========================================== Notation: Paper Symbols to Code Variables ========================================== This page maps the mathematical symbols used in Pichler et al. (2022) to the variable names used in the Python implementation. The **Source** column indicates where each value lives at runtime: ``state`` (current-step ``self.state[...]``), ``prior`` (previous-step ``self.prior[...]``), ``params`` (scalar or tensor parameters in ``params[...]``), ``hyper`` (``params.hyper[...]``), ``scenario`` (per-step scenario tensor), or ``local`` (method-local variable). Time-Varying Variables ---------------------- Stored in ``self.state[...]`` and ``self.prior[...]``. Addressable via :py:attr:`macrostat.models.PichlerEtAl2022DIO.VariablesPichlerEtAl2022DIO`. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`x_{i,t}` - ``GrossOutput`` - state - Realised gross output of industry *i*. * - :math:`d_{i,t}` - ``AggregateDemand`` - state - Total demand for output of industry *i*. * - :math:`c^d_{i,t}` - ``ConsumptionDemand`` - state - Household consumption demand for good *i*. * - :math:`c_{i,t}` - ``RealizedConsumption`` - state - Realised household consumption of good *i*. * - :math:`l_{i,t}` - ``LabourCompensation`` - state - Labour compensation paid by industry *i*. * - :math:`\pi_{i,t}` - ``Profits`` - state - Profit of industry *i*. * - :math:`x^{\text{cap}}_{i,t}` - ``ProductiveCapacity`` - state - Labour-based productive capacity. * - :math:`x^{\text{inp}}_{i,t}` - ``InputCapacity`` - state - Input-based productive capacity. * - :math:`S_{ji,t}` - ``Inventories`` - state - Inventory of input *j* held by industry *i*. * - :math:`Z_{ji,t}` - ``IntermediateConsumption`` - state - Intermediate delivery of *j* received by *i*. * - :math:`O_{ji,t}` - ``IntermediateOrders`` - state - Intermediate order from *i* to *j*. * - :math:`\tilde{c}^d_t` - ``TotalConsumptionDemand`` - state - Household-aggregate consumption demand. * - :math:`s_t` - ``Savings`` - state - Household savings stock. Data Parameters --------------- Tensor-valued parameters initialised from external CSV files. Addressable via ``model.parameters.data["..."]["value"]``. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`A_{ji}` - ``TechnicalCoefficients`` - params - Technical coefficient: input *j* per unit output of *i*. * - :math:`A^{\text{ess}}_{ji}` - ``CriticalInputMatrix`` - params - Mask × technical coefficient for critical inputs only. * - :math:`Z^{(0)}_{ji}` - ``IntermediateConsumptionMatrix`` - params - Initial intermediate-consumption matrix. * - :math:`x_{i,0}` - ``InitialGrossOutput`` - params - Initial gross output by industry. * - :math:`l_{i,0}` - ``InitialLabourCompensation`` - params - Initial labour compensation by industry. * - :math:`c_{i,0}` - ``InitialHouseholdConsumption`` - params - Initial household consumption by good. * - :math:`\pi_{i,0}` - ``InitialProfits`` - params - Initial industry profits. * - :math:`f_{i,0}` - ``InitialOtherFinalDemand`` - params - Initial other final demand by good. * - :math:`n_i` - ``InventoryTargetDays`` - params - Days-of-cover inventory target. * - :math:`e_i / x_i` - ``OtherCostCoefficients`` - params - Other-cost share (taxes, imports) of output. * - :math:`c^{\text{other}}` - ``HouseholdOtherCostCoefficient`` - params - Household share spent on non-modelled goods. Scalar Parameters ----------------- Real-valued scalar parameters in ``model.parameters.values["..."]["value"]``. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`\tau` - ``InventoryAdjustmentSpeed`` - params - Speed of inventory adjustment toward target. * - :math:`\gamma_H` - ``HiringRate`` - params - Daily hiring rate. * - :math:`\gamma_F` - ``FiringRate`` - params - Daily firing rate. * - :math:`\rho` - ``ConsumptionPersistence`` - params - AR(1) persistence in aggregate consumption demand. * - :math:`m` - ``PropensityToConsume`` - params - Marginal propensity to consume out of labour income. * - :math:`b` - ``BenefitRate`` - params - Unemployment benefit replacement rate. * - :math:`\Delta s` - ``SavingsRedirection`` - params - Fraction of savings redirected to consumption. Hyperparameters --------------- Configuration values in ``model.parameters.hyper["..."]``. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`N` - ``n_sectors`` - hyper - Number of input-output sectors. * - :math:`T` - ``timesteps`` - hyper - Simulation horizon (days). * - — - ``production_function`` - hyper - One of ``leontief``, ``strongly_critical``, ``half_critical``, ``weakly_critical``, ``linear``. * - — - ``hiring_firing`` - hyper - Whether labour adjusts via hire/fire rates. * - — - ``firm_priority`` - hyper - Whether firms are rationed before households on tight supply. Scenario Tensors ---------------- Per-step exogenous paths in ``scenario["..."]``. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`f^d_{i,t}` - ``other_final_demand`` - scenario - Exogenous other final demand (govt + exports + investment). * - :math:`\epsilon^c_{i,t}` - ``consumption_shock`` - scenario - Multiplicative shock to consumption demand. * - :math:`\epsilon^x_{i,t}` - ``productivity_shock`` - scenario - Multiplicative shock to productive capacity. Auxiliary Sets -------------- Index sets derived from data parameters. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`\mathcal{V}_i` - ``critical_inputs[i]`` - local - Indices *j* with :math:`A^{\text{ess}}_{ji} > 0`. * - :math:`\mathcal{U}_i` - ``important_inputs[i]`` - local - Indices *j* with :math:`A_{ji} > 0` and :math:`A^{\text{ess}}_{ji} = 0`. Selected Local Variables ------------------------ Names used inside ``BehaviorPichlerEtAl2022DIO`` methods. .. list-table:: :header-rows: 1 :widths: 12 28 12 48 * - Paper symbol - Code variable - Source - Meaning * - :math:`\delta l_{i,t}` - ``delta_labour`` - local - Net hire/fire flow this step. * - :math:`\tilde{c}^d_t` - ``total_consumption_demand`` - local - Aggregate of ``ConsumptionDemand`` across goods at time *t*. * - :math:`\log \tilde{c}^d_{t-1}` - ``log_total_consumption_demand`` - local - Lagged log of aggregate consumption demand (AR(1) regressor). * - :math:`\sum_j Z_{ji,t}` - ``intermediate_consumption_total`` - local - Column sum of intermediate deliveries by industry. * - :math:`\sum_j A_{ji}` - ``col_sum_technical_coefficients`` - local - Column sum of the technical-coefficient matrix. * - :math:`\sum_j S_{ji,t}` - ``col_sum_inventory_matrix`` - local - Column sum of inventories.