Behavioral Equations PichlerEtAl2022DIO#

Step Equations#

  1. Accounting

Firm profits and household savings. Industry profits are total output minus intermediate purchases, labour compensation, and other expenses (taxes, imports). Household savings are computed as total income minus total realized consumption (including non-modeled import and tax expenditures). Replicates Eq. 2.

\begin{align} \pi_{i,t} = x_{i,t} - \sum_{j=1}^{N} Z_{ji,t} - l_{i,t} - e_{i,t} \end{align}

  1. Aggregate Demand

Total demand aggregation. Total demand for the output of industry i is the sum of intermediate orders from all other industries, household consumption demand, and exogenous other final demand (government, exports, investment). Replicates Eq. 3.

\begin{align} d_{i,t} = \sum_{j=1}^{N} O_{ij,t} + c^d_{i,t} + f^d_{i,t} \end{align}

  1. Compute Production

Production function and output-level choice. Realized output is the minimum of three constraints: labour capacity, input-based capacity (from inventories and the chosen production function), and demand. The input-based capacity depends on the production_function hyperparameter; five functional forms range from Leontief (all inputs binding) through partially binding Leontief variants to linear (perfect substitutes). The partially binding Leontief distinguishes critical, important, and non-critical inputs based on an industry analyst survey. Replicates Eqs. 8-14.

\begin{align} x_{i,t} &= \min\{x^{\text{cap}}_{i,t},\; x^{\text{inp}}_{i,t},\; d_{i,t}\} \\[6pt] x^{\text{inp}}_{i,t} &= \begin{cases} \displaystyle\min_{\{j:\,A_{ji}>0\}} \frac{S_{ji,t}}{A_{ji}} & \text{leontief} \\ \displaystyle\min_{j \in \mathcal{V}_i \cup \mathcal{U}_i} \frac{S_{ji,t}}{A_{ji}} & \text{strongly\_critical} \\ \displaystyle\min\!\left\{ \min_{j\in\mathcal{V}_i}\frac{S_{ji,t}}{A_{ji}},\; \tfrac{1}{2}\!\left( \min_{k\in\mathcal{U}_i}\frac{S_{ki,t}}{A_{ki}} + x^{\text{cap}}_{i,0} \right) \right\} & \text{half\_critical} \\ \displaystyle\min_{j \in \mathcal{V}_i} \frac{S_{ji,t}}{A_{ji}} & \text{weakly\_critical} \\ \displaystyle\sum_j S_{ji,t} \big/ \sum_j A_{ji} & \text{linear} \end{cases} \end{align}

  1. Consumption Demand

Muellbauer consumption function with fear-of-infection. Total household consumption demand follows an adapted version of Muellbauer (2020), combining persistence of past consumption with current and permanent labour income. A fear-of-infection factor \((1 - \tilde{\epsilon}^D_t)\) scales aggregate demand. Consumption is allocated across industries via time-varying preference coefficients. Replicates Eqs. 5-6.

\begin{align} \tilde{c}^d_t &= (1 - \tilde{\epsilon}^D_t)\, \exp\!\left( \rho \log \tilde{c}^d_{t-1} + \frac{1-\rho}{2}\log(m\tilde{l}_t) + \frac{1-\rho}{2}\log(m\tilde{l}^p_t) \right) \\ c^d_{i,t} &= \theta_{i,t}\,\tilde{c}^d_t \end{align}

  1. Hire Fire

Sluggish labour adjustment towards a target workforce. Firms adjust their labour force depending on which production constraint is binding. If capacity is binding, the firm tries to hire; if demand or input constraints bind, it fires. Adjustment is sluggish – firms can only move a fraction of the way toward their target each period. During lockdown, labour is additionally capped by the exogenous supply shock. Replicates Eqs. 19-20.

\begin{align} \Delta l_{i,t} &= \frac{l_{i,0}}{x_{i,0}} \left[\min\{x^{\text{inp}}_{i,t},\, d_{i,t}\} - x^{\text{cap}}_{i,t}\right] \\ l_{i,t} &= \begin{cases} l_{i,t-1} + \gamma_H \Delta l_{i,t} & \text{if } \Delta l_{i,t} \ge 0 \\ l_{i,t-1} + \gamma_F \Delta l_{i,t} & \text{if } \Delta l_{i,t} < 0 \end{cases} \end{align}

  1. Intermediate Orders

Inventory-gap ordering of intermediate inputs. Each industry places orders for intermediate inputs based on two components: (1) a naive expectation that demand will equal last period’s level, scaled by the technical coefficients; and (2) a correction term that moves inventories toward a target of \(n_i\) days of each input, at a speed governed by \(\tau\). Replicates Eq. 4.

\begin{align} O_{ji,t} = A_{ji}\,d_{i,t-1} + \frac{1}{\tau}\left(n_i Z_{ji,0} - S_{ji,t-1}\right) \end{align}

  1. Inventory Update

Inventory accumulation from deliveries minus usage. After production and rationing, each industry updates its inventory of every input. Inventories increase by deliveries received and decrease by inputs consumed in production (at rates given by the technical coefficients). A floor of zero prevents negative stocks for non-critical inputs that may be fully depleted. Replicates Eq. 18.

\begin{align} S_{ji,t+1} = \max\{S_{ji,t} + Z_{ji,t} - A_{ji}\,x_{i,t},\; 0\} \end{align}

  1. Productive Capacity

Labour-scaled production capacity. Each industry has a finite production capacity that scales linearly with available labour relative to the pre-pandemic baseline. Initially every industry employs \(l_{i,0}\) workers and produces at full capacity \(x^{\text{cap}}_{i,0} = x_{i,0}\). Replicates Eq. 7.

\begin{align} x^{\text{cap}}_{i,t} = \frac{l_{i,t}}{l_{i,0}}\, x^{\text{cap}}_{i,0} \end{align}

  1. Rationing

Proportional rationing of output across buyers. When output falls short of demand, industry i rations its output proportionally across all customers. Each buyer receives a share of their order equal to the ratio of output to demand. An optional firm_priority mode gives precedence to intermediate demand over final consumption. Replicates Eqs. 15-17.

\begin{align} Z_{ji,t} &= O_{ji,t}\,\frac{x_{j,t}}{d_{j,t}}, & c_{i,t} &= c^d_{i,t}\,\frac{x_{i,t}}{d_{i,t}}, & f_{i,t} &= f^d_{i,t}\,\frac{x_{i,t}}{d_{i,t}} \end{align}