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 macrostat.models.PichlerEtAl2022DIO.VariablesPichlerEtAl2022DIO.

Paper symbol	Code variable	Source	Meaning
\(x_{i,t}\)	`GrossOutput`	state	Realised gross output of industry i.
\(d_{i,t}\)	`AggregateDemand`	state	Total demand for output of industry i.
\(c^d_{i,t}\)	`ConsumptionDemand`	state	Household consumption demand for good i.
\(c_{i,t}\)	`RealizedConsumption`	state	Realised household consumption of good i.
\(l_{i,t}\)	`LabourCompensation`	state	Labour compensation paid by industry i.
\(\pi_{i,t}\)	`Profits`	state	Profit of industry i.
\(x^{\text{cap}}_{i,t}\)	`ProductiveCapacity`	state	Labour-based productive capacity.
\(x^{\text{inp}}_{i,t}\)	`InputCapacity`	state	Input-based productive capacity.
\(S_{ji,t}\)	`Inventories`	state	Inventory of input j held by industry i.
\(Z_{ji,t}\)	`IntermediateConsumption`	state	Intermediate delivery of j received by i.
\(O_{ji,t}\)	`IntermediateOrders`	state	Intermediate order from i to j.
\(\tilde{c}^d_t\)	`TotalConsumptionDemand`	state	Household-aggregate consumption demand.
\(s_t\)	`Savings`	state	Household savings stock.

Data Parameters#

Tensor-valued parameters initialised from external CSV files. Addressable via model.parameters.data["..."]["value"].

Paper symbol	Code variable	Source	Meaning
\(A_{ji}\)	`TechnicalCoefficients`	params	Technical coefficient: input j per unit output of i.
\(A^{\text{ess}}_{ji}\)	`CriticalInputMatrix`	params	Mask × technical coefficient for critical inputs only.
\(Z^{(0)}_{ji}\)	`IntermediateConsumptionMatrix`	params	Initial intermediate-consumption matrix.
\(x_{i,0}\)	`InitialGrossOutput`	params	Initial gross output by industry.
\(l_{i,0}\)	`InitialLabourCompensation`	params	Initial labour compensation by industry.
\(c_{i,0}\)	`InitialHouseholdConsumption`	params	Initial household consumption by good.
\(\pi_{i,0}\)	`InitialProfits`	params	Initial industry profits.
\(f_{i,0}\)	`InitialOtherFinalDemand`	params	Initial other final demand by good.
\(n_i\)	`InventoryTargetDays`	params	Days-of-cover inventory target.
\(e_i / x_i\)	`OtherCostCoefficients`	params	Other-cost share (taxes, imports) of output.
\(c^{\text{other}}\)	`HouseholdOtherCostCoefficient`	params	Household share spent on non-modelled goods.

Scalar Parameters#

Real-valued scalar parameters in model.parameters.values["..."]["value"].

Paper symbol	Code variable	Source	Meaning
\(\tau\)	`InventoryAdjustmentSpeed`	params	Speed of inventory adjustment toward target.
\(\gamma_H\)	`HiringRate`	params	Daily hiring rate.
\(\gamma_F\)	`FiringRate`	params	Daily firing rate.
\(\rho\)	`ConsumptionPersistence`	params	AR(1) persistence in aggregate consumption demand.
\(m\)	`PropensityToConsume`	params	Marginal propensity to consume out of labour income.
\(b\)	`BenefitRate`	params	Unemployment benefit replacement rate.
\(\Delta s\)	`SavingsRedirection`	params	Fraction of savings redirected to consumption.

Hyperparameters#

Configuration values in model.parameters.hyper["..."].

Paper symbol	Code variable	Source	Meaning
\(N\)	`n_sectors`	hyper	Number of input-output sectors.
\(T\)	`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["..."].

Paper symbol	Code variable	Source	Meaning
\(f^d_{i,t}\)	`other_final_demand`	scenario	Exogenous other final demand (govt + exports + investment).
\(\epsilon^c_{i,t}\)	`consumption_shock`	scenario	Multiplicative shock to consumption demand.
\(\epsilon^x_{i,t}\)	`productivity_shock`	scenario	Multiplicative shock to productive capacity.

Auxiliary Sets#

Index sets derived from data parameters.

Paper symbol	Code variable	Source	Meaning
\(\mathcal{V}_i\)	`critical_inputs[i]`	local	Indices j with \(A^{\text{ess}}_{ji} > 0\).
\(\mathcal{U}_i\)	`important_inputs[i]`	local	Indices j with \(A_{ji} > 0\) and \(A^{\text{ess}}_{ji} = 0\).

Selected Local Variables#

Names used inside BehaviorPichlerEtAl2022DIO methods.

Paper symbol	Code variable	Source	Meaning
\(\delta l_{i,t}\)	`delta_labour`	local	Net hire/fire flow this step.
\(\tilde{c}^d_t\)	`total_consumption_demand`	local	Aggregate of `ConsumptionDemand` across goods at time t.
\(\log \tilde{c}^d_{t-1}\)	`log_total_consumption_demand`	local	Lagged log of aggregate consumption demand (AR(1) regressor).
\(\sum_j Z_{ji,t}\)	`intermediate_consumption_total`	local	Column sum of intermediate deliveries by industry.
\(\sum_j A_{ji}\)	`col_sum_technical_coefficients`	local	Column sum of the technical-coefficient matrix.
\(\sum_j S_{ji,t}\)	`col_sum_inventory_matrix`	local	Column sum of inventories.