Linear goal program
For the list of variables, their symbols, and their definitions, see the Variables page.
The Macro model solves, in each year, a linear goal program (LGP) that seeks the following, potentially competing, objectives:
- Full utilization
- Meeting or exceeding a "normal" level of final demand
- Meeting or exceeding normal export demand
- Minimizing the gap between normal and realized imports
As with any goal program, the individual objectives are expressed in terms of a gap between a target level and the modeled level. The objective function is a weighted sum of the separate objectives:
\[\min \underline{w}_u\sum_{i = 1}^{n_s} \underline{\sigma}^u_i\Delta u_i + \underline{w}_X\sum_{k = 1}^{n_p} \underline{\sigma}^K_k\Delta s_{X,k} + \underline{w}_F\sum_{k = 1}^{n_p} \underline{\sigma}^F_k\Delta s_{F,k} + \underline{w}_M\sum_{k = 1}^{n_p} \left(\psi^+_k + \psi^-_k\right).\]
All goal variables are scaled so that they take values between zero and one.
In the objective function, there are four "category weights", $\underline{w}_u$, $\underline{w}_X$, $\underline{w}_F$, and $\underline{w}_M$. The category weights are specified in the configuration file.
In addition, there are sector or product-specific weights for for utilization, exports, and final demand, but not for imports. The reason is that while weights are needed to capture the relative importance of certain goods in output, the export basket, and household final demand, import flexibility simply allows for demand to be met when domestic production is insufficient, a consideration that does not privilege one product over another.
The sector or product weights are given as weighted average of two possible weighting schemes: the base-year shares of the different sectors or products in output (for utilization), exports, or final demand, or equal weights. The allocations between the two possible weighting schemes are specified by the product and sector weight parameters $\underline{\varphi}_u$, $\underline{\varphi}_X$, and $\underline{\varphi}_F$, which are also set in the configuration file.
When the report-diagnostics parameter is set to true in the configuration file, Macro will export the structure of the LGP for each year to the diagnostics folder in a set of files with names like model_0_yyyy.txt. The output looks something like this:
Min 1.7333333333333334 ugap[1] + 0.5333333333333333 ugap[2] + 0.6000000000000001 ugap[3] + ...
Subject to
eq_util[1] : ugap[1] + u[1] == 1.0
eq_util[2] : ugap[2] + u[2] == 1.0
eq_util[3] : ugap[3] + u[3] == 1.0
eq_util[4] : ugap[4] + u[4] == 1.0
...Each equation in the file is labeled, for example by eq_util. For reference, the label is provided for each equation listed below, along with a motivation and a mathematical expression.
Domestic goods market equilibrium
eq_totsupply: Equilibrium in the domestic goods market is met when domestic supply equals total demand (the sum of final demand and demand for intermediate goods, exports, and investment goods), net of imports, and corrected by margins. This is enforced by the condition
\[q_{s,k} = q_{d,k} - m^+_k + m^-_k + X_k + F_k + I_k - M_k.\]
eq_no_dom_prod: If the country does not produce a particular product $k$, then $q_{s,k}$ must equal zero. This is enforced by the condition
\[q_{s,k}\underline{d}_k = 0.\]
Supply of intermediate goods
eq_intdmd: Intermediate goods are required for production and are given by technical coefficients (calculated from the use table and possibly updated dynamically). Intermediate demand is determined by
\[q_{d,k} - \sum_{i = 1}^{n_s} \overline{D}_{ki}\overline{z}_i u_i = 0.\]
Capacity utilization
eq_io: A core equation determines capacity utilization. It states that the value of domestic supply equals the value of domestic production:
\[\sum_{k=1}^{n_p} \underline{S}_{ik} \overline{p}_{b,k}q_{s,k} - \overline{P}_g\overline{z}_i u_i = 0.\]
eq_util: Deviations in utilization for sector $i$ below the target value of $u_i = \underline{u}_i^\text{max}$ are penalized in the goal program. (Note that $\underline{u}_i^\text{max} = 1$ unless it is set exogenously. See optional exogenous parameters and the format for the maximum capacity utilization file.) The deviation variable is calculated by imposing the condition
\[u_i + \Delta u_i = \underline{u}_i^\text{max},\quad 0 \leq u_i,\Delta u_i \leq 1.\]
Investment goods allocation
eq_inv_supply: Total investment demand $\overline{I}$ is supplied by a variety of sectors with shares $\theta_k$. This is enforced by the condition
\[I_k = \underline{\theta}_k \overline{I}.\]
Final demand
eq_F: Final demand is expressed as a multiple of the normal level,
\[F_k = s_{F,k} \overline{F}^\text{norm}_k.\]
eq_fshare: The multiplier satsifies
\[s_{F,k} + \Delta s_{F,k} = 1, \quad 0\leq s_{F,k}, \Delta s_{F,k} \leq 1.\]
Trade
eq_X: As with final demand, exports are expressed as a multiple of the normal level,
\[X_k = s_{X,k} \overline{X}^\text{norm}_k.\]
eq_xshare: The multiplier satsifies
\[s_{X,k} + \Delta s_{X,k} = 1, \quad 0\leq s_{X,k}, \Delta s_{X,k} \leq 1.\]
eq_M: Imports are given by a multiplier applied to total domestic demand, plus a possible deviation:
\[M_k = \overline{f}_k\left(q_{d,k} + F_k + I_k\right) + \left(\psi^+_k - \psi^-_k\right) \overline{M}^\text{ref}_k.\]
Margins
eq_margpos: Positive margins are a fraction of total supply (domestic plus imports),
\[m^+_k = \underline{\chi}^+_k \left(q_{s,k} + M_k\right).\]
eq_margneg: Margins are allocated across products that correspond to the supply of those goods,
\[m^-_k = \underline{\chi}^-_k \sum_{l = 1}^{n_p} m^+_l.\]