Variables

The Macro model contains three types of variables:

  • The goal program variables are solved in each time step (equal to one year) using a linear goal program.
  • The dynamic parameters appear as parameters in the goal program, but are updated between runs. They are shown with an overline to make them easier to identify in the equations.
  • As the dynamic parameters are calculated, intermediate variables are introduced to simplify the equations and for reporting purposes.

In addition to model variables, there are exogenous parameters, some of which are optional. Exogenous parameters are shown with an underline to make them easier to identify in the equations. Some "exogenous" parameters, such as the target profit rate $\underline{r}^*$, are in fact calculated from a combination of initial values and exogenous parameters, but are then held fixed.

Variables and parameters may be labeled by sector, by product, or both:

SymbolDefinition
$n_s$Number of sectors
$n_p$Number of products
$i$, $j$Sector indices, taking values from $1\ldots n_s$
$k$, $l$Product indices, taking values from $1\ldots n_p$

Goal program variables

SymbolDefinition
$u_i$Capacity utilization in sector $i$
$\Delta u_i$Gap between full and realized utilization in sector $i$
$s_{X,k}$Exports relative to the normal level for product $k$
$\Delta s_{X,k}$Gap between normal and realized exports as a share for product $k$
$s_{F,k}$Final demand (household and government combined) relative to the normal level for product $k$
$\Delta s_{F,k}$Gap between normal and realized final demand as a share for product $k$
$\psi^\pm_k$Import excess $(+)$ or deficit $(-)$ relative to the normal level as a multiplier on the reference level for product $k$
$X_k$Exports of product $k$
$F_k$Final demand for product $k$
$I_k$Investment demand for product $k$
$M_k$Imports of product $k$
$q_{s,k}$Domestic production of product $k$
$q_{d,k}$Intermediate demand for product $k$
$m^\pm_k$Positive $(+)$ and negative $(-)$ margins for product $k$

Dynamic parameters

SymbolDefinition
$\overline{z}_i$Potential output for sector $i$
$\overline{p}_{b,k}$Basic price for product $k$ as an index relative to the initial year
$\overline{P}_g$Price level of output as an index relative to the initial year
$\overline{I}$Demand for investment goods
$\overline{X}^\text{norm}_k$Normal level of export demand for product $k$
$\overline{F}^\text{norm}_k$Normal level of final demand for product $k$
$\overline{M}^\text{ref}_k$Reference level of imports for product $k$
$\overline{f}_k$Normal level of imports of good $k$ as a fraction of domestic demand
$\overline{D}_{ki}$Intermediate demand for product $k$ per unit of output from sector $i$

Intermediate variables

SymbolDefinition
$g_i$Total output from sector $i$
$\hat{Y}$Real GDP growth rate
$p_{d,k}$Domestic price for product $k$ as an index relative to the initial year
$p_{w,k}$World price index for product $k$
$p_{x,k}$Export-weighted average of world and domestic prices for product $k$
$\pi_{b,k}$Basic price inflation rate for product $k$
$\pi_{d,k}$Domestic price inflation rate for product $k$
$\pi_F$Inflation rate for domestic final demand
$\pi_\text{GDP}$Inflation rate for the GDP price level
$W_i$Nominal wage bill in sector $i$
$\hat{L}$Growth rate of employment
$\hat{w}$Growth rate of the nominal wage
$\hat{\lambda}$Growth rate of labor productivity
$\omega_i$Wage share in sector $i$
$\gamma_i$Rate of net investment demand in sector $i$
$\gamma_{i0}$The autonomous (smoothed) component of the rate of net investment demand in sector $i$
$i_b$Central bank interest rate
$i_{b0}$Central bank interest rate when GDP growth and inflation are at their targets
$\Pi_i$Gross profits in sector $i$
$r_i$Gross rate of profit in sector $i$
$\gamma^\text{wage}$Growth rate of the total wage bill
$\gamma^\text{world}_\text{smooth}$Smoothed growth rate of world GDP (also termed gross world product, GWP)
$f_k$Current level of imports of good $k$ as a fraction of domestic demand

Exogenous parameters

SymbolDefinition
$\underline{w}_u$In the linear goal program, the category weight penalizing low utilization
$\underline{w}_F$In the linear goal program, the category weight penalizing departure from normal final demand
$\underline{w}_X$In the linear goal program, the category weight penalizing departure from normal exports
$\underline{w}_M$In the linear goal program, the category weight penalizing imports over domestic supply
$\underline{\sigma}^u_i$In the linear goal program, sector weight for utilization for sector $i$
$\underline{\sigma}^X_k$In the linear goal program, product weight for exports for product $k$
$\underline{\sigma}^F_k$In the linear goal program, product weight for final consumption demand for product $k$
$\underline{\varphi}_u$For utilization sector weights, weight of value share vs. constant share
$\underline{\varphi}_F$For final demand sector weights, weight of value share vs. constant share
$\underline{\varphi}_X$For export sector weights, weight of value share vs. constant share
$\underline{S}_{ik}$Sector $i$'s share of domestic production of product $k$
$\underline{D}^\text{init}_{ki}$Initial value for intermediate demand for product $k$ per unit of output from sector $i$
$\underline{d}_k$Equal to $\underline{d}_k = 1$ if the country does not produce product $k$ and $\underline{d}_k = 0$ if it does produce it
$\underline{\varepsilon}_i$Initial energy cost share for sector $i$ if energy excluded, otherwise zero
$\underline{e}$Exchange rate
$\underline{\pi}_{w,k}$Inflation rate for the world price of product $k$
$\underline{\mu}_i$Profit margin in sector $i$
$\underline{\alpha}^\text{KV}$, $\underline{\alpha}^\text{KV}_i$Kaldor-Verdoorn law coefficient, for the whole economy or by sector $i$
$\underline{\beta}^\text{KV}$, $\underline{\beta}^\text{KV}_i$Kaldor-Verdoorn law intercept, for the whole economy or by sector $i$
$\underline{L}_{i0}$Initial employment level for sector $i$, if labor productivity is specified by sector
$\underline{h}$Inflation pass-through to the nominal wage (wage indexation parameter)
$\underline{k}$Response of the real wage to labor supply constraints
$\underline{\hat{N}}$Growth rate of the working-age population
$\underline{\chi}^\pm_k$Allocation coefficients for positive $(+)$ and negative $(-)$ margins for product $k$
$\underline{i}^\text{init}_{b0}$Initial value for $i_{b0}$
$\underline{i}^\text{min}_{b0}, \underline{i}^\text{max}_{b0}$Minimum and maximum values for $i_{b0}$
$\underline{b}_\text{xr}$Sensitivity of $i_{b0}$ to changes in the exchange rate
$\underline{T}_\text{xr}$Adaptation time for $i_{b0}$ to adjust to the target
$\underline{\rho}_Y$Taylor coefficient on the GDP growth rate
$\underline{\rho}_\pi$Taylor coefficient on the inflation rate
$\hat{\underline{Y}}^*_\text{min}, \hat{\underline{Y}}^*_\text{max}$Minimum and maximum target growth rates for the Taylor rule
$\underline{\pi}^*$Taylor rule target inflation rate
$\underline{\pi}_d^\text{init}$Initial domestic price inflation rate
$\underline{\gamma}_0$Initial autonomous investment rate in the investment function
$\underline{\xi}$Rate of adjustment of autonomous demand to realized growth rate
$\underline{\alpha}_\text{util}$Response of induced investment to capacity utilization (utilization investment sensitivity)
$\underline{\alpha}_\text{profit}$Response of induced investment to profitability (profit rate investment sensitivity)
$\underline{\alpha}_\text{bank}$Response of induced investment to borrowing costs (interest rate investment sensitivity)
$\underline{\alpha}_\text{netx}$Response of induced investment to the ratio of net exports to GDP (due to perceived lender's risk)
$\underline{r}^*$Target rate of gross profit (in the investment function)
$\underline{\delta}_i$Depreciation rate for sector $i$
$\underline{v}_i$Capital-output ratio in sector $i$
$\underline{\theta}_k$Share of product $k$ in the total supply of investment goods
$\underline{\gamma}^\text{world}$Growth rate of world GDP (also termed gross world product, GWP)
$\underline{\eta}^\text{exp}_k$Elasticity of normal export demand for product $k$ with respect to a change in GWP
$\underline{\eta}^\text{wage}_k$Elasticity of normal final demand for product $k$ with respect to a change in the wage bill
$\underline{\phi}^\text{exp}_k, \underline{\phi}^\text{imp}_k$Elasticities of export and import demand with respect to relative price changes

Optional exogenous parameters

The following are optional exogenous parameters:

SymbolDefinition
$\underline{I}_\text{en}$Energy sector investment, retrieved from LEAP
$\underline{I}_\text{exog}$Other exogenous investment, not associated with a sector (default is 0.0)
$\underline{z}_i^\text{exog}$Exogenously specified potential output (default is that Macro calculates potential output)
$\underline{u}_i^\text{max}$In the linear goal program, the maximum capacity utilization (default is 1.0)
$\underline{a}_i$Rate constant for endogenous intermediate demand coefficients
$\underline{\vartheta}_i$Exponent for endogenous intermediate demand coefficients