format: sfcr-notebook-yaml formatVersion: 1 id: bmw-notebook title: BMW Browser Notebook metadata: version: 1 template: bmw cells: - markdown: id: overview title: Overview source: | This browser notebook adapts the BMW vignette into executable model, run, and chart cells. It focuses on a Newton baseline and two scenario comparisons. more: | The **BMW** model — Godley & Lavoie's _"model with private bank money"_ (_Monetary Economics_, 2007, chapter 7) — is the first model in the book where money is **inside money** created by private banks rather than the **outside money** issued by government in SIM (chapter 3). It is the bridge from a pure government-money economy to a genuine credit economy. Two big things arrive in chapter 7 that SIM did not have: - **Banks and bank money.** A banking sector makes **loans** to firms (`Ld`/`Ls`) and accepts **deposits** from households (`Mh`/`Ms`). Deposits are now the economy's money, and they are _created by lending_: when a firm borrows to invest, the loan simultaneously creates a deposit. This is Godley & Lavoie's illustration of the "loans make deposits" view of endogenous money. - **Fixed capital and investment.** Firms now own a stock of fixed capital `K`, which **depreciates** and must be replaced. Investment `Id` is driven toward a **target capital stock** `KT` set by a stock-flow norm (`kappa`), so the model has real accumulation, not just a circular flow. The economy still has three sectors — **households**, **production firms**, and **banks** — but now the firms hold real assets (capital) financed by bank loans, and households hold their wealth as bank deposits. Banks are pure intermediaries: they pay the same rate on deposits that they charge on loans (`rm = rl`), so they make zero profit and have zero net worth. - matrix: id: balance-sheet sourceRunCellId: baseline-run title: BMW balance sheet description: Balance-sheet matrix for the BMW model, following the sfcr article presentation. note: Source structure adapted from the sfcr BMW article balance-sheet display. more: | This is the BMW **balance sheet** (Table 7.1 in Godley & Lavoie). Compared with SIM's single-asset world, three things stand out: - **A real asset exists.** Production firms own fixed capital `+K`; it appears only once, in the firms' column, and it is the one item with no offsetting financial counterpart — it is the economy's tangible net worth. - **Inside money replaces outside money.** Households hold their wealth as bank **deposits** `+Mh`; banks record the same amount as a liability `-Ms`. Firms' capital is financed by bank **loans** `-Ld`, which the banks hold as an asset `+Ls`. - **Banks net to zero.** With `Ms = Ls` the banking column's assets and liabilities cancel, so banks carry no net worth — exactly the role of a pure intermediary. Reading across each financial row sums to zero (one sector's asset is another's liability), and the net-worth row closes each column. Household net worth `Vh` equals their deposits; the firms' net worth `V` is their capital less their loans. In the stationary state these accounting links pin every financial stock to the size of the capital stock (see the baseline panels below). columns: [Households, Production firms, Banks, Sum] sectors: [Households, Firms, Banks, ""] accountingKind: balance-sheet rows: - [Deposits, Money deposits, +Mh, "", -Ms, "0"] - [Loans, Loans, "", -Ld, +Ls, "0"] - [Investment, Fixed capital, "", +K, "", +K] - [Balance, Balance (net worth), -Vh, -V, "0", "0"] - [Sum, Sum, "0", "0", "0", "0"] - matrix: id: transactions-flow sourceRunCellId: baseline-run title: BMW transactions-flow matrix description: Transactions-flow matrix for the BMW model, shown in the same accounting style as the sfcr article. note: Signs and row structure follow the BMW transactions-flow matrix in the sfcr article. more: | The **transactions-flow matrix** (Table 7.2 in Godley & Lavoie) is the heart of the SFC method. Every cell is a payment _out of_ one sector (a minus) that is simultaneously a receipt _into_ another (a plus), so: - **every row sums to zero** — each transaction has a payer and a payee; - **every column sums to zero** — each sector's outlays plus its saving exhaust its receipts (its budget constraint). BMW adds rows that SIM never had, and they are what make it a credit economy: - **Investment and depreciation.** Firms buy investment goods (`Is`/`Id`) and set aside **amortization funds** `AF` out of the value of output to replace worn-out capital. `AF` is firms' internal saving — it lowers the wage bill rather than becoming profit. - **Interest flows.** Firms pay interest `rl[-1]*Ld[-1]` on last period's loans; households receive interest `rm[-1]*Mh[-1]` on last period's deposits. Because `rm = rl`, the bank passes loan interest straight through to depositors. - **Loan and deposit changes.** The bottom rows, `d(Ld)` and `d(Mh)`, are where the flow matrix meets the balance sheet: new lending and new deposits are the _flows_ that change the _stocks_. Splitting firms and banks into **current** and **capital** sub-columns shows the financing story directly: firms' capital account borrows `+d(Ld)` to cover the part of investment not paid for out of amortization funds, `Id - AF`. columns: [Households, Firms_current, Firms_capital, Banks_current, Banks_capital, Sum] sectors: [Households, Firms, Firms, Banks, Banks, ""] accountingKind: transactions-flow rows: - [Consumption, Consumption, -Cs, +Cd, "", "", "", "0"] - [Investment, Investment, "", +Is, -Id, "", "", "0"] - [Wages, Wages, +WBs, -WBd, "", "", "", "0"] - [Depreciation, Depreciation, "", -AF, +AF, "", "", "0"] - [Loans, Interest loans, "", "-rl[-1]*Ld[-1]", "", "+rl[-1]*Ls[-1]", "", "0"] - [Deposits, Interest on deposits, "+rm[-1]*Mh[-1]", "", "", "-rm[-1]*Ms[-1]", "", "0"] - [Loans, Ch. loans, "", "", +d(Ld), "", -d(Ls), "0"] - [Deposits, Ch. deposits, -d(Mh), "", "", "", +d(Ms), "0"] - [Sum, Sum, "0", "0", "0", "0", "0", "0"] - matrix: id: account-transactions collapsed: false sourceRunCellId: baseline-run title: BMW account transactions description: Account-level transaction matrix for BMW with hierarchical sector and account columns. note: Columns group balance-sheet accounts by sector. Use Expand all / Collapse all or click sector headers to show or hide account columns. more: | This account-level view expands the transactions-flow matrix into the individual balance-sheet accounts behind each sector, so you can trace a single event through every ledger it touches. It is the most literal expression of Godley's **quadruple-entry** principle: one transaction moves four account entries at once. Use _Expand all_ / _Collapse all_, or click a sector header, to reveal or hide the account columns grouped under **Households**, **Firms**, and **Banks**. The asset / liability / equity badges on each column show how every flow lands on one side of some sector's books and is matched on another's — the visual proof that the accounting is watertight. columns: [Deposits (Mh), Net_Worth (Vh), Deposits (Mf), Capital, Loans, Net_Worth (Vf), Firm.Loans, HH.Deposits, Firm.Deposits, Net_Worth (Vb), Sum] sectors: [Households(HH), Households(HH), Firms, Firms, Firms, Firms, Banks, Banks, Banks, Banks, ""] columnBadges: [asset, equity, asset, asset, liability, equity, asset, liability, liability, equity, ""] accountingKind: account-transactions rows: - [Initial, Initial values, "", "", "", "", "", "", "", "", "", "", "0"] - [Wages, Wages, WBd, WBd, -WBd, "", "", "", "", -WBd, +WBd, "", "0"] - [Consumption, Consumption, -Cs, -Cs, +Cd, "", "", +Cd, "", +Cs, -Cd, "", "0"] - [Investment, Investment, "", "", +Is, -Is, "", "", "", "", "", "", "0"] - [Depreciation, Depreciation, "", "", -AF, +AF, "", "", "", "", "", "", "0"] - [Loans, Interest loans, "", "", -rl'*Ld', "", "", -rl'*Ld', "", "", +rl'*Ld', "", "0"] - [Deposits, Interest on deposits, +rm'*Mh', +rm'*Mh', "", "", "", "", "", rm'•Mh', "", "", "0"] - [Loans, Ch. loans, "", "", d(Ld), "", d(Ls), "", +d(Ls), "", d(Ld), "", "0"] - [Deposits, Ch. deposits, "", "", "", "", "", +d(K), "", "", "", "", "0"] - [Sum, Sum, Mh, Vh, Mf, "", Ls, "", "", "", "", "", "0"] - sequence: id: transaction-flow-sequence title: BMW transaction flow sequence source: kind: matrix matrixCellId: transactions-flow participantColumnOrder: - Households - Firms_current - Firms_capital - Banks_current - Banks_capital description: Canvas-rendered sequence view generated from the transactions-flow matrix at the selected simulation period. note: Use Reset and Next step to manually reveal flows in order. more: | This is the same transactions-flow matrix read **column by column** as each sector's budget identity, animated step by step. Following the columns shows the _monetary circuit_ of a credit economy within a single period: - firms borrow and spend on investment and wages; - wages and deposit interest become household income; - households consume (returning money to firms) and bank the rest as deposits; - firms use sales plus amortization funds to repay/extend loans. Use **Reset** and **Next step** to reveal the flows in order and watch how new bank lending and new household deposits are two sides of the same act of money creation. - equations: id: equations-bmw title: BMW model modelId: bmw collapsed: false more: | These are the equations of model BMW, following Godley & Lavoie chapter 7. They fall into a few natural groups: - **Equilibrium by quantity adjustment.** `Cs = Cd`, `Is = Id`, `Ns = Nd`, and `Ls = Ld`: supply equals demand not through prices but because firms and banks simply meet whatever is demanded. Output is `Y = Cs + Is`. - **Firms, capital and investment.** Capital accumulates as `K = lag(K) + (Id - DA)*dt`, where `DA = delta*lag(K)` is depreciation. Investment chases a **target capital stock** `KT = kappa*lag(Y)` via `Id = gamma*(KT - lag(K)) + DA`: firms close a fraction `gamma` of the gap to target each period and always replace what depreciates. In a stationary state `KT = K`, so `Id = DA` and **net investment is zero**. - **Financing and the wage bill.** Firms keep amortization funds `AF = delta*lag(K)` and pay interest `lag(rl)*lag(Ld)` before paying wages, so `WBd = Y - lag(rl)*lag(Ld) - AF`. There are no pure profits. New borrowing covers the rest of investment: `Ld = lag(Ld) + (Id - AF)*dt`. - **Households.** Disposable income is wages plus deposit interest, `YD = WBs + lag(rm)*lag(Mh)`. The behavioural core is the consumption function `Cd = alpha0 + alpha1*YD + alpha2*lag(Mh)` — an autonomous term `alpha0`, a propensity `alpha1` to consume out of income, and a smaller propensity `alpha2` to consume out of accumulated deposit wealth. Unspent income is banked: `Mh = lag(Mh) + Households.Deposits*dt`. - **Banks.** The deposit rate tracks the loan rate, `rm = rl`, and deposit and loan supplies grow with new lending: `Ms = lag(Ms) + d(Ls)*dt`. **The redundant equation.** As in every SFC model, one equation is implied by all the others. Here it is `Ms = Mh`: the deposits banks issue always equal the deposits households hold. It is never imposed on the solver — it _emerges_, and checking that it holds is how you know the accounting is leak-free (see the solver note). rows: - Equalize supply to demand. - [Cs, Cd, "Consumption goods supply", $/year, flow, definition] - Production Firms - [Is, Id, "Supply of investment goods", $/year, flow, definition] - [Y, Cs + Is, "Income = GDP", $/year, flow, identity, eq-5-Y] - [WBd, Y - lag(rl) * lag(Ld) - AF, "Wage bill - demand", $/year, flow, identity, eq-6-WBd] - [AF, delta * lag(K), "Amortization funds", $/year, flow, definition, eq-7-AF] - [DA, delta * lag(K), "Depreciation allowance", $/year, flow, definition, eq-18-DA] - [Ld, lag(Ld) + (Id - AF) * dt, "Demand for bank loans", $, stock, accumulation] - [KT, kappa * lag(Y), "Target stock of capital", $, stock, target, eq-19-KT] - [K, K' + (Id - DA) • dt, "Stock of capital", $, stock, accumulation, eq-17-K] - [Id, gamma * (KT - lag(K)) + DA, "Demand for investment goods", $/year, flow, behavioral, eq-20-Id] - [Mf, Mf' + Firms.Deposits * dt, "Stock accumulation from Firms.Deposits (Mf)", $, stock, accumulation, eq-mf] - Wage bill - [Ns, Nd, "Supply of labor", items/year, flow, definition] - [WBs, W * Ns, "Wage bill - supply", $/year, flow, identity, eq-13-WBs] - [Nd, Y / pr, "Demand for labor", items/year, flow, definition, eq-14-Nd] - [W, WBd / Nd, "Wage rate", $/item, aux, definition, eq-15-W] - Households. - [YD, WBs + lag(rm) * lag(Mh), "Disposable income of households", $/year, flow, identity, eq-9-YD] - [Cd, alpha0 + alpha1 * YD + alpha2 * lag(Mh), "Consumption goods demand by households", $/year, flow, behavioral, eq-16-Cd] - [Mh, Mh' + Households.Deposits * dt, "Stock accumulation from Households.Deposits (Mh)", $, stock, accumulation, eq-10-Mh] - [Vh, Vh' + Households.Net_Worth * dt, "Stock accumulation from Households.Net_Worth (Vh)", $, stock, accumulation, eq-vh] - Banks - [rm, rl, "Rate of interest on bank deposits", 1/year, aux, definition, eq-12-rm] - [Ls, lag(Ls) + d(Ld) * dt, "Supply of bank loans", $, stock, accumulation] - [Ms, lag(Ms) + d(Ls) * dt, "Supply of bank deposits", $, stock, accumulation, eq-11-Ms] - sequence: id: sequence-cld title: Causal loop diagram source: kind: cld modelId: bmw collapsed: false more: | The **causal loop diagram** turns the equation list into a directed graph of who drives whom. It is the quickest way to see BMW's central feedback loop: higher output raises the target capital stock `KT`, which lifts investment `Id`, which (through `Y = Cs + Is`) raises output again — the accelerator — while the consumption-out-of-wealth term `alpha2*lag(Mh)` and depreciation provide the damping that pulls the economy to a stationary state. - solver: id: solver-bmw title: Solver options modelId: bmw collapsed: true more: | BMW is solved period by period with a **Newton** method: lagged stocks (`lag(K)`, `lag(Ld)`, `lag(Mh)`, …) are known from the previous period, and the within-period equations are solved simultaneously for the current flows and end-of-period stocks. The solver's **hidden variables** `Ms` (left) and `Mh` (right) implement the redundant-equation check. Because one equation in the system is redundant, the equality `Ms = Mh` is _not_ used to find the solution — instead the solver verifies, to tolerance `hiddenTolerance`, that the independently computed deposit supply and deposit demand agree every period. If they ever diverged, the model's accounting would be leaking. This is the numerical embodiment of Godley & Lavoie's consistency requirement. method: newton tolerance: "1e-10" maxIterations: 100 defaultInitialValue: "1e-15" hiddenLeftVariable: Ms hiddenRightVariable: Mh hiddenTolerance: "0.00001" relativeHiddenTolerance: false - externals: id: parameters-bmw title: Externals modelId: bmw collapsed: true more: | These are the exogenous parameters and policy levers of BMW: - `rl` = 0.025 — the loan rate of interest, set exogenously by the banks; the deposit rate follows it via `rm = rl`. - `alpha0` = 20 — autonomous consumption (the variable shocked in Scenario 1). - `alpha1` = 0.75 — propensity to consume out of disposable income (shocked in Scenario 2). - `alpha2` = 0.1 — propensity to consume out of accumulated deposit wealth. - `delta` = 0.1 — the depreciation rate of fixed capital. - `gamma` = 0.15 — the speed at which investment closes the gap to target capital. - `kappa` = 1 — the target capital-output ratio (`KT = kappa*lag(Y)`). - `pr` = 1 — labour productivity. These numbers determine the whole stationary state. Combining the steady-state conditions (`KT = K`, `Id = DA`, constant deposits so `Cd = YD`) gives a clean closed form for stationary output: `Y* = alpha0 / ((1 - delta*kappa)*(1 - alpha1) - alpha2*kappa)` With these values the denominator is `0.9*0.25 - 0.1 = 0.125`, so `Y* = 20 / 0.125 = 160` — exactly the level the baseline converges to. rows: - [rl, 0.025, "Rate of interest on bank loans, set exogenously", 1/year, aux] - [alpha0, 20, "Exogenous component in consumption", $/year, aux] - [alpha1, 0.75, "Propensity to consume out of income", "", aux] - [alpha2, 0.1, "Propensity to consume out of wealth", 1/year, aux] - [delta, 0.1, "Depreciation rate", 1/year, aux] - [gamma, 0.15, "Speed of adjustment of capital to its target value", 1/year, aux] - [kappa, 1, "Capital-output ratio", year, aux] - [pr, 1, "Labor productivity", $/item, aux] - initial-values: id: initial-values-bmw title: Initial values modelId: bmw collapsed: true more: | BMW carries no inherited stocks: capital, loans and deposits all start from (effectively) zero, set by the solver's tiny `defaultInitialValue`. The economy therefore begins far below its stationary state and has to _build up_ its capital stock and money stock from scratch — which is what produces the long, smooth climb in the baseline run. rows: [] - run: id: baseline-run title: Baseline run with Newton note: This Newton baseline provides the reference path for the BMW scenarios. description: Run the baseline for 50 periods using the Newton solver to generate the reference path for the BMW scenarios. mode: baseline periods: 50 resultKey: bmw_newton sourceModelId: bmw more: | The baseline integrates BMW forward for 50 periods from near-zero stocks, with all parameters held at the values above. With no capital and no deposits inherited from the past, the economy starts well below its stationary state and **accumulates** its way up: firms borrow to build the capital stock, that lending creates the deposits households hold as wealth, and the system converges to a stationary state in which net investment is zero and all stocks stop growing. This run is the reference path for both scenarios. - chart: id: chart-1 title: Baseline headline variables description: Chart of key variables in the baseline run to show the model's dynamic behavior over time. note: This chart shows the time paths of selected variables in the baseline run, which serves as the reference for the scenario comparisons. more: | In the baseline, output `Y` and disposable income `YD` rise quickly at first and then converge as the capital stock approaches its target and net investment fades to zero. The mechanism is the interaction of the accelerator (investment chasing `KT = kappa*lag(Y)`) with consumption out of accumulated deposits (`alpha2*lag(Mh)`): both strengthen as stocks build, until saving and replacement investment exactly balance. The numbers match the steady-state arithmetic of the model: - output converges to `Y* = alpha0 / ((1 - delta*kappa)*(1 - alpha1) - alpha2*kappa) = 160`; - consumption to `Cd = Y*(1 - delta*kappa) = 144`, which also equals disposable income `YD = 144` (in the stationary state households consume all their income); - gross investment to `Id = DA = delta*K = 16`, i.e. **net investment is zero**; - the wage rate settles at `W = 0.875` — pinned because, with `kappa = 1`, `WBd = Y - rl*K - delta*K = Y*(1 - rl - delta)` while `Nd = Y`, so `W = 1 - rl - delta = 1 - 0.025 - 0.1`. variables: - Y - YD - Cd - Id - W axisMode: shared axisSnapTolarance: 0.1 niceScale: true sourceRunCellId: baseline-run - table: id: table-1 title: Baseline variable summary note: Table of selected variables for the baseline run, showing values at each period. description: This table shows the values of key variables in the baseline run across all periods. more: | The table makes the convergence concrete. Watch the capital stock `K` climb and flatten as `Id` falls toward `DA` (the moment net investment hits zero), and watch the household deposit stock `Mh` rise alongside it. The most striking feature of the stationary state is that **every stock ends up equal**: `K = Ld = Ls = Mh = Ms = 160`. This is forced by the accounting. Capital is fully loan-financed, so `Ld = K`; banks net to zero, so `Ms = Ls = Ld`; and the redundant equation gives `Mh = Ms`. With `kappa = 1` the common stock also equals output, `K = kappa*Y = Y = 160`. The whole financial superstructure of the economy is exactly the size of its capital stock. variables: - Y - Id - DA - K - Ld - Ls - Mh - Ms sourceRunCellId: baseline-run - markdown: id: scenario-1-note title: Scenario 1 source: Scenario 1 increases autonomous consumption expenditure by raising alpha0 from period 5 to period 50. more: | Scenario 1 is a permanent **upward shift in autonomous consumption**: `alpha0` is raised from 20 to 30 from period 5 onward, holding every other parameter fixed. Households decide to spend more independently of their income and wealth, and the question is where the economy settles as a result. - run: id: scenario-1-run title: "Scenario 1: autonomous consumption shock" note: This scenario applies a shock to autonomous consumption by increasing alpha0 for a period of time description: allowing us to see the effect on the economy compared to the baseline. mode: scenario scenario: shocks: - rangeInclusive: - 5 - 50 variables: alpha0: kind: constant value: 30 startPeriodInclusive: 5 endPeriodInclusive: 50 baselineRunCellId: baseline-run periods: 50 resultKey: bmw_s1 sourceModelId: bmw baselineStartPeriod: 40 more: | The scenario re-solves the model with `alpha0` stepped up to 30 from period 5, plotted against the baseline as a reference. Because only `alpha0` changes, the contrast isolates the pure effect of a stronger consumption stance on output, investment and the economy's stocks. - chart: id: scenario-1-chart title: "Scenario 1: output, investment and stocks" more: | Raising `alpha0` from 20 to 30 moves the economy to a **new, higher** stationary state. Since stationary output is `Y* = alpha0 / 0.125 = 8*alpha0` here, output rises from `160` to `8*30 = 240`. In fact the entire stationary state scales up **in exact proportion** (×1.5): consumption `Cd` goes `144 → 216`, gross investment `Id` goes `16 → 24`, and the capital, loan and deposit stocks all rise `160 → 240`. Two of Godley & Lavoie's points show up clearly: - **Real and financial stocks move together.** More consumption demand pulls up output, the target capital stock, and hence investment; the extra investment is loan-financed, so loans and the deposits backing them grow in lockstep. - **The wage rate is unchanged at `W = 0.875`.** Because `W = 1 - rl - delta` depends only on the interest and depreciation rates, a pure demand shift changes the _scale_ of the economy but not this intensive price. variables: - Cd - Id - K - W axisMode: shared axisSnapTolarance: 0.1 niceScale: true sharedRange: includeZero: true sourceRunCellId: scenario-1-run - run: id: scenario-2-run title: "Scenario 2: propensity-to-save shock" description: Scenario 2 lowers alpha1 from period 5 to period 50 to examine the effect of a higher propensity to save. mode: scenario scenario: shocks: - rangeInclusive: - 5 - 50 variables: alpha1: kind: constant value: 0.7 startPeriodInclusive: 5 endPeriodInclusive: 50 baselineRunCellId: baseline-run periods: 50 resultKey: bmw_s2 sourceModelId: bmw more: | Scenario 2 is a **paradox-of-thrift** experiment: the propensity to consume out of income `alpha1` is lowered from 0.75 to 0.70 from period 5 onward — households decide to _save more_. The run is solved from the baseline's stationary state and plotted against the baseline, so the chart shows the transition to the new, thriftier equilibrium. The result is the classic Keynesian paradox, intact in a full SFC model: trying to save a larger fraction of income makes the economy **smaller**. Stationary output falls from `160` to `Y* = 20 / ((1 - 0.1)*(1 - 0.70) - 0.1) = 20 / 0.17 ≈ 117.6`, dragging the capital stock, loans and deposits down with it (all settle near `117.6`). Less spending means a smaller desired capital stock, less investment, less lending, and ultimately less income. - chart: id: scenario-2-chart title: Scenario 2 headline variables more: | With the higher saving propensity, consumption `Cd` and disposable income `YD` both settle at a permanently **lower** level (≈ `105.9`, against `144` in the baseline) as the economy contracts to its smaller stationary state. The wage rate `W` again stays at `0.875`: as in Scenario 1, the demand-side parameter changes the size of the economy but not the interest-and-depreciation-determined wage. The contraction is the mirror image of Scenario 1 — a reminder that in BMW, as in SIM, the long-run level of activity is governed by the spending decisions embedded in the consumption function. variables: - Cd - YD - W sharedRange: includeZero: true sourceRunCellId: scenario-2-run - sequence: id: equation-dependency-graph title: BMW equation dependency graph source: kind: dependency modelId: bmw showAccountingStrips: true showExogenous: false description: Dependency view of the BMW equations organized by sector strips and accounting bands. note: Use this alongside the transaction-flow sequence to compare accounting flows with equation structure. collapsed: true more: | The **dependency graph** lays the equations out by sector strip and accounting band, showing the order in which variables are computed within a period and how the household, firm and bank blocks connect. Read it alongside the transactions-flow sequence to compare the _equation_ structure (what determines what) with the _accounting_ structure (what pays what) — two views of the same stock-flow-consistent model.