format: sfcr-notebook-yaml formatVersion: 1 id: sim-notebook title: SIM Browser Notebook metadata: version: 1 template: sim cells: - markdown: id: overview title: Overview source: | This notebook implements the SIM model from Godley and Lavoie chapter 3. It includes the baseline model, a government-spending shock scenario, accounting matrices, and result views for the main income, consumption, and stock variables. more: | The **SIM** model — Godley & Lavoie's _"Simplest model with government money"_ (_Monetary Economics_, 2007, chapter 3) — is the smallest complete stock-flow consistent (SFC) model. It strips the economy down to the bare minimum needed to illustrate how money, income and wealth fit together without leaks. The economy has three sectors — **households**, **production firms**, and **government** — and a single financial asset: **government money** (`Hh` held by households, `Hs` issued by government). To keep things simple the model assumes away almost everything else: - a pure **service/labour** economy: firms hire labour and sell the output in the same period, so there are no inventories and no fixed capital; - **no banks, no loans, no interest, and no private money**; - **no profits**: all of firms' sales revenue is paid out as wages. Money is **created** when the government spends (`Gd`) and **destroyed** when it taxes (`TXd`). Because every flow in the model has a matching counter-flow, the system is **"watertight"**: there are no black holes where money leaks out. This is the central pedagogical point of chapter 3 — _all_ monetary flows come from somewhere and go somewhere. ![The SIM money circuit: government spending and wages inject money into households, while consumption and taxes flow back to firms and government; the gap accumulates as household money.](figures/sim-circuit.svg) - matrix: id: balance-sheet sourceRunCellId: baseline-run title: SIM balance sheet description: Compact balance sheet for the SIM model. Household money holdings mirror government money liabilities. more: | This is the SIM **balance sheet** (Table 3.1 in Godley & Lavoie). At any point in time the only asset in the whole economy is government money. Households hold it as wealth (`+Hh`); the government records the same amount as a liability (`-Hs`). Production firms own nothing and owe nothing — in SIM they hold no financial assets, because all revenue is paid straight out as wages. Two accounting conventions make the table read cleanly: - **assets are positive, liabilities negative**, so each row sums _across_ sectors to zero — one sector's asset is another's liability; - the **net-worth (balance) row** closes each column to zero. Money is "outside" wealth: it is simultaneously private net worth and public debt. There is no real capital to back it, which is exactly what makes SIM the _simplest_ possible closed financial system. columns: [Households, Government, Sum] sectors: [Households, Government, ""] accountingKind: balance-sheet rows: - [Money, Money stock, +Hh, -Hs, "0"] - [Balance, Net worth, -Hh, +Hs, "0"] - [Sum, Sum, "0", "0", "0"] - matrix: id: transactions-flow sourceRunCellId: baseline-run title: SIM transactions-flow matrix description: Transactions-flow matrix for the SIM model, arranged around households, production, and government. more: | The **transactions-flow matrix** (Table 3.2 in Godley & Lavoie) is the heart of the SFC method, and the single most important object in chapter 3. 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). This is Godley's **quadruple-entry** principle: a single economic event touches four entries at once. When the government spends `Gd`, that is an outlay for the government, a receipt for firms, which pays wages to households, who in turn acquire money. Nothing is lost in transit. The final row, the **change in the money stock**, is where stocks and flows meet: household saving `d(Hh)` is exactly mirrored by the change in government-issued money `-d(Hs)`. That mirroring is the model's consistency check (see the solver note below). columns: [Households, Production, Government, Sum] sectors: [Households, Firms, Government, ""] accountingKind: transactions-flow rows: - [Consumption, Consumption, -Cd, +Cs, "", "0"] - [Government, Government expenditure, "", +Gs, -Gd, "0"] - [Income, Wages, +W * Ns, -W * Ns, "", "0"] - [Taxes, Taxes, -TXs, "", +TXd, "0"] - [Money, Change in money stock, +d(Hh), "", -d(Hs), "0"] - [Sum, Sum, "0", "0", "0", "0"] - sequence: id: transaction-flow-sequence title: SIM transaction flow sequence source: kind: matrix matrixCellId: transactions-flow description: Sequence view generated from the SIM transactions-flow matrix. more: | This is the same transactions-flow matrix read **column by column** as each sector's budget identity. Following the columns shows the _circuit_ of money within a single period: government spending and wage payments inject money; consumption and taxes withdraw it; whatever households do not spend or pay in tax accumulates as the change in their money stock. - sequence: id: equation-dependency-graph title: SIM equation dependency graph source: kind: dependency modelId: sim showAccountingStrips: true showExogenous: false stripMapping: transactionMatrixCellId: transactions-flow balanceMatrixCellId: balance-sheet description: Dependency view of the SIM equations. Lagged stock terms keep the dynamic stock accumulation outside same-period dependency cycles. - equations: id: equations-sim title: SIM equations modelId: sim collapsed: true more: | These eleven equations are model SIM exactly as written in Godley & Lavoie (equations 3.1–3.11). They fall into a few natural groups: - **Equilibrium-by-assumption** (`Cs = Cd`, `Gs = Gd`, `TXs = TXd`, `Ns = Nd`): supply equals demand not through a market-clearing price but by _quantity adjustment_ — firms simply produce whatever is demanded. There are no prices and no rationing in SIM. - **Income and taxes**: disposable income `YD = W * Ns - TXs`, with taxes levied as a flat fraction of the wage bill, `TXd = theta * W * Ns`. - **The behavioural core — the consumption function** `Cd = alpha1 * YD + alpha2 * lag(Hh)`. Households consume a fraction `alpha1` out of _current_ disposable income and a smaller fraction `alpha2` out of the _wealth_ they accumulated last period (G&L require `0 < alpha2 < alpha1 < 1`). This single equation is what gives SIM its dynamics: spending out of past wealth is the channel through which the economy converges to a steady state. - **Stock accumulation**: `Hh = lag(Hh) + YD - Cd` (households bank what they do not consume) and `Hs = lag(Hs) + Gd - TXd` (the government deficit issues new money). - **Output and employment**: `Y = Cs + Gs` and `Nd = Y / W`. **The redundant equation.** Counting unknowns against equations, the system has one equation more than it strictly needs. Godley & Lavoie show that the missing / redundant relation is `Hh = Hs` (equivalently `d(Hh) = d(Hs)`): household money holdings always equal government-issued money. It is never imposed on the solver — it _emerges_, and checking it holds is how you know the accounting is watertight. rows: - [TXs, TXd, "Taxes paid by households.", "", "", identity, eq-0-txs] - [YD, W * Ns - TXs, "Disposable income.", "", "", definition, eq-1-yd] - [Cd, alpha1 * YD + alpha2 * lag(Hh), "Household consumption demand.", "", "", behavioral, eq-2-cd] - [Hh, YD - Cd + lag(Hh), "Household money holdings.", "", "", accumulation, eq-3-hh] - [Ns, Nd, "Labor supply.", "", "", identity, eq-4-ns] - [Nd, Y / W, "Labor demand.", "", "", definition, eq-5-nd] - [Cs, Cd, "Consumption supply.", "", "", identity, eq-6-cs] - [Gs, Gd, "Government goods supply.", "", "", identity, eq-7-gs] - [Y, Cs + Gs, "Output.", "", "", identity, eq-8-y] - [TXd, theta * W * Ns, "Tax demand.", "", "", definition, eq-9-txd] - [Hs, Gd - TXd + lag(Hs), "Government money supply.", "", "", accumulation, eq-10-hs] - solver: id: solver-sim title: Solver options modelId: sim collapsed: true more: | SIM is solved period by period: lagged stocks (`lag(Hh)`, `lag(Hs)`) 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** `Hh` and `Hs` implement the redundant-equation check described above. Because one equation in the system is redundant, the equality `Hh = Hs` is _not_ used to find the solution — instead the solver verifies, to tolerance `hiddenTolerance`, that the independently computed household money stock and government money stock agree every period. If they ever diverged, the model's accounting would be leaking. This is the numerical embodiment of Godley & Lavoie's "watertight" claim. method: broyden tolerance: "1e-8" maxIterations: 350 defaultInitialValue: "1e-15" hiddenLeftVariable: Hh hiddenRightVariable: Hs hiddenTolerance: "1e-5" relativeHiddenTolerance: false - externals: id: parameters-sim title: Externals modelId: sim collapsed: true more: | These are the exogenous parameters and policy levers of SIM, with the values used throughout Godley & Lavoie chapter 3: - `Gd` = 20 — government expenditure, the model's main exogenous driver and the variable shocked in Scenario 1. - `W` = 1 — the wage rate. With `W = 1`, output `Y`, the wage bill, and employment `N` all coincide numerically, which keeps the arithmetic transparent. - `alpha1` = 0.6 — propensity to consume out of current disposable income. - `alpha2` = 0.4 — propensity to consume out of accumulated money wealth. - `theta` = 0.2 — the (flat) tax rate. Two ratios determined by these numbers govern the whole solution. The **long-run fiscal multiplier** is `1 / theta = 5`, so steady-state output is `Y* = Gd / theta = 100`. The **steady-state wealth norm** is `Hh* / YD* = (1 - alpha1) / alpha2 = 1`, so in the stationary state households hold money equal to one year of disposable income. rows: - [Gd, 20, "Government expenditure.", "", "", ext-0-gd] - [W, 1, "Wage rate.", "", "", ext-1-w] - [alpha1, 0.6, "Propensity to consume from current disposable income."] - [alpha2, 0.4, "Propensity to consume from lagged money holdings."] - [theta, 0.2, "Tax rate."] - initial-values: id: initial-values-sim title: Initial values modelId: sim collapsed: true rows: [] - run: id: baseline-run title: Baseline run mode: baseline periods: 60 resultKey: sim_baseline sourceModelId: sim description: Baseline SIM run with government expenditure fixed at 20. more: | The baseline integrates the model forward for 60 periods starting from zero stocks, with `Gd` held constant at 20. With no money inherited from the past, the economy starts well below its steady state and climbs toward it. This run illustrates Godley & Lavoie's central dynamic result: a constant injection of government money drives the economy to a **stationary state** in which stocks stop growing and the government budget is balanced. - chart: id: chart-1 title: Baseline income, consumption, and money more: | In the baseline, government spending `Gd` is held constant at `20`. Output `Y` and disposable income `YD` rise quickly at first and then converge to a steady state. The mechanism is the consumption-out-of-wealth term `alpha2 * lag(Hh)`: as households accumulate money, that extra spending shrinks each period until, in the limit, saving out of income exactly offsets nothing more is added — `Hh` flattens. The numbers match the chapter exactly. Output converges to `Y* = Gd / theta = 20 / 0.2 = 100`; disposable income to `YD* = Y* * (1 - theta) = 80`; consumption to `Cd = YD* = 80`; and the money stock to `Hh* = 80` (one year of disposable income, the wealth norm `(1 - alpha1) / alpha2 = 1`). In the steady state `TXd = Gd = 20`: **the budget is balanced and no new money is created.** variables: - Y - YD - Cd - Hh axisMode: shared referenceTrace: none timeRangeInclusive: - 1 - 60 sourceRunCellId: baseline-run - table: id: table-1 title: Baseline SIM summary more: | The table makes the convergence concrete. Watch `Hh` flatten as the increments `YD - Cd` shrink toward zero, and watch `TXd` rise from below `Gd` (a deficit that creates money while the economy is growing) up to exactly `Gd = 20` (the balanced budget of the stationary state). The gap `Gd - TXd` in any row _is_ that period's newly created money — and it is what households are simultaneously saving. variables: - Y - YD - Cd - Hh - TXd - Gd sourceRunCellId: baseline-run - markdown: id: scenario-note title: Scenario 1 source: Scenario 1 increases government expenditure `Gd` from `20` to `30` from period 5 through period 60, matching the introductory SIM scenario in the R `sfcr` README. more: | Scenario 1 reproduces Godley & Lavoie's first SIM experiment: a permanent **step increase in the fiscal stance**. Holding the tax _rate_ `theta` fixed and raising government spending `Gd` from 20 to 30 moves the economy to a new, higher stationary state. It is the cleanest way to see the long-run multiplier `1/theta` at work. - run: id: scenario-run title: "Scenario 1: government spending shock" mode: scenario scenario: shocks: - rangeInclusive: - 5 - 60 variables: Gd: kind: constant value: 30 startPeriodInclusive: 5 endPeriodInclusive: 60 baselineRunCellId: baseline-run periods: 60 resultKey: sim_scenario_gd_30 sourceModelId: sim description: Scenario run with government expenditure raised to 30 from period 5 onward. more: | The scenario re-solves the model with `Gd` stepped up to 30 from period 5 onward, and is plotted against the baseline as a reference. Because only `Gd` changes, the contrast isolates the pure effect of fiscal policy on output, income, and private wealth. - chart: id: scenario-chart title: Scenario 1 output and fiscal variables more: | Scenario 1 raises `Gd` from `20` to `30` starting in period 5. The economy transitions to a **new, higher** steady state. The long-run multiplier is `1 / theta = 5`, so the `+10` rise in spending lifts steady-state output by `+50`: `Y*` moves from `100` to `Gd / theta = 30 / 0.2 = 150`. Disposable income settles at `YD* = 120` and the household money stock `Hh` rises to a correspondingly higher level of `120`. Note the two-speed adjustment Godley & Lavoie emphasise: the _impact_ multiplier in the shock period is much smaller than the eventual `5`, because consumption out of newly accumulated wealth feeds through only gradually. Output approaches the new stationary state asymptotically rather than jumping to it. variables: - Y - YD - Cd - Hh - Gd axisMode: shared referenceTrace: baseline timeRangeInclusive: - 1 - 60 sourceRunCellId: scenario-run - table: id: scenario-table title: Scenario 1 SIM summary more: | Compare the closing rows against the baseline: every stationary-state value is exactly `1.5x` its baseline counterpart, since both `Y* = Gd / theta` and `Hh* = YD*` scale linearly with `Gd`. The budget rebalances at the higher level — `TXd` climbs to meet the new `Gd = 30` — confirming that a _permanent_ change in the fiscal stance leaves the long-run budget balanced while permanently raising output, income, and the stock of money. variables: - Gd - Y - YD - Cd - Hh - TXd - Hs sourceRunCellId: scenario-run - markdown: id: growth-note title: "Appendix 3.4: Government deficits in a growing economy" source: | Appendix 3.4 of Godley and Lavoie asks whether sustained government deficits are a problem. In a _growing_ economy they are normal: if government expenditure grows at a constant rate `g`, all stocks and flows eventually grow at `g`, the stock of money (government debt) grows at `g` too, and the government therefore runs a _permanent_ deficit while the debt-to-income ratio settles at a constant level. Here `Gd` grows at 3% per period, starting from the stationary baseline. more: | This section reproduces **Appendix 3.4** of Godley & Lavoie, _"Government deficits in a growing economy"_. The stationary state studied above has a balanced budget (`TXd = Gd`) and a _constant_ money stock — but a stationary economy is a special case. The appendix asks the natural follow-up question: **are continual government deficits sustainable?** The answer is that in a growing economy a permanent deficit is not just sustainable, it is _required_. The reasoning is pure stock-flow accounting: - private wealth is held entirely as government money, so the stock of money _is_ the public debt: `Hh = Hs`; - if real activity grows at a steady rate `g`, the wealth households wish to hold grows at `g`, so the stock of money/debt must grow at `g`; - a _growing_ stock of government money can only be created by a flow of new money each period — that flow is exactly the government deficit `Gd - TXd`. So a steadily growing economy needs a steadily positive deficit. What matters for sustainability is not the deficit itself but the **debt-to-income ratio**, which settles to a constant. - run: id: growth-run title: "Appendix 3.4: government spending growing at 3%" mode: scenario scenario: shocks: - rangeInclusive: - 1 - 60 variables: Gd: kind: series values: - 20 - 20.6 - 21.218 - 21.8545 - 22.5102 - 23.1855 - 23.881 - 24.5975 - 25.3354 - 26.0955 - 26.8783 - 27.6847 - 28.5152 - 29.3707 - 30.2518 - 31.1593 - 32.0941 - 33.057 - 34.0487 - 35.0701 - 36.1222 - 37.2059 - 38.3221 - 39.4717 - 40.6559 - 41.8756 - 43.1318 - 44.4258 - 45.7586 - 47.1313 - 48.5452 - 50.0016 - 51.5017 - 53.0467 - 54.6381 - 56.2772 - 57.9656 - 59.7045 - 61.4957 - 63.3405 - 65.2408 - 67.198 - 69.2139 - 71.2903 - 73.429 - 75.6319 - 77.9009 - 80.2379 - 82.645 - 85.1244 - 87.6781 - 90.3085 - 93.0177 - 95.8082 - 98.6825 - 101.643 - 104.6923 - 107.833 - 111.068 - 114.4001 startPeriodInclusive: 1 endPeriodInclusive: 60 baselineRunCellId: baseline-run periods: 60 resultKey: sim_growth_gd_3pc sourceModelId: sim description: Government expenditure grows at 3% per period, taking the economy onto a steady-growth path with a permanent deficit and a constant debt-to-income ratio. more: | To put a growing economy inside SIM — which has no intrinsic growth engine — we let the only exogenous driver, government expenditure `Gd`, grow at a constant `g = 3%` per period (a `series` shock), starting from the stationary baseline state. Because every flow in SIM is ultimately pinned to `Gd`, this pulls the whole system onto a **steady-growth path** on which output, income, taxes and the money stock all eventually grow at `3%`. - chart: id: growth-chart title: "Appendix 3.4: output, spending, taxes, and money under growth" more: | Government spending `Gd` grows at `3%` per period, and after an initial adjustment `Y`, `TXd` and the money stock `Hh` all settle onto the same `3%` growth path. Two features reproduce Godley & Lavoie's Appendix 3.4 point: - **The budget never balances.** `Gd` stays permanently above `TXd`: the government runs a deficit in _every_ period. That deficit is precisely the new money being issued to satisfy households' growing demand for wealth. - **The deficit is benign.** Although the debt stock `Hh` rises without bound in absolute terms, it grows at the same `3%` as income, so the debt-to-income ratio `Hh / YD` is constant. With `g = 0.03` it settles near `(1 - alpha1)(1 + g) / (g + alpha2) ≈ 0.96` — slightly _below_ the stationary norm of `1`, because in a growing economy households are forever rebuilding wealth against a rising income. variables: - Y - Gd - TXd - Hh axisMode: shared referenceTrace: none timeRangeInclusive: - 1 - 60 sourceRunCellId: growth-run - table: id: growth-table title: "Appendix 3.4: growing-economy summary" more: | Read down the columns and two ratios reveal the steady-growth state. First, the period-to-period ratio of any column (e.g. `Y` to its previous value) approaches `1.03`, confirming a common `3%` growth rate. Second, `Gd - TXd` is positive in every row — the permanent deficit — yet `Hh / YD` (and `Hs`, since `Hh = Hs`) holds steady. This is the heart of Appendix 3.4: **a constant deficit ratio and a constant debt ratio are perfectly consistent with one another once the economy is growing.** variables: - Gd - TXd - Y - YD - Hh - Hs sourceRunCellId: growth-run