A Simple SFC Model (ODE Form)
Three-Sector Stock-Flow Consistent Model with Differential Equations
Brief Description
This is a simple SFC model that consists of three sectors: firms, households, and banks. This version presents the model using differential equations suitable for implementation in CircuitJS1.
- Firms undertake investment by using retained profits and loans
- A part of firms’ profits is distributed to households
- Households accumulate savings in the form of deposits
- Banks provide firm loans by creating deposits
- Banks’ profits are distributed to households
In the model, loans are endogenously created when firms receive credit from banks. The model is calibrated using data for the US economy over the period 1960-2010.
Balance Sheet Matrix
| Households | Firms | Commercial banks | Total | |
|---|---|---|---|---|
| Deposits | +D | -D | 0 | |
| Loans | -L | +L | 0 | |
| Capital | +K | +K | ||
| Total (net worth) | +D | +VF | 0 | +K |
Key insight: Every financial asset has a corresponding liability. Banks have zero net worth.
Transactions Flow Matrix
| Households | Firms | Banks | Total | |||
|---|---|---|---|---|---|---|
| Current | Capital | Current | Capital | |||
| Consumption | -C | +C | 0 | |||
| Investment | +I | -I | 0 | |||
| Wages | +W | -W | 0 | |||
| Firms’ profits | +DP | -TP | +RP | 0 | ||
| Banks’ profits | +BP | -BP | 0 | |||
| Interest on deposits | +intD·D | -intD·D | 0 | |||
| Interest on loans | -intL·L | +intL·L | 0 | |||
| Change in deposits | -dD/dt | +dD/dt | 0 | |||
| Change in loans | +dL/dt | -dL/dt | 0 | |||
| Total | 0 | 0 | 0 | 0 | 0 | 0 |
Flow Definitions
Income Flows
Wages: Households ← Firms \[W = s_W \cdot Y\]
Consumption: Households → Firms \[C = c_1 \cdot W + c_2 \cdot Y_C + c_3 \cdot D\]
Investment: Internal to Firms (financed by retained profits + loans) \[I = g_K \cdot K\]
Profit Flows
Total Profits: Firms generate \[TP = Y - W - int_L \cdot L\]
Retained Profits: Kept by Firms \[RP = s_F \cdot TP\]
Distributed Profits: Firms → Households \[DP = TP - RP\]
Interest Flows
Interest on Loans: Firms → Banks \[Int_L = int_L \cdot L\]
Interest on Deposits: Banks → Households \[Int_D = int_D \cdot D\]
Bank Profits
- Bank Profits: Banks → Households \[BP = Int_L - Int_D = int_L \cdot L - int_D \cdot D\]
Capital Income
- Household Capital Income: \[Y_C = DP + BP + Int_D\]
Differential Equations (Stock Dynamics)
The stock variables evolve according to these differential equations:
Household Deposits
\[\frac{dD}{dt} = W + Y_C - C\]
Expanding: \[\frac{dD}{dt} = W + DP + BP + int_D \cdot D - C\]
Firm Loans
\[\frac{dL}{dt} = I - RP\]
Firms borrow the difference between investment and retained profits.
Capital Stock
\[\frac{dK}{dt} = I\]
Capital accumulates through investment.
Firm Net Worth
\[\frac{dV_F}{dt} = RP - int_L \cdot L + \frac{dK}{dt} - \frac{dL}{dt}\]
Simplifying: \[\frac{dV_F}{dt} = TP - DP = RP\]
Complete System of Equations
Parameters
| Symbol | Description | Value |
|---|---|---|
| sW | Wage share of output | 0.60 |
| c1 | Propensity to consume from wages | 0.90 |
| c2 | Propensity to consume from capital income | 0.75 |
| c3 | Propensity to consume from deposits | 0.47 |
| gK | Investment rate (growth of capital) | 0.03 |
| sF | Retention rate of profits | 0.18 |
| intL | Interest rate on loans | 0.05 |
| intD | Interest rate on deposits | 0.02 |
Algebraic Equations (Flows)
\[Y = C + I\]
\[W = s_W \cdot Y\]
\[C = c_1 \cdot W + c_2 \cdot Y_C + c_3 \cdot D\]
\[I = g_K \cdot K\]
\[TP = Y - W - int_L \cdot L\]
\[RP = s_F \cdot TP\]
\[DP = TP - RP\]
\[BP = int_L \cdot L - int_D \cdot D\]
\[Y_C = DP + BP + int_D \cdot D\]
Differential Equations (Stocks)
\[\frac{dD}{dt} = W + DP + BP + int_D \cdot D - C\]
\[\frac{dL}{dt} = I - RP\]
\[\frac{dK}{dt} = I\]
Redundant Equation (Consistency Check)
\[D = L\]
This identity confirms stock-flow consistency: all deposits are backed by loans.
CircuitJS1 Implementation
This model can be implemented in CircuitJS1 using:
- Godley Tables: For tracking balance sheets of each sector
- ODE Elements: For integrating the stock dynamics
- Arithmetic Elements: For computing flows from stocks
Stock Variables (use ODE/Integrator elements)
D- Household depositsL- Firm loans
K- Capital stock
Flow Computations (use arithmetic elements)
Connect the differential equations using multipliers, adders, and the stock values.
Model Dynamics
Steady State
At steady state, all growth rates are equal: \[g_Y = g_K = g_D = g_L = g\]
The steady-state leverage ratio: \[\frac{L}{K} = \frac{g_K}{s_F \cdot (1 - s_W) - int_L \cdot (L/K)}\]
Stability Analysis
The model exhibits stable dynamics when:
- Consumption propensities ensure positive saving
- Investment doesn’t exceed available financing
- Interest rates don’t cause explosive debt dynamics
Key Insights
Endogenous Money Creation
\[\frac{dL}{dt} = \frac{dD}{dt}\]
When banks extend loans, they simultaneously create deposits. This is the essence of endogenous money.
Stock-Flow Consistency
Every flow has a source and destination:
- Horizontal: Financial assets = Financial liabilities
- Vertical: All income is spent or saved
- Quadruple entry: Each transaction affects 4 cells in the matrices
Profit Distribution
Firm profits flow to: 1. Retained profits (RP) → Internal financing for investment 2. Distributed profits (DP) → Household income
Bank profits flow entirely to households (as bank owners).
References
- Godley, W. & Lavoie, M. (2007) Monetary Economics: An Integrated Approach
- Caverzasi, E. & Godin, A. (2015) “Post-Keynesian stock-flow-consistent modelling: a survey”
- Nikiforos, M. & Zezza, G. (2017) “Stock-flow Consistent Macroeconomic Models: A Survey”
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