From 44965d735c0d280279e5955a17cd2401cedd308d Mon Sep 17 00:00:00 2001 From: jdebacker Date: Tue, 3 Sep 2024 20:23:57 -0400 Subject: [PATCH 1/6] update firm calib --- docs/book/content/calibration/firms.md | 25 +++++++++++++++++++++++-- 1 file changed, 23 insertions(+), 2 deletions(-) diff --git a/docs/book/content/calibration/firms.md b/docs/book/content/calibration/firms.md index 61a23e4..2a05f2f 100644 --- a/docs/book/content/calibration/firms.md +++ b/docs/book/content/calibration/firms.md @@ -5,10 +5,31 @@ 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: + +: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 [Julius Pain, Mpho Rapapali and Daan Steenkamp, "Industry TFP estimates for South Africa", Chapter 5 in Occasional Bulletin of Economic Notes, OBEN/20/02, South African Reserve Bank, Nov. 2020](https://econpapers.repec.org/scripts/redir.pf?u=https%3A%2F%2Fwww.resbank.co.za%2Fcontent%2Fdam%2Fsarb%2Fpublications%2Foccasional-bulletin-of-economic-notes%2F2020%2F10412%2FOBEN%25202002%2520%28Industry%2520TFP%2520estimates%2520for%2520South%2520Africa%29%2520-%2520November%25202020.pdf;h=repec:rbz:oboens:10412) who identify TFP for various sectors in South Africa. \ No newline at end of file From 35aa02298470959bd98dca2de15bd1fe06771297 Mon Sep 17 00:00:00 2001 From: jdebacker Date: Thu, 5 Sep 2024 20:27:57 -0400 Subject: [PATCH 2/6] remove some specifics from the description of demographics --- docs/book/content/calibration/demographics.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/book/content/calibration/demographics.md b/docs/book/content/calibration/demographics.md index ce09f62..9f09f24 100644 --- a/docs/book/content/calibration/demographics.md +++ b/docs/book/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 --- From ce5da0d7314fc9f4892039cf2ab097f8caa891e2 Mon Sep 17 00:00:00 2001 From: jdebacker Date: Fri, 6 Sep 2024 09:18:42 -0400 Subject: [PATCH 3/6] describe r_gov params --- docs/book/OGZAF_references.bib | 15 +++++++++++++++ docs/book/content/calibration/macro.md | 13 +++++++++++++ 2 files changed, 28 insertions(+) diff --git a/docs/book/OGZAF_references.bib b/docs/book/OGZAF_references.bib index 5c1a5ea..91a992a 100755 --- a/docs/book/OGZAF_references.bib +++ b/docs/book/OGZAF_references.bib @@ -738,3 +738,18 @@ @ARTICLE{Zhang:1997 month = {December}, pages = {2187-2209}, } + +@Article{LMW2023, + author={Li, Delong and Magud, Nicolas E. and Werner, Alejandro}, + title={{The long-run impact of sovereign yields on corporate yields in emerging markets}}, + journal={Journal of International Money and Finance}, + year=2023, + volume={130}, + number={C}, + pages={}, + month={}, + keywords={Bonds; Emerging markets; Sovereign risk; Transfer risk; Liquidity premium; JEL Classification Number}, + doi={10.1016/j.jimonfin.2022.1}, + abstract={We analyze the long-run impact of sovereign yields on corporate yields of the same country, finding that, for emerging markets, the average pass-through is around one. The pass-through is larger in countries with greater sovereign risk and where sovereign bonds are more liquid. The pass-through is also greater for corporate bonds with lower ratings, shorter maturities, and those issued by financial companies and government-related firms. Our results support theoretical arguments that corporate and sovereign yields are linked together through credit risk and liquidity premiums. Consequently, high sovereign risk can slowdown growth by persistently increasing private sector borrowing costs.}, + url={https://ideas.repec.org/a/eee/jimfin/v130y2023ics0261560622001516.html} +} \ No newline at end of file diff --git a/docs/book/content/calibration/macro.md b/docs/book/content/calibration/macro.md index 64bb348..3898148 100644 --- a/docs/book/content/calibration/macro.md +++ b/docs/book/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). + From 320b25264aa2a35ddadf2cff83f56fcd988f0b9d Mon Sep 17 00:00:00 2001 From: jdebacker Date: Fri, 6 Sep 2024 10:45:29 -0400 Subject: [PATCH 4/6] update exog params --- .../calibration/exogenous_parameters.md | 103 +++++++++++++----- 1 file changed, 78 insertions(+), 25 deletions(-) diff --git a/docs/book/content/calibration/exogenous_parameters.md b/docs/book/content/calibration/exogenous_parameters.md index 2ec415d..b6163a7 100644 --- a/docs/book/content/calibration/exogenous_parameters.md +++ b/docs/book/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 | From 569404773ea89dd6bdc76d5715f7564f984eb011 Mon Sep 17 00:00:00 2001 From: jdebacker Date: Fri, 6 Sep 2024 10:49:13 -0400 Subject: [PATCH 5/6] fix firm eq --- docs/book/content/calibration/firms.md | 2 ++ docs/book/content/calibration/taxes.md | 2 +- 2 files changed, 3 insertions(+), 1 deletion(-) diff --git a/docs/book/content/calibration/firms.md b/docs/book/content/calibration/firms.md index 2a05f2f..a4ef222 100644 --- a/docs/book/content/calibration/firms.md +++ b/docs/book/content/calibration/firms.md @@ -7,12 +7,14 @@ The [OG-Core firm theory documentation](https://pslmodels.github.io/OG-Core/cont 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 $m$. diff --git a/docs/book/content/calibration/taxes.md b/docs/book/content/calibration/taxes.md index eab6ac0..1bdb2ec 100644 --- a/docs/book/content/calibration/taxes.md +++ b/docs/book/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 From d64b42ecd24320c4f09714eb459983203bb61b4e Mon Sep 17 00:00:00 2001 From: jdebacker Date: Sat, 7 Sep 2024 21:17:44 -0400 Subject: [PATCH 6/6] update tfp cite --- docs/book/OGZAF_references.bib | 15 +++++++++++++++ docs/book/content/calibration/firms.md | 2 +- 2 files changed, 16 insertions(+), 1 deletion(-) diff --git a/docs/book/OGZAF_references.bib b/docs/book/OGZAF_references.bib index 91a992a..3cb3dbf 100755 --- a/docs/book/OGZAF_references.bib +++ b/docs/book/OGZAF_references.bib @@ -752,4 +752,19 @@ @Article{LMW2023 doi={10.1016/j.jimonfin.2022.1}, abstract={We analyze the long-run impact of sovereign yields on corporate yields of the same country, finding that, for emerging markets, the average pass-through is around one. The pass-through is larger in countries with greater sovereign risk and where sovereign bonds are more liquid. The pass-through is also greater for corporate bonds with lower ratings, shorter maturities, and those issued by financial companies and government-related firms. Our results support theoretical arguments that corporate and sovereign yields are linked together through credit risk and liquidity premiums. Consequently, high sovereign risk can slowdown growth by persistently increasing private sector borrowing costs.}, url={https://ideas.repec.org/a/eee/jimfin/v130y2023ics0261560622001516.html} +} + +@Article{PRS2020, + author={Pain, Julius and Rapapali, Mpho and Steenkamp, Daan}, + title={{Industry TFP estimates for South Africa}}, + journal={Occasional Bulletin of Economic Notes}, + year=2020, + volume={}, + number={}, + pages={}, + month={November}, + keywords={}, + doi={}, + abstract={}, + url={https://econpapers.repec.org/scripts/redir.pf?u=https%3A%2F%2Fwww.resbank.co.za%2Fcontent%2Fdam%2Fsarb%2Fpublications%2Foccasional-bulletin-of-economic-notes%2F2020%2F10412%2FOBEN%25202002%2520%28Industry%2520TFP%2520estimates%2520for%2520South%2520Africa%29%2520-%2520November%25202020.pdf;h=repec:rbz:oboens:10412} } \ No newline at end of file diff --git a/docs/book/content/calibration/firms.md b/docs/book/content/calibration/firms.md index a4ef222..8cc5f55 100644 --- a/docs/book/content/calibration/firms.md +++ b/docs/book/content/calibration/firms.md @@ -34,4 +34,4 @@ We use a default value of $\gamma =0.40$, which corresponds to one minus labor's ### 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 [Julius Pain, Mpho Rapapali and Daan Steenkamp, "Industry TFP estimates for South Africa", Chapter 5 in Occasional Bulletin of Economic Notes, OBEN/20/02, South African Reserve Bank, Nov. 2020](https://econpapers.repec.org/scripts/redir.pf?u=https%3A%2F%2Fwww.resbank.co.za%2Fcontent%2Fdam%2Fsarb%2Fpublications%2Foccasional-bulletin-of-economic-notes%2F2020%2F10412%2FOBEN%25202002%2520%28Industry%2520TFP%2520estimates%2520for%2520South%2520Africa%29%2520-%2520November%25202020.pdf;h=repec:rbz:oboens:10412) who identify TFP for various sectors in South Africa. \ 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