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  Demography

Overview

The demography module determines the following population quantities; all of them are considered to be simple exogenous processes not affected by the rest of the model:

  • Global population trend

  • Area's total population

  • Area's working age population

  • Area's labor force (labor participation)

Endogenously within the model, we then model per-worker labor supply (e.g. per-worker hours).

Total Population

$$ \newcommand{\tsum}{\textstyle\sum} \newcommand{\extern}[1]{\mathrm{\mathbf{{#1}}}} \newcommand{\local}{\mathrm{local}} \newcommand{\roc}[1]{\overset{\scriptsize\Delta}{#1{}}} \newcommand{\ss}{\mathrm{ss}} \newcommand{\aa}{\mathrm{aa}} \newcommand{\bb}{\mathrm{bb}} \newcommand{\E}{\mathrm{E}} \newcommand{\ref}{{\mathrm{ref}}} \newcommand{\blog}{\mathbf{log}\ } \newcommand{\bmax}{\mathbf{max}\ } \newcommand{\bDelta}{\mathbf{\Delta}} \newcommand{\bPi}{\mathbf{\Pi}} \newcommand{\bU}{\mathbf{U}} \newcommand{\newl}{\\[8pt]} \newcommand{\betak}{\mathit{zk}} \newcommand{\betay}{\mathit{zy}} \newcommand{\gg}{\mathrm{gg}} \newcommand{\tsum}{\textstyle{\sum}} \newcommand{\xnf}{\mathit{nf}} \newcommand{\ratio}[2]{\Bigl[\textstyle{\frac{#1}{#2}}\Bigr]} \newcommand{\unc}{\mathrm{unc}} \notag $$

![[math]]

Global population trend, \(nn_t\), is a unit root process common to all areas. The level of the global population trend does not correspond to any particular demographic indicator; rather, we can think of \(nn\) as a driving force for population growth  

\[ \begin{gathered} \log \roc{nn}{}^\gg_t = \rho_{nn}^\gg\, \log \roc{nn}{}^\gg_{t-1} + (1-\rho_{nn}^\gg)\, \log \roc{nn}{}^\gg_\ss \\[10pt] \end{gathered} \]


Area's total population

\[ nn_t = nr_t \cdot nn_t^\gg \]
\[ \log(nr_t) = \rho_{nr} \log nr_{t-1} + (1-\rho_{nr}) \log nr_\ss \]

Labor Market Population

Area's working age population

\[ \frac{nw_t}{nn_t} = \rho_{nw} \, \frac{nw_{t-1}}{nn_{t-1}} + (1-\rho_{nw}) \, \ratio{nw}{nn}_\ss + \epsilon_{nw,t} \]


Area's labor force (participation)

\[ \frac{\xnf_t}{nw_t} = \rho_{\xnf} \, \frac{\xnf_{t-1}}{nw_{t-1}} + (1-\rho_{\xnf}) \, \ratio{\xnf}{nw}_\ss + \epsilon_{\xnf,t} \]

GEES Demography module

Declare quantities

!variables

    "Total population" nn
    "Total population, Y/Y" roc_nn
    "Area specific component in total population" nr
    "Working age population" nw
    "Labor force" nf


!log-variables !all-but


!parameters(:demography :steady)

    "S/S Population, Relative to global population component !! $\mathit{nr}_\ss$" ss_nr
    "S/S Share of working age population in total population !! $\tratio{\mathit{nw}}{\mathit{nn}}_\ss$" ss_nw_to_nn
    "S/S Labor participation rate !! $\tratio{\mathit{nf}}{\mathit{nw}}_\ss$" ss_nf_to_nw


!parameters(:demography :dynamic)

    "A/R Total population relative to global population component !! $\rho_\mathit{nr}$" rho_nr
    "A/R Share of working age population in total population !! $\rho_\mathit{nw}$" rho_nw
    "A/R Labor participation rate !! $\rho_\mathit{nf}$" rho_nf


!shocks

    "Shock to relative population component" shk_nr
    "Shock to share of working age population" shk_nw_to_nn
    "Shock to labor participation rate" shk_nf_to_nw

Define equations

!equations

    "Total population relative to global population component"
    log(nr) = rho_nr * log(nr{-1}) + (1-rho_nr) * log(ss_nr) + shk_nr ...
    !! nr = ss_nr;


    "Total population"
    nn = nr * gg_nt;


    "Total population, Y/Y"
    roc_nn = nn / nn{-1} ...
    !! roc_nn = gg_ss_roc_nt;


    "Share of working age population"
    nw/nn = ...
        + rho_nw * nw{-1}/nn{-1} ...
        + (1-rho_nw) * ss_nw_to_nn ...
        + shk_nw_to_nn ...
    !! nw = ss_nw_to_nn * nn;


    "Labor participation rate"
    nf/nw = ...
        + rho_nf * nf{-1}/nw{-1} ...
        + (1-rho_nf) * ss_nf_to_nw ...
        + shk_nf_to_nw ...
    !! nf = ss_nf_to_nw * nw;

Postprocessing equations outside model

!postprocessor(:demography)

    nw_to_nn = nw / nn;
    nf_to_nw = nf / nw;