Households

Overview

Each area's household sector is modeled as a single representative household with an exogenous time-varying number of household members, \(nn_t\). The household enters a net position in debt instruments (e.g. loans, deposits, fixed-income securities, etc.) with the local financial sector, \(bh_t\), and holds a portfolio of claims on production capital in all areas (including the local area), \(\tsum_a s_{a,t} \, ex_{a,t}\, pk^a_t \, k^a_t\); the latter is our way to mimic corporate equity holdings with cross-border exposures. During each period, the household purchases consumption goods, \(ch_t\), supplies per-worker hours worked, \(nh_t\), rents production capital, \(k^a_t\), out to producers in the respective area, chooses the utilization rate of local production capital, \(u_t\), invests in creating addition local capital, \(i_t\), pays lump-sum taxes (or receives lump-sum transfers) of two types, \(txls1_t\) and \(txls2_t\), and collects period profits from local producers, local exporters, and the local financial sectors (of whom all the household is the ultimate owner).

The household chooses the following quantities

total consumption, \(ch_t\),
per-capita hours worked, \(nh_t\),
shares of claims on production capital possibly from all areas, \(s_{a,t}\in[0,\,1]\), \(a\in A\),
the utilization rate of local production capital, \(u_t\),
investment in local production capital, \(i_t\),
net financial position with the local financial sector, \(bh_t\),

to maximize its infinite lifetime utility function subject to a dynamic budget constaint. The household derives utility from consumption, disutility from work, and utility from its wealth (net worth).

Algebra Model source code

Household preferences

The representative household preferences are described by a time-separable utility function over an infinite life horizon, \(t=0,\dots,\infty\). The period utility function consists of a consumption utility component, \(\bU^{ch}_t\), a work disutility component, \(\bU^{nh}_t\), and a current wealth (net worth) utility component, \(\bU^{netw}_t\). The individual utility function components are each evaluated on a per-capita basis, and the overall period utility is multiplied by the total number of household members

\[ \begin{equation} \E_0 \sum_{t=0}^{\infty} \beta^t \Bigl( \bU^{ch}_t - \bU^{nh}_t + \bU^{netw}_t \Bigr)\, nn_t \end{equation} \]

The respective components of the utility function related to consumption, work and wealth, respectively, are given as follows

\[ \newcommand{\Uch}{\kappa_{ch}\, \blog \frac{ch_t - \extern{ch}_t^\ref}{nn_t}} \bU^{ch}_t \equiv \Uch \]

\[ \newcommand{\Unh}{\frac{1}{1+\eta}\ nh_t{}^{1+\eta}} \bU^{nh}_t \equiv \Unh \]

\[ \newcommand{\Unetw}{\nu_1 \left( \blog \frac{netw_t}{pc_t\, \extern{ch}_t} - \nu_0 \frac{netw_t}{pc_t\, \extern{ch}_t}\right) } \bU^{netw}_t \equiv \Unetw \]

where

\(ch_t^\ref\) is the reference point in household consumption proportional to the level of real current labor income net of type 1 lump-sum taxes (or transfers) and externalized from the household optimization

\[ ch_t^\ref \equiv \chi_{curr} \, \frac{curr_t}{pc_t} + \chi_c \, c_{t-1} \]

\(\kappa_{ch} \equiv 1 - ch^\ref_\ss\, ch_\ss{}^{-1}\) is a steady-state correction constant ensuring that the marginal utility of consumption equals \(nn_t \, ch_\ss{}^{-1}\) in steady state, a feature of modeling convenience,
\(curr_t\) is current labor income net of Type 1 lump sum taxes (or transfers)

\[ curr_t \equiv w_t \, nh_t \, \xnf_t - tx1_t \]

\(netw_t\) is the nominal net worth given by the sum of the value of the production capital portfolio, the net financial position of the household to the local financial sector, \(bh_t\) (a positive balance means net claims of the financial sector on the household), and the net worth of the local financial sector (whose ultimate owner the household is), \(bb_t\),

\[ netw_t \equiv \tsum\nolimits_a s_{a,t} \, ex_{a,t} \, pk_{a,t} \, k_{a,t} - bh_t + bb_t \]

\(ex_{a,t}\) is the cross rate between local currency and area \(a\)'s currency (with movements up meaning depreciation of local currency)

\[ ex_{a,t} = \frac{e_{\local, t}}{e_{a,t}}, \quad ex_{\local,t}=1 \]

Dynamic budget constraint

The dynamic budget constraint facing the household sector describes a stock-flow relationship between the household assets and liabilities (stocks) on the one hand, and current receipts and current outlays (flows) on the other hand. The household assets and liabilities consist of

a net position with the local financial sector, \(-bh_t\) (a positive balance means net claims of the financial sector on the household),
claims on production capital (local and ccross-border capital), \(\sum\nolimits_a s_{a,t} \, ex_{a,t}\, pk_{t}^a \, k_{t}^a\), and

The change in the household assets and liabilities is equal to the revaluation of capital claims, and the total amount of current receipts and outlays:

revaluation of claims on production capital (both from the nominal exchange rate and the capital price), \(\sum\nolimits_a s_{a,t-1} \, \bDelta\!\left( ex_{a,t} \, pk_{t}^a \right) \, k_{t-1}^a\),
interest receipts or outlays on the net position with the local financial sector, \(\left( rh_{t-1} -1 \right) bh_{t-1}\)
current receipts from capital rentals net of capital utilization costs, \(\sum\nolimits_a s_{a,t} \,ex_{a,t}\, pu_{t}^a \, k_{t}^a - \Xi_{u,t}\),
current receipts from labor income, \(w_t \, nh_t \, nf_t\),
current receipts from selling newly installed capital, \(pk_t\,i_t\),
profits from loccal producers, \(\Pi_{y,t}\), exporters, \(\Pi_{x,t}\), labor unions, \(\Pi_{l,t}\), and the financial sector, \(\Pi_{b,t}\),
current outlays on consumption goods, \(-pc_t \, ch_t\),
current outlays on investment goods, \(-pi_t \, i_t\),

\[ \begin{gathered} \tsum\nolimits_a s_{a,t} \, ex_{a,t} \, pk_{t}^a \, k_{t}^a - bh_t \ \cdots \newl =\ \tsum\nolimits_a s_{a,t-1}\, ex_{a,t} \left(1-\delta^a\right) pk^a_{t}\, k^a_{t-1} - bh_{t-1} \ \cdots \newl + \ \tsum\nolimits_a s_{a,t}\, ex_{a,t} \, pu_{t}^a \, u^a_t k^a_{t} - \left(rh_{t-1} - 1\right) bh_{t-1} \ \cdots \newl + \ w^\mathrm{flex} \, nh_t \, nf_t + \left( w_t - w^\mathrm{flex} \right) \, \extern{nh_t} \, \extern{nf_t} \ \cdots \newl - \ pc_t \, ch_t + \left(pk_t - pi_t\right) i_t - tx1_t - tx2_t \ \cdots \newl + \ \bPi_{y,t} + \bPi_{x,t} + \bPi_{b,t} + \bPi_{l,t} - \Xi_{i,t} - \Xi_{k,t} - \Xi_{u,t} + \extern{\Xi}_{h,t} \end{gathered} \]

Lagrange multiplier associated with the budget constraint is denoted by \(vh_t\) (shadow value of nominal household wealth)

Labor market

The representative household delegates wage negotiation to a trade union agent
Each period, the household tells the trade union its target level for nominal wage, \(w^\mathrm{flex}\),
The union solves to following optimizing problem including a wage inflation adjustment cost:

\[ \min\nolimits_{\{w_t\}} \mathrm E_t \, \sum_{t=0}^\infty \beta^t\,vh_t \, \left[ (\log w_t - \log w^\mathrm{flex} )^2 + \tfrac{\xi_w}{2} \, (\log \roc{w}{}_t - \log \roc{w}{}_t^{\ref} )^2 \right] \]

where

\(\xi_w\) is an adjustment cost parameter
\(\roc{w}{}_t^{\ref} \equiv \roc{\extern{w}}_{t-1}\) is a reference rate of wage inflation representing the backward indexation present in wage negotiation

The resulting first-order condition (a wage setting equation) is given by

\[ \log w_t - \log w^\mathrm{flex} = \xi_w ( \log \roc{w}_t - \log {\roc{w}}{}^\ref_t ) - \xi_w \, \mathrm E_t \left[ \beta \tfrac{vh_{t+1}\,w_t}{vh_{t}\,w_{t+1}} ( \log \roc{w}_{t+1} - \log \roc{w}{}^\ref_t ) \right] \]

Short-term adjustment costs

The optimizing behavior of the representative household is subjected to two types of costly short-term adjustment processes:

an investment adjustment/installation cost
a capital utilization cost.

The investment adjustment/installation cost comprises two components: departures from a steady-state investment-to-capital ratio, and departures from a steady-state rate of change in investment

\[ \Xi_{i,t} \equiv \tfrac{1}{2} \, \xi_{i1} \, pi_t \, \extern{i}_t \, \Bigl( \blog i_t - \blog \extern{i}^\ref_t \Bigr)^2 + \tfrac{1}{2} \, \xi_{i2} \, pi_t \, \extern{i}_t \, \Bigl( \mathbf{\Delta log\ } i_t - \blog \kappa_{i} \Bigr)^2 \\[5pt] \]

where \(i^\ref_t\) is a point of reference derived from the steady-state investment-to-capital ratio applied to the stock of capital last period,

\[ i^\ref_t \equiv \frac{i_\ss}{k_\ss} \, k_{t-1} \, \roc{\imath}_\ss \]

and \(\kappa_{i} \equiv \roc{\imath}_\ss\) is a steady-state adjustment constant ensuring that the cost term disappears in steady-state.

The cost of capital utilization give rise to a cyclical response in the rate of utilization of the existing stock of capital. The cost function is given by

\[ \Xi_{u,t} \equiv s_{\local,t} \, py_t \, k_t \, \frac{\upsilon_0}{1+\upsilon_1} \, u_t{}^{1+\upsilon_1} \]

Capital accumulation

The household purchases investment goods, converts them to newly installed production capital (paying the adjustment/installation cost in the process) and adds these to the existing stock of capital

\[ k_t = (1-\delta)\ k_{t-1} + i_t \]

Lagrangian

The Lagrangian for the constrained optimization problem facing the representative household consists of the lifetime utility function and a sequence of dynamic budget constraints for each time from now until infinity, \(t=0, \dots, \infty\). Note that we use \(w^{flex}_t\) in place of \(w_t\) in the Lagrangian.

\[ \begin{gathered} \bmax_{\{ch_t, bh_t, s_{a,t}, i_{t}, nh_t, u_t \}} \newl \sum\nolimits_{t} \beta^t \Bigl[ \Uch + \Unh + \Unetw \Bigr] \, nn_t \ \cdots \end{gathered} \]

\[ \begin{multline} +\ \sum\nolimits_t \beta^t vh_t\, \Bigl\{ - \tsum\nolimits_{a} s_{a,t} \, ex_{a,t} \, pk_{t}^a \, k_{t}^a + bh_t \Bigr. \ \cdots \newl +\ \tsum\nolimits_a s_{a,t-1}\, ex_{a,t} \left(1-\delta^a\right) pk^a_{t}\, k^a_{t-1} - bh_{t-1} \ \cdots \newl + \ \left(\betak_{t-1}{}\right)^t \tsum\nolimits_a s_{a,t}\, ex_{a,t} \, pu_{t}^a \, u^a_t k^a_{t} - \left(rh_{t-1} - 1\right) bh_{t-1} \ \cdots \newl + \ w^{flex}_t \, nh_t \, \xnf_t - pc_t \, ch_t + \left(pk_t - pi_t\right) i_t - tx1_t - tx2_t . \ \cdots \newl + \bPi_{y,t} + \bPi_{x,t} + \bPi_{b,t} + \bPi_{l,t} -\ \Xi_{i,t} - \Xi_{k,t} - \Xi_{u,t} + \extern{\Xi}_{h,t} \Bigr. \Bigr\} \end{multline} \]

where \(vh_t\) is the Lagrange multiplier on time-\(t\) budget constraint, and \(\betak\) is the additional discount factor applied to the value of corporate equity holdings to compensate for the risk aversion of households.

Optimality conditions

The optimal (utility maximizing) choices of the representative household are described by the following first-order conditions.

Consumption, \(ch_t\)

\[ vh_t \, pc_t = \kappa_{ch}\, \frac{1}{ch_t - \extern{ch}^\ref_t} \, nn_t \]

Per-worker hours worked depending on the hypothetical flexible wage rate, \(w^{flex}_t\)

\[ vh_t \, w^{flex}_t = nh_t{}^\eta \]

Net position with the financial sector, \(bh_t\) (an intertemporal no-arbitrage condition)

\[ vh_t = \beta \, vh_{t+1} \, rh_t + \nu_1 \frac{1}{pc_t \, \extern{ch}_t} \left( \frac{pc_t \, \extern{ch}_t}{netw_t} - \nu_0 \right) \]

Utilization rate of production capital, \(u_t\)

\[ \upsilon_0 \, u_t{}^{\upsilon_1} \, py_t = pu_t \]

Investment in local production capital, \(i_t\)

\[ pk_t = pi_t \Bigl[ 1 + \xi_{i1} \bigl( \blog i_t - \blog \extern{i}^\ref_t \bigr) + \xi_{i2} \bigl(\bDelta \blog i_t - \kappa_i\bigr) - \xi_{i2} \, \beta\, zk_t \, \bigl(\bDelta \blog i_{t+1} - \kappa_i\bigr) \Bigr] \]

Claims on area \(a\)'s production capital, \(s_{a,t}\), \(\forall a \in A\)

\[ \beta \, vh_{t+1} \, rh_t \, pk^a_t \, ex_t^a = vh_t \, pu^a_t \, ex_t^a \, u^a_t + \beta \, \betak_{t} \, vh_{t+1} \left(1-\delta^a\right) pk^a_{t+1}\, ex_{t+1}^a \]

The last set of equations defines arbitrage-free conditions (AFCs) for a corporate equity portfolio choice. We need to further address the following two characteristics of these NACs:

As is common in macro models, the AFCs themselves do not determine the actual portfolio shares, \(s_{a,t}\), only the relationship between the price of production capital, the cash flows it generates, and the houseshold discount factor. The actual shares are then calibrated and kept fixed in the baseline version of the model.
Since we allow for cross-border holdings, each area's capital is subject to muliple AFCs, each relating to the household residing in a different area and exhibiting, in general, different preferences. We therefore create aggregate AFCs by taking the weighted average with the weights equal the portfolio shares. The aggregate AFCs for the capital markets are described in the Global equilibrium section.

GEES Households module

Define quantities

!variables(:households)

    "Private consumption" ch
    "Per-capita private consumption" ch_to_nn
    "Private consumption reference level" ref_ch_to_ch
    "Real discount factor" rdf
    "Per-worker labor supply" nh
    "Auxiliary calibration variable" ss_nh_to_ref_nh
    "Private investment" ih
    "Uncertainty in capital" zk
    "Uncertainty in profits" zy
    "Ex-post return on capital" rk
    "Production capital" k
    "Production capital services" uk
    "Portfolio of claims on production capital" kk
    "Household net worth" netw
    "Nominal wage rate" w
    "Real labor income" rli
    "Inflation expectations" E_roc_pc

    "Price of production capital" pk
    "Price of production capital services" pu
    "Real consumer wage rate" w_to_pc
    "Target nominal wage rate" ww

    "Nominal wage rate, Y/Y" roc_w
    "Household investment, Y/Y" roc_ih
    "Production capital, Y/Y" roc_k


!log-variables !all-but

    ref_ch_to_ch
    ss_nh_to_ref_nh


!parameters(:households :steady)

    "S/S Uncertainty discount factor on capital !! $\mathit{zk}_\mathrm{ss}$" ss_zk
    "S/S Uncertainty discount factor on production cash flows !! $\mathit{zy}_\mathrm{ss}$" ss_zy
    "Household discount factor !! $\beta$" beta
    "Depreciation of production capital !! $\delta$" delta
    "Inverse short-run elasticity of labor supply !! $\eta$" eta_short
    "Inverse long-run elasticity of labor supply !! $\eta$" eta_long
    "Utility location parameter of labor supply !! $\eta_0$" eta0
    "Reference level of long-run per-worker labor supply !! $\mathit{nh}_\mathrm{ref}$" ref_nh
    "Intercept in current wealth utility !! $\nu_0$" nu_0
    "Slope of current wealth utility !! $\nu_1$" nu_1
    "Level parameter in cost of utilization of production capital !! $\upsilon_0$" upsilon_0
    "Inverse elasticity of cost of utilization of production capital !! $\upsilon_1$" upsilon_1


!parameters(:households :dynamic)

    "A/R in uncertainty discount factor on capital !! $\rho_\mathit{zk}$" rho_zk
    "A/R in uncertainty discount factor on production cash flows !! $\rho_\mathit{zy}$" rho_zy
    "Point of reference in consuptions switch !! $\chi$" chi
    "Past consumption in reference consumption parameter !! $\chi_\mathit{ch}$" chi_ch
    "Current income in reference consumption parameter !! $\chi_\mathit{curr}$" chi_curr
    "Type 1 investment adjustment cost parameter !! $\xi_\mathit{ih,1}$" xi_ih1
    "Type 2 investment adjustment cost parameter !! $\xi_\mathit{ih,2}$" xi_ih2
    "Pressure relief valve for interest rate lower bound !! $\theta_\mathir{rh}$" theta_rh
    "Adjustment cost in wage setting" xi_w
    "A/R in long-run labor supply" rho_nh


!shocks(:households)

    "Shock to intertemporal preferences" shk_beta
    "Shock to private consumption" shk_ch
    "Shock to private investment" shk_ih
    "Shock to wage rate" shk_w
    "Shock to current income" shk_curr
    "Shock to uncertainty discount factor on capital" shk_zk
    "Shock to uncertainty discount factor on production cash flows" shk_zy

Define substitutions

!substitutions

    % Current income definition
    curr := (w*nh*nf - txls2_to_nc*pc*ch);

    % Investment adjustment marginal cost
    ref_ih := (&ih/&k*&roc_k)*k{-1};
    adj_ih1 := log(ih) - log($ref_ih$);
    adj_ih2 := log(ih/ih{-1}/&roc_ih) - beta*log(ih{+1}/ih/&roc_ih);

Define equations

!equations(:households)

%% Household consumption-saving choice

    "Optimal choice of household consumption"
    vh*pc*(ch - chi*ref_ch_to_ch*ch)*exp(-shk_ch) = nn*(1 - chi*&ref_ch_to_ch) ...
    !! vh*pc*ch = nn;

    "Point of reference in household consumption"
    ref_ch_to_ch = [ chi_curr*($curr$)/pc + chi_ch*ch{-1}*gg_ss_roc_a ] / ch;

    "Optimal choice of net position with local financial sector"
    vh = beta*vh{+1}*rh + nn/(pc*ch)*[ nu_1*(pc*ch/netw - nu_0) - gg_nu ] * exp(shk_beta) ...
    !! 1 = beta*rh/gg_ss_roc_a/ss_roc_pc + nu_1*nch_to_netw_minus_nu_0 - gg_nu;

    "Auxiliary equation for steady-state calibration of nu_0"
    nch_to_netw_minus_nu_0 = pc * ch / netw - nu_0;

    rdf = beta*vh{+1}*pc{+1}/(vh*pc);

    "Risk adjusted household rate"
    rh = r + theta_rh*(unc_r - r) + zh;

    "Current net worth of households"
    netw = kk + dg_to_ngdp*ngdp + nfa_to_ngdp*ngdp;

    "Uncertainty discount factor on capital"
    log(zk) = rho_zk*log(zk{-1}) + (1-rho_zk)*log(ss_zk) + shk_zk ...
    !! zk = ss_zk;

    "Uncertainty discount factor on production cash flows"
    log(zy) = rho_zy*log(zy{-1}) + (1-rho_zy)*log(ss_zy) + shk_zy ...
    !! zy = ss_zy;

    "Real labor income"
    rli = w*nh*nf / pc;


%% Labor supply

    "Short-run and long-run labor supply"
    vh * ww = eta0 * (nh/&nh)^eta_short * (ss_nh_to_ref_nh)^eta_long ...
    !! vh * ww = eta0 *  (ss_nh_to_ref_nh)^eta_long;

    "Auxiliary calibration equation for long-run labor supply"
    ss_nh_to_ref_nh = ...
        + rho_nh * ss_nh_to_ref_nh{-1} ...
        + (1 - rho_nh) * &ss_nh_to_ref_nh ...
    !! ss_nh_to_ref_nh = &nh / ref_nh;

    "Real consumer wage rate"
    w_to_pc = w / pc;

    "Wage rigidities"
    log(ww) - log(w) = ...
        + xi_w*(log(roc_w) - log(roc_w{-1})) ...
        - beta/&roc_w*xi_w*(log(roc_w{+1}) - log(roc_w)) ...
        + shk_w ...
    !! w = ww;

%     log(w_to_pc) = ...
%         + 0.5 * [log(w_to_pc{-1}) + log(&roc_w/&roc_pc)] ...
%         + (1-0.5) * log(ww/pc) ...

%% Supply of production capital

    "Optimal choice of investment in local production capital"
    pk * exp(shk_ih) = pih*[ 1 + xi_ih1*($adj_ih1$) + xi_ih2*($adj_ih2$) ] ...
    !! pk = pih;
    %     % Investment adjustment marginal cost
    % ref_ih := (&ih/&k*&roc_k)*k{-1};
    % adj_ih1 := log(ih) - log($ref_ih$);
    % adj_ih2 := log(ih/ih{-1}/&roc_ih) - beta*log(ih{+1}/ih/&roc_ih);

    "Accumulation of production capital"
    k = (1-delta)*k{-1} + ih;

    "Definition of utilized production capital"
    uk = u * k;

    "Optimal choice of utilization rate of production capital"
    py * upsilon_0 * u^upsilon_1 = pu;

    "Auxiliary equation for steady-state calibration of upsilon_0"
    upsilon_1_py_to_pu = upsilon_0 * py / pu;

    "Ex-post return on capital"
    rk*pk{-1} = u{-1}*pu{-1}*rh{-1} + (1-delta)*pk;


%% Rates of change

    !for w, ih, k !do
        roc_? = ? / ?{-1};
    !end

%% One-period-ahead expectations

    E_roc_pc = roc_pc{+1};


## Postprocessing equations outside model


```matlab

!postprocessor(:households)

    ne = nh * nf;
    rrh = rh / E_roc_pc;