Step Equations
Bankrupt Firms
Zero out production, assets, wage of bankrupt firms; record
the defaulted asset total.
\begin{align}
D^{\text{def}}_t &= -\sum_i \text{enter}_{i,t}\,A_{i,t}, \\
Y_{i,t},\,A_{i,t},\,\alpha_{i,t},\,W_{i,t} &\leftarrow 0
\quad\text{if}\quad \text{enter}_{i,t} = 1.
Bankrupt firms are zeroed simultaneously across production,
assets, alive flag, and wage; the negative-asset sum surfaces as
the system-level defaulted total, which the loan rate phase uses
to set the bankruptcy-adjusted spread.
\end{align}
Compute Employment
Update aggregate employment and unemployment scalars.
\begin{align}
e_t &= Y_{tot} / N, \\
u_t &= 1 - e_t.
Aggregate employment is the total production normalised to the
labour-force ceiling; unemployment is the residual that
downstream wage and revival phases consume.
\end{align}
Compute Gamma And Ren
Bank-side \(\Gamma\) (real-rate gap above baseline) and
per-firm \(\text{ren}\) ratio (\(\Gamma\) times solvency).
\begin{align}
\Gamma_t &= \Gamma_0 + \text{ReLU}\big(\alpha_\Gamma\,(\bar\rho^l_t - \hat\pi_t) - \Gamma_0\big), \\
\text{ren}_{i,t} &= \Gamma_t \cdot \frac{A_{i,t}}{W_{i,t} Y_{i,t} + \epsilon}.
baseline; :math:`\text{ren}_{i,t}` scales each firm's gross
solvency by that bank-side aggressiveness, driving the hiring,
wage, and price asymmetries downstream.
\end{align}
Compute Inflation And Employment
Inflation from the current/prior average price; aggregate
employment update.
\begin{align}
\pi_t &= \bar P_t - 1, \\
e_t &= Y_{tot} / N, \\
u_t &= 1 - e_t.
Re-deriving employment after revival closes the labour-market
identity used by the next period's wage and consumption phases.
\end{align}
Compute Lowest Price
Lowest price among surviving firms with positive price.
\begin{align}
P^{\min}_t = \min_{i \in \{i: \text{stay}_{i,t}\,P_{i,t} > 0\}} P_{i,t}.
Dead firms are masked to a large sentinel so the minimum picks
up only live, positively-priced firms. The result anchors the
household demand allocation in the next phase.
\end{align}
Compute Moments
Recompute wage and price moments after bankruptcy zeroing.
\begin{align}
W_{tot} &= \sum_i \text{stay}_{i,t}\,W_{i,t}\,Y_{i,t}, \\
\bar W_t &= W_{tot} / (Y_{tot} + \epsilon), \\
\bar P_t &= \sum_i \alpha_{i,t}\,P_{i,t}\,Y_{i,t} / (Y_{tot} + \epsilon), \\
W^{\max}_t &= \max_i \alpha_{i,t}\,W_{i,t}.
The post-bankruptcy moments anchor the bank-side and
household-side decisions: maximum wage drives the unemployment
logit, average price drives the demand allocation, average wage
feeds the consumption budget.
\end{align}
Compute Ushare
Per-firm share of the unemployed labour pool via a logit
weight on the wage gap to the maximum wage.
\begin{align}
z_{i,t} &= \beta\,(W_{i,t} - W^{\max}_t) / \bar W_t, \\
w_{i,t} &= \alpha_{i,t}\,e^{z_{i,t}}, \\
Z_t &= \sum_i w_{i,t}, \\
u^{\text{share}}_{i,t} &= \alpha_{i,t}\,\frac{u_t\,N\,(1 - \text{bust}_t)\,e^{z_{i,t}}}{Z_t + \epsilon}.
Higher-wage live firms attract a larger share of the unemployed
pool, allowing them to expand production faster when demand
pressure builds.
\end{align}
Demand Production Imbalance
Compute the per-firm demand-production gap and the
corresponding excess-demand and excess-supply indicator masks.
\begin{align}
\Delta Y_{i,t} &= \text{stay}_{i,t}\,(D_{i,t} - Y_{i,t}), \\
\mathbf{1}^D_{i,t} &= \mathbb{1}\{\Delta Y_{i,t} > 0\}, \\
\mathbf{1}^S_{i,t} &= \alpha_{i,t}\,\mathbb{1}\{D_{i,t} - Y_{i,t} \le 0\}.
The masks partition the surviving firm cross-section into
demand-constrained and supply-constrained sub-populations consumed
by the subsequent production, price, and wage updates.
\end{align}
Expectation Adjustments
Anticipated inflation pass-through into prices and wages for
surviving firms.
\begin{align}
P_{i,t} &\leftarrow P_{i,t}\,(1 + \hat\pi_t) \quad\text{if stay}_{i,t}, \\
W_{i,t} &\leftarrow W_{i,t}\,(1 + \hat\pi_t\,w_f) \quad\text{if stay}_{i,t}.
Surviving firms transmit a fraction :math:`w_f` of the expected
inflation to wages and the full amount to prices, building in a
wedge that drives the loan-rate response.
\end{align}
Find Surviving Firms
Identify firms whose assets plus a fraction \(\Theta\) of
payroll are positive; survivors stay alive next period, the rest
enter bankruptcy.
The differentiable indicator uses Behavior.diffwhere() so
gradient flows through the solvency threshold. The bankruptcy
indicator is the squared complement, giving a smoother gradient
at the boundary.
\begin{align}
\text{stay}_{i,t} &= \alpha_{i,t-1} \cdot \text{diffwhere}\big(A_{i,t} + \Theta\, W_{i,t} Y_{i,t},\, 1,\, 0\big), \\
\text{enter}_{i,t} &= \alpha_{i,t-1}\,(1 - \text{stay}_{i,t})^2.
Firms whose end-of-period assets cover a fraction :math:`\Theta`
of the next period's wage bill survive; the rest are flagged for
bankruptcy.
\end{align}
Firm Accounting
Realise per-firm EBIT and profits, draw down household savings
by the aggregate EBIT, and update firm assets by the
alive-weighted profit stream.
\begin{align}
\text{ebit}_{i,t} &= P_{i,t}\,\min(D_{i,t}, Y_{i,t}) - W_{i,t}\,Y_{i,t}, \\
\Pi_{i,t} &= \text{ebit}_{i,t}
+ \rho^l_t\,\min(A_{i,t}, 0)
+ \rho^d_t\,\max(A_{i,t}, 0), \\
A_{i,t} &\leftarrow A_{i,t} + \alpha_{i,t}\,\Pi_{i,t}, \\
S_t &\leftarrow S_t - \sum_i \alpha_{i,t}\,\text{ebit}_{i,t}.
The accounting balances aggregate EBIT against the household
savings drawdown and pushes profit signals into firm assets,
carrying the previous-period balance-sheet effects into the next
bankruptcy check.
\end{align}
Household Consumption
Update the household consumption propensity, set the budget,
and allocate firm-level demand by a logit weight on the price
gap to the lowest live price.
\begin{align}
c_t &= \mathrm{clamp}\big(c_0\,[1 + \alpha_c\,(\hat\pi_t - \bar\rho^d_t)],\,0,\,1\big), \\
B_t &= c_t\,[W_{tot} + \max(S_t, 0)], \\
z_{i,t} &= \beta\,(P^{\min}_t - P_{i,t}) / \bar P_t, \\
D_{i,t} &= \alpha_{i,t}\,\frac{B_t\,e^{z_{i,t}}}{(\sum_j \alpha_{j,t}\,e^{z_{j,t}} + \epsilon)\,P_{i,t}}, \\
D_t^{tot} &= \sum_i \alpha_{i,t}\,D_{i,t}.
Real-rate gaps raise the consumption propensity; the logit
allocation sends a larger share of demand to lower-priced live
firms, transmitting price competition into the real economy.
\end{align}
Monetary Policy
Central bank policy: Taylor-like rule on EWMA inflation only.
\begin{align}
\rho^{CB}_t = \rho^\star + \phi_\pi\,(\pi^{ema}_t - \pi^\star).
The CB raises the policy rate when EWMA inflation runs above
target and lowers it when it runs below, feeding into the next
period's loan-rate setting and the household propensity gap.
\end{align}
Pay Dividends
Pay dividends from firms with positive assets and profits;
credit household savings, debit firm assets.
\begin{align}
m_{i,t} &= \alpha_{i,t}\,\mathbb{1}\{A_{i,t} > 0\}\,\mathbb{1}\{\Pi_{i,t} > 0\}, \\
d_{i,t} &= m_{i,t}\,A_{i,t}\,\delta, \\
S_t &\leftarrow S_t + \sum_i d_{i,t}, \\
A_{i,t} &\leftarrow A_{i,t} - d_{i,t}.
The mask gates dividends on both balance-sheet sign and
profitability, recycling part of the surviving firms' surplus
back to households as a closed-loop income source.
\end{align}
Price Adjustment
Price update from the frozen-noise buffer.
Excess-demand firms with below-average price scale \(P\) up
by \((1 + r_p)\); excess-supply firms with above-average
price scale down by \((1 - r_p)\).
\begin{align}
r_p &= \gamma_p\,u^p_{t,i}, \\
P_{i,t} &\leftarrow P_{i,t}\,(1 + r_p)
\quad\text{if}\quad \text{stay}_{i,t}\,\mathbf{1}^D_{i,t}\,(P_{i,t} < \bar P_t), \\
P_{i,t} &\leftarrow P_{i,t}\,(1 - r_p)
\quad\text{if}\quad \mathbf{1}^S_{i,t}\,(P_{i,t} > \bar P_t).
Below-average price-setters facing excess demand raise prices and
above-average price-setters facing excess supply cut prices,
pulling the cross-section toward the average.
\end{align}
Production Adjustment
Adjust production: increase when in excess-demand regime,
decrease when in excess-supply regime.
\begin{align}
\eta^+_{i,t} &= \mathrm{clamp}\big(\eta_0 r\,(1 + \text{ren}_{i,t}),\,0,\,1\big), \\
\eta^-_{i,t} &= \mathrm{clamp}\big(\eta_0\,(1 - \text{ren}_{i,t}),\,0,\,1\big), \\
Y_{i,t} &\leftarrow Y_{i,t}
+ \text{stay}_{i,t}\,\mathbf{1}^D_{i,t}\,\min(\eta^+_{i,t}\,\Delta Y_{i,t},\,u^{\text{share}}_{i,t}) \\
&\phantom{\leftarrow Y_{i,t}}\;
+ \text{stay}_{i,t}\,\mathbf{1}^S_{i,t}\,\eta^-_{i,t}\,\Delta Y_{i,t}.
Hiring is capped by the firm's share of the unemployed labour
pool, and firing is asymmetric to hiring through the
modulation.
\end{align}
Recompute Firm Totals Consumption
Ordering-critical re-aggregation after firm accounting and
dividends.
\begin{align}
D^{tot}_t &= \sum_i \alpha_{i,t}\,D_{i,t}, \\
S^{f+}_t &= \sum_i \text{stay}_{i,t}\,\max(A_{i,t}, 0), \\
A^{tot}_t &= \sum_i \alpha_{i,t}\,A_{i,t}.
The recompute refreshes the aggregate stocks the revival and
balance-sheet-adjustment phases will read, after firm-level
profits and dividends have moved cash around.
\end{align}
Recompute Firm Totals Revival
Ordering-critical re-aggregation after revival; absorb the
default loss into surviving firms’ assets, then refresh every
macro aggregate that downstream phases will read.
\begin{align}
A_{i,t} &\leftarrow A_{i,t} - A_{i,t}\,\frac{D^{\text{def}}_t}{\tilde S^{f+}_t}
\quad\text{if}\quad \alpha_{i,t}\,S^{f+}_t\,A_{i,t} > 0, \\
W_{tot} &= \sum_i \alpha_{i,t}\,W_{i,t}\,Y_{i,t}, \\
Y_{tot} &= \sum_i \alpha_{i,t}\,Y_{i,t}, \\
A_{tot} &= \sum_i \alpha_{i,t}\,A_{i,t}, \\
D^{f-}_t &= \sum_i \alpha_{i,t}\,\max(-A_{i,t}, 0), \\
W^{\max}_t &= \max_i \alpha_{i,t}\,W_{i,t}, \\
\text{bust}_t &= (N - \sum_i \alpha_{i,t})/N, \\
\bar P_t &= \sum_i \alpha_{i,t}\,P_{i,t}\,Y_{i,t} / (Y_{tot} + \epsilon), \\
\bar W_t &= W_{tot} / (Y_{tot} + \epsilon).
The proportional default absorption keeps the aggregate-asset
identity intact across revival; the refreshed moments feed the
inflation phase and the next period's wage and price updates.
\end{align}
Recompute Firm Totals Stayalive
Ordering-critical re-aggregation after the wage/price update.
Enforces non-negative production for surviving firms via
Behavior.diffwhere(), then recomputes macro totals from the
stay-alive mask. Must fire before bankrupt_firms() zeroes
out the bankrupt slice.
\begin{align}
Y_{i,t} &\leftarrow \text{diffwhere}\big(\text{stay}_{i,t},\,\max(Y_{i,t}, 0),\,Y_{i,t}\big), \\
Y_{tot} &= \sum_i \text{stay}_{i,t}\,Y_{i,t}, \\
W_{tot} &= \sum_i \text{stay}_{i,t}\,W_{i,t}\,Y_{i,t}, \\
\bar W_t &= W_{tot} / (Y_{tot} + \epsilon), \\
\bar P_t &= \sum_i \text{stay}_{i,t}\,P_{i,t}\,Y_{i,t} / (Y_{tot} + \epsilon), \\
S^{f+}_t &= \sum_i \text{stay}_{i,t}\,\max(A_{i,t}, 0), \\
D^{f-}_t &= \sum_i \text{stay}_{i,t}\,\max(-A_{i,t}, 0).
The clamp keeps production weakly positive for solvent firms; the
production-weighted averages and total payroll feed the
subsequent bank-rate and household-consumption phases.
\end{align}
Renormalize Prices
Rescale all nominal quantities by the lagged average price so
that \(\bar P_t \equiv 1\) going into the wage/price update.
Every nominal series — firm prices, wages, assets, profits,
household savings, the wage moments, and the money stock — is
divided by \(\bar P_{t-1}\) in lock-step. The carry-over
\(\bar P\) then collapses to 1.
\begin{align}
P_{i,t} &\leftarrow P_{i,t} / \bar P_{t-1}, \\
W_{i,t} &\leftarrow W_{i,t} / \bar P_{t-1}, \\
A_{i,t} &\leftarrow A_{i,t} / \bar P_{t-1}, \\
\Pi_{i,t} &\leftarrow \Pi_{i,t} / \bar P_{t-1}, \\
S_t &\leftarrow S_t / \bar P_{t-1}, \\
\bar W_t, M_t^{0}, W^{\max}_t &\leftarrow (\cdot) / \bar P_{t-1}, \\
\bar P_t &\leftarrow 1.
Lock-step rescaling preserves every nominal-quantity ratio while
zeroing out trend inflation in the carry-over level, so subsequent
wage and price update phases work in unit price units.
\end{align}
Revive Firms
Bernoulli revival of dead firms; revived firms receive a
fresh production / price / wage / asset endowment.
\begin{align}
m^{\text{rev}}_{i,t} &= (1 - \alpha_{i,t})\,\mathbb{1}\{u^r_{t,i} < \phi\}, \\
Y_{i,t} &\leftarrow \max(u_t, 0)\,u^{r,Y}_{t,i}
\quad\text{if}\quad m^{\text{rev}}_{i,t} = 1, \\
P_{i,t} &\leftarrow \bar P_t,\quad
W_{i,t} \leftarrow \bar W_t
\quad\text{if}\quad m^{\text{rev}}_{i,t} = 1, \\
A_{i,t} &\leftarrow W_{i,t}\,Y_{i,t}
\quad\text{if}\quad m^{\text{rev}}_{i,t} = 1, \\
\Pi_{i,t} &\leftarrow 0
\quad\text{if}\quad m^{\text{rev}}_{i,t} = 1, \\
D^{\text{def}}_t &= \sum_i m^{\text{rev}}_{i,t}\,A_{i,t}, \\
S^{f+}_t &\leftarrow S^{f+}_t + D^{\text{def}}_t, \\
\alpha_{i,t} &\leftarrow \alpha_{i,t} + m^{\text{rev}}_{i,t}.
Revival keeps the firm count steady and seeds new entrants near
the average wage and price, preventing the population from
collapsing.
\end{align}
Set Interest Rates
Set loan rate (CB rate plus bankruptcy-adjusted spread) and
deposit rate (residual closing the bank’s balance sheet); then
accrue interest on household savings.
\begin{align}
\rho^l_t &=
\begin{cases}
\rho^{CB}_t + (1 - f)\,D^{\text{def}}_t / D^{f-}_t & \text{if}\ D^{f-}_t > 0, \\
\rho^{CB}_t & \text{otherwise},
\end{cases} \\
I_t &= \rho^l_t\,D^{f-}_t, \\
\rho^d_t &=
\begin{cases}
(I_t - D^{\text{def}}_t) / (S_t + S^{f+}_t) & \text{if}\ S_t + S^{f+}_t > 0, \\
0 & \text{otherwise},
\end{cases} \\
S_t &\leftarrow (1 + \rho^d_t)\,S_t.
The loan-rate spread compensates the bank for default losses; the
deposit rate is whatever residual balances total bank cashflow
against accrued interest, so the bank's balance sheet closes
without an exogenous funding source.
\end{align}
Solve Rounding Errors
Patch the monetary-closure residual to absorb floating-point
drift from earlier denominator regularisations.
\begin{align}
R_t &= S_t + S^{f+}_t - D^{f-}_t - D^{\text{def}}_t - M^0_t, \\
S_t &\leftarrow
\begin{cases}
S_t - R_t & \text{if}\ |R_t| > 10^{-9}, \\
S_t & \text{otherwise}.
\end{cases}
The patch enforces the money-stock identity
floating-point tolerance at every period.
\end{align}
Update Averages
EWMA update of inflation, interest rate, and unemployment
registers; convex combination of EWMA inflation and CB target for
the expectation used downstream.
The four EWMA registers share the same memory weight
\(\omega\). The expected-inflation pipe used by every
downstream phase mixes the CB target with the EWMA term in a
fixed convex combination.
\begin{align}
\pi^{ema}_t &= \omega\,\pi_{t-1} + (1-\omega)\,\pi^{ema}_{t-1}, \\
\bar\rho^d_t &= \omega\,\rho^d_{t-1} + (1-\omega)\,\bar\rho^d_{t-1}, \\
\bar\rho^l_t &= \omega\,\rho^l_{t-1} + (1-\omega)\,\bar\rho^l_{t-1}, \\
\bar u_t &= \omega\,u_{t-1} + (1-\omega)\,\bar u_{t-1}, \\
\hat\pi_t &= \tau^T \pi^\star + \tau^R \pi^{ema}_t.
Smoothing the four registers in lock-step makes downstream
bank-side and household-side decisions react to slow-moving
aggregate signals rather than period noise.
\end{align}
Wage Adjustment
Smooth wage update from the frozen-noise buffer.
Excess-demand profitable firms raise wages; excess-supply
loss-making firms lower wages. The cashflow-per-production
ceiling caps wage increases at the firm’s affordable level so
the asset balance does not turn nominally infeasible.
\begin{align}
r_w &= \gamma_p\,r\,u^w_{t,i}, \\
W_{i,t} &\leftarrow W_{i,t}\,[1 + (1 + \text{ren}_{i,t})\,r_w\,e_t]
\quad\text{if}\quad \text{stay}_{i,t}\,\mathbf{1}^D_{i,t}\,(\Pi_{i,t} > 0), \\
W_{i,t} &\leftarrow \min\big(W_{i,t},\,\text{cashflow}_{i,t}/Y_{i,t}\big)
\quad\text{(ceiling on the wage rise)}, \\
W_{i,t} &\leftarrow W_{i,t}\,[1 - (1 - \text{ren}_{i,t})\,r_w\,u_t]
\quad\text{if}\quad \text{stay}_{i,t}\,\mathbf{1}^S_{i,t}\,(\Pi_{i,t} < 0).
Wage growth is gated on both regime (demand/supply) and firm
profitability; the cashflow ceiling prevents wage-driven
insolvency, and the :math:`(1 \pm \text{ren})` weighting
introduces the same hiring/firing asymmetry seen in production.
\end{align}
