth are key aspects of our model, we calibrate these processes so that we can represent earners in the top 1 percent of the distribution of lifetime income.

 
We calibrate deterministic productivity paths such that each lifetime income group has a different life-cycle profile of earnings. The distribution of income and wealth are often focal components of macroeconomic models. These calibrations require the use of microeconomic data on household incomes, but this level of data is not readily available for South Africa from public sources or surveys. To overcome this, we start with the proposition that estimated productivity curves calibrated for the OG-USA model, generated from micro-level earnings data, represent a generalized relationship between age and lifetime income [DeBacker et al., 2017]. As such, our objective is to generate the curves for the U.S. and then adjust their generalized shapes to produce those for South Africa. In other words, our strategic approach is to begin with the lifecycle labor productivity profiles estimated from detailed U.S. data and then adjust these to match the distribution of income in South Africa. This is done in two ways (in this order):

diff --git a/content/calibration/exogenous_parameters.html b/content/calibration/exogenous_parameters.html
index 7fa90c4..e7ced1c 100644
--- a/content/calibration/exogenous_parameters.html
+++ b/content/calibration/exogenous_parameters.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -419,41 +419,291 @@ 
Exogenous Parameters

   ```{code-cell} ogzaf-dev
   :tags: [hide-cell]
   from myst_nb import glue
-  import ogzaf.parameter_tables as pt
-  from ogzaf import Specifications
+  import ogcore.parameter_tables as pt
+  from ogcore import Specifications
+  import ogzaf
+  import importlib
+  import json
   p = Specifications()
-  table = pt.param_table(p, table_format=None, path=None)
+  with importlib.resources.open_text(
+      "ogzaf", "ogzaf_default_parameters.json"
+      ) as file:
+          defaults = json.load(file)
+  table = pt.param_table(p, table_format="md", path=None)
   glue("param_table", table, display=False)
   ```
   -->
-

-
+
Table 1 List of exogenous parameters and baseline calibration values.#

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Symbol
Description
Value
Symbol
Description
Value

 

 
 
\(S\)
Maximum periods in economically active household life
80
\(\texttt{start_year}\)
Initial year
2025

 
\(E\)
Number of periods of youth economically outside the model
\(\text{round} \frac{S}{4}\)=20
\(\omega_{s,t}\)
Population by age over time
Too large to report here, see default parameters JSON

 
\(T_1\)
Number of periods to steady state for initial time path guesses
160
\(i_{s,t}\)
Immigration rates by age
Too large to report here, see default parameters JSON

 
\(T_2\)
Maximum number of periods to steady state for nonsteady-state equilibrium
160
\(\rho_{s,t}\)
Mortality rates by age
Too large to report here, see default parameters JSON

 
\(\nu\)
Dampening parameter for TPI
0.4
\(e_{j,s,t}\)
Deterministic ability process
Too large to report here, see default parameters JSON
\(\lambda_{j}\)
Lifetime income group percentages
Too large to report here, see default parameters JSON
\(J\)
Number of lifetime income groups
7
\(S\)
Maximum periods in economically active individual life
80
\(E\)
Number of periods of youth economically outside the model
20
\(T\)
Number of periods to steady-state
320
\(R\)
Retirement age
[65.000…65.000]
\(\tilde{l}\)
Maximum hours of labor supply
1.000
\(\beta\)
Discount factor
[0.960…0.960]
\(\sigma\)
Coefficient of constant relative risk aversion
1.500
\(\nu\)
Frisch elasticity of labor supply
0.400
\(b\)
Scale parameter in utility of leisure
0.573
\(\upsilon\)
Shape parameter in utility of leisure
2.856
\(\chi^{n}_{s}\)
Disutility of labor level parameters
Too large to report here, see default parameters JSON
\(\chi^{b}_{j}\)
Utility of bequests level parameters
[80.000…80.000]
\(\texttt{use_zeta}\)
Whether to distribute bequests between lifetime income groups
0.00E+00
\(\zeta\)
Distribution of bequests
Too large to report here, see default parameters JSON
\(Z_{t}\)
Total factor productivity
Too large to report here, see default parameters JSON
\(\gamma\)
Capital share of income
[0.401…0.401]
\(\varepsilon\)
Elasticity of substitution between capital and labor
[1.000…1.000]
\(\delta\)
Capital depreciation rate
0.050
\(g_{y}\)
Growth rate of labor augmenting technological progress
0.00E+00
\(\texttt{tax_func_type}\)
Functional form used for income tax functions
linear
\(\texttt{analytical_mtrs}\)
Whether use analytical MTRs or estimate MTRs
0.00E+00
\(\texttt{age_specific}\)
Whether use age-specific tax functions
0.00E+00
\(\tau^{p}_{t}\)
Payroll tax rate
[0.000…0.000]
\(\tau^{BQ}_{t}\)
Bequest (estate) tax rate
[0.200…0.200]
\(\tau^{b}_{t}\)
Entity-level business income tax rate
Too large to report here, see default parameters JSON
\(\delta^{\tau}_{t}\)
Rate of depreciation for tax purposes
Too large to report here, see default parameters JSON
\(\tau^{c}_{t,s,j}\)
Consumption tax rates
Too large to report here, see default parameters JSON
\(H\)
Coefficient on linear term in wealth tax function
[0.100…0.100]
\(M\)
Constant in wealth tax function
[1.000…1.000]
\(P\)
Coefficient on level term in wealth tax function
[0.000…0.000]
\(\texttt{budget_balance}\)
Whether have a balanced budget in each period
0.00E+00
\(\texttt{baseline_spending}\)
Whether level of spending constant between the baseline and reform runs
0.00E+00
\(\alpha^{T}_{t}\)
Transfers as a share of GDP
[0.041…0.041]
\(\eta_{j,s,t}\)
Distribution of transfers
Too large to report here, see default parameters JSON
\(\alpha^{G}_{t}\)
Government spending as a share of GDP
[0.267…0.267]
\(t_{G1}\)
Model period in which budget closure rule starts
20
\(t_{G2}\)
Model period in which budget closure rule ends
256
\(\rho_{G}\)
Budget closure rule smoothing parameter
0.100
\(\bar{\alpha}_{D}\)
Steady-state Debt-to-GDP ratio
1.200
\(\alpha_{D,0}\)
Initial period Debt-to-GDP ratio
0.740
\(\tau_{d,t}\)
Scale parameter in government interest rate wedge
[0.245…0.245]
\(\mu_{d,t}\)
Shift parameter in government interest rate wedge
[-0.034…-0.034]
\(\texttt{avg_earn_num_years}\)
Number of years over which compute average earnings for pension benefit
35
\(\texttt{AIME_bkt_1}\)
First AIME bracket threshold
749.000
\(\texttt{AIME_bkt_2}\)
Second AIME bracket threshold
4517.000
\(\texttt{PIA_rate_bkt_1}\)
First AIME bracket PIA rate
0.00E+00
\(\texttt{PIA_rate_bkt_2}\)
Second AIME bracket PIA rate
0.00E+00
\(\texttt{PIA_rate_bkt_3}\)
Third AIME bracket PIA rate
0.00E+00
\(\texttt{PIA_maxpayment}\)
Maximum PIA payment
0.00E+00
\(\texttt{PIA_minpayment}\)
Minimum PIA payment
0.00E+00
\(\theta_{adj,t}\)
Adjustment to replacement rate
[1.000…1.000]
\(r^{*}_{t}\)
World interest rate
[0.040…0.040]
\(D_{f,0}\)
Share of government debt held by foreigners in initial period
0.237
\(\zeta_{D, t}\)
Share of new debt issues purchased by foreigners
[0.237…0.237]
\(\zeta_{K, t}\)
Share of excess capital demand satisfied by foreigners
[0.900…0.900]
\(\xi\)
Dampening parameter for TPI
0.400
\(\texttt{maxiter}\)
Maximum number of iterations for TPI
250
\(\texttt{mindist_SS}\)
SS solution tolerance
1.00E-09
\(\texttt{mindist_TPI}\)
TPI solution tolerance
1.00E-05

 

 
 

diff --git a/content/calibration/firms.html b/content/calibration/firms.html
index 17d27cf..37951c3 100644
--- a/content/calibration/firms.html
+++ b/content/calibration/firms.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -412,7 +412,8 @@ 
 Contents 

                 
@@ -431,13 +432,32 @@ 
 Contents 

 

 
Aggregate Production Function and Capital Accumulation#

 
The OG-Core firm theory documentation outlines the constant returns to scale, constant elasticity of substitution production function of the representative firm.  This function has two parameters; the elasticity of substitution and capital’s share of output.

+
The production function is given as:

+

+(1)#\[\begin{split}  \begin{split}
+    Y_{m,t} &= F(K_{m,t}, K_{g,m,t}, L_{m,t}) \\
+    &\equiv Z_{m,t}\biggl[(\gamma_m)^\frac{1}{\varepsilon_m}(K_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + (\gamma_{g,m})^\frac{1}{\varepsilon_m}(K_{g,m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + \\
+    &\quad\quad\quad\quad\quad(1-\gamma_m-\gamma_{g,m})^\frac{1}{\varepsilon_m}(e^{g_y t}L_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m}\biggr]^\frac{\varepsilon_m}{\varepsilon_m-1} \quad\forall m,t
+  \end{split}\end{split}\]

+
This production function has the following parameters:

+
    
+
  • \(\varepsilon_m\) is the elasticity of substitution between capital, labor, and infrastructure in sector \(m\).
    • 
+
  • \(\gamma_m\) is the share of capital in sector \(m\).
    • 
+
  • \(\gamma_{g,m}\) is the share of government capital in sector \(m\).
    • 
+
  • \(Z_{m,t}\) is the total factor productivity in sector \(m\) at time \(t\).
    • 
+

 

 
Elasticity of substitution#

 
OG-ZAF’s default parameterization has an elasticity of substitution of \(\varepsilon=1.0\), which implies a Cobb-Douglas production function.

 

-

-
Capital’s share of output#

-
Here, we use a default value of \(\gamma =0.40\).

+

+
Factor shares of output#

+
In the default calibration, we set infrastructure’s share of output to \(\gamma_{g,m}=0.0\) for all sectors.  This parameter is hard to identify from national accounts data and would entail an empirical study to tease out the relationship between infrastructure and output.

+
We use a default value of \(\gamma =0.40\), which corresponds to one minus labor’s share of output, where labor’s share of output is found as 0.60 in in the UN ILOSTAT database.

+

+

+
Total factor productivity#

+
In the case of the single prodcution sector, we can normalize \(Z_{m,t}=1.0\).  In the case of multiple production sectors, we use {cite}`PRS2020 who identify TFP for various sectors in South Africa.

 

 

 

@@ -508,7 +528,8 @@ 
Capital’s share of output
 
  • Aggregate Production Function and Capital Accumulation
 
    • 
 
diff --git a/content/calibration/households.html b/content/calibration/households.html
index 240c4d9..137e82f 100644
--- a/content/calibration/households.html
+++ b/content/calibration/households.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/calibration/macro.html b/content/calibration/macro.html
index f144b1b..d89ac01 100644
--- a/content/calibration/macro.html
+++ b/content/calibration/macro.html
@@ -34,7 +34,7 @@
     
     
     
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+    
   
   
   
@@ -418,7 +418,10 @@ 
     Contents 
    
 
 
 
  • Government Debt, Spending and Transfers
@@ -460,6 +463,13 @@ 
    Government Debt, Spending and Transfers
 
    Government Debt#
    
 
    The path of government debt is endogenous.  But the initial value is exogenous.  To avoid converting between model units and dollars, we calibrate the initial debt to GDP ratio, rather than the dollar value of the debt.  This is the model parameter \(\alpha_D\).  We compute this from the ratio of publicly held debt outstanding to GDP.  Based on 2023 values, this gives us a ratio of 0.59.
    
+
    
+
    Interest rates on government debt#
    
+
    We assume that there is a wedge between the real rate of return on private capital and the real interest rate on government debt.  We model this wedge a scale and level shift.  Specifically, we assume that the real interest rate on government debt, \(r_{gov,t}\), is related to the real rate of return on private capital, \(r_{t}\), by the following equation:
    
+
    
+(2)#\[    r_{gov,t} = (1-\tau_{d,t})r_t + \mu_d\]
    
+
    where \(\tau_d\) is the scale parameter and \(\mu_d\) is the level shift parameter.  We set the values of these two parameters to 0.245 and -0.034, respectively.  These are found by using the estimated relationship between corporate and sovereign yields in [Li et al., 2023] (Table 8, Column 2) and simulating a series of corporate yields given a series of sovereign yields between 2% and 12%.  We then estimate the scale and level shift parameters that best fit these simulated data using ordinary least squares.
    
+
    
 
 
    
 
    Aggregate transfers#
    
@@ -546,7 +556,10 @@ 
    Government expenditures
 
    • 
 
  • Government Debt, Spending and Transfers
diff --git a/content/calibration/matching_lwi.html b/content/calibration/matching_lwi.html
index 23649eb..01e296c 100644
--- a/content/calibration/matching_lwi.html
+++ b/content/calibration/matching_lwi.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/calibration/taxes.html b/content/calibration/taxes.html
index 328a1f9..9f61d02 100644
--- a/content/calibration/taxes.html
+++ b/content/calibration/taxes.html
@@ -34,7 +34,7 @@
     
     
     
-    
+    
   
   
   
@@ -429,9 +429,9 @@ 
     Contents 
    
 
    The government is not an optimizing agent in OG-ZAF. The government levies taxes on household income, corporate income, and value added. With these resources, the government provides transfers to households, spends resources on public goods, and makes rule-based adjustments to stabilize the economy in the long-run. The government can run budget deficits or surpluses in a given year and must, therefore, be able to accumulate debt or savings.  The spending and debt parameters are discussed in Chapter Calibration of Macroeconomic Parameters.  Taxes are discussed in this chapter.
    
 
    
 
    Personal income taxes#
    
-
    The government sector influences households through two terms in the household budget constraint (12)—government transfers \(TR_{t}\) and through the total tax liability function \(T_{s,t}\), which can be decomposed into the effective tax rate times total income EqTaxCalcLiabETR2. In this chapter, we detail the household tax component of government activity \(T_{s,t}\) in OG-ZAF.
    
+
    The government sector influences households through two terms in the household budget constraint (14)—government transfers \(TR_{t}\) and through the total tax liability function \(T_{s,t}\), which can be decomposed into the effective tax rate times total income. In this chapter, we detail the household tax component of government activity \(T_{s,t}\) in OG-ZAF.
    
 
    
-(12)#\[\begin{split}  c_{j,s,t} + b_{j,s+1,t+1} &= (1 + r_{hh,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \\
+(14)#\[\begin{split}  c_{j,s,t} + b_{j,s+1,t+1} &= (1 + r_{hh,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \\
   &\quad\quad\zeta_{j,s}\frac{BQ_t}{\lambda_j\omega_{s,t}} + \eta_{j,s,t}\frac{TR_{t}}{\lambda_j\omega_{s,t}} + ubi_{j,s,t} - T_{s,t}  \\
   &\quad\forall j,t\quad\text{and}\quad s\geq E+1 \quad\text{where}\quad b_{j,E+1,t}=0\quad\forall j,t\end{split}\]
    
 
    The total tax function, \(T_{s,t}\), is a function of personal income taxes, taxes on bequests, and wealth taxes.  In the default calibration, wealth and bequest taxes are set to zero in OG-ZAF. Personal income taxes are modeled as linear taxes and set to average effective and marginal tax rates.  The OG-Core documentation details more detailed ways to match the progressivity of the tax system.  But given limited data for South Africa, we start with simple linear tax rates of 22% for effective tax rates on personal income, a 25% marginal tax rate on capital income, and a 31% marginal tax rate on labor income.
    
diff --git a/content/citations.html b/content/citations.html
index 2618db0..d1265d8 100644
--- a/content/citations.html
+++ b/content/citations.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/contributing/contributor_guide.html b/content/contributing/contributor_guide.html
index e383021..34214d2 100644
--- a/content/contributing/contributor_guide.html
+++ b/content/contributing/contributor_guide.html
@@ -34,7 +34,7 @@
     
     
     
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diff --git a/content/intro/intro.html b/content/intro/intro.html
index 081a508..31ad3c2 100644
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diff --git a/genindex.html b/genindex.html
index aade11f..4da7e91 100644
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diff --git a/py-modindex.html b/py-modindex.html
index b80e706..fd0f694 100644
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diff --git a/search.html b/search.html
index c6dfca7..0a61f60 100644
--- a/search.html
+++ b/search.html
@@ -32,7 +32,7 @@
     
     
     
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diff --git a/searchindex.js b/searchindex.js
index 59dcf3e..4fb9115 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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\ No newline at end of file
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