A Simple SFC Model

Three-Sector Stock-Flow Consistent Model

Authors

Affiliations

Yannis Dafermos

University of the West of England

Maria Nikolaidi

University of Greenwich

Published

August 1, 2019

Brief Description

This is a simple SFC model that consists of three sectors: firms, households, and banks.

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

The balance sheet matrix shows the stock positions of all sectors:

	Households	Firms	Commercial banks	Total
Deposits	+D		-D	0
Loans		-L	+L	0
Capital		+K		+K
Total (net worth)	+D	+V_F	0	+K

Key observations:

Household wealth consists entirely of deposits
Firm net worth (V_F) equals capital minus loans: V_F = K - L
Banks have zero net worth (assets = liabilities)
Total economy net worth equals physical capital stock

Transactions Flow Matrix

The transactions flow matrix shows all flows between sectors:

	Households	Firms		Commercial 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	+int_DD_-1			-int_DD_-1		0
Interest on loans		-int_LL_-1		+int_LL_-1		0
Change in deposits	-ΔD				+ΔD	0
Change in loans			+ΔL		-ΔL	0
Total	0	0	0	0	0	0

Key observations:

Every row sums to zero (stock-flow consistency)
Every column sums to zero (budget constraints satisfied)
Current and capital accounts are separated for firms and banks

Model Equations

Households

\[\text{Wage income: } W = s_w Y \tag{1}\]

\[\text{Capital income: } Y_C = DP + BP + int_D D_{-1} \tag{2}\]

\[\text{Consumption: } C = c_1 W_{-1} + c_2 Y_{C-1} + c_3 D_{-1} \tag{3}\]

\[\text{Deposits (identity): } D = D_{-1} + W + Y_C - C \tag{4}\]

Firms

\[\text{Output: } Y = C + I \tag{5}\]

\[\text{Total profits (identity): } TP = Y - W - int_L L_{-1} \tag{6}\]

\[\text{Retained profits: } RP = s_F TP_{-1} \tag{7}\]

\[\text{Distributed profits (identity): } DP = TP - RP \tag{8}\]

\[\text{Investment: } I = s_K K_{-1} \tag{9}\]

\[\text{Capital stock: } K = K_{-1} + I \tag{10}\]

\[\text{Loans (identity): } L = L_{-1} + I - RP \tag{11}\]

Banks

\[\text{Profits (identity): } BP = int_L L_{-1} - int_D D_{-1} \tag{12}\]

\[\text{Deposits (redundant identity): } D_{red} = L \tag{13}\]

Auxiliary Equations

\[\text{Potential output: } Y^* = vK \tag{14}\]

\[\text{Capacity utilisation: } u = Y / Y^* \tag{15}\]

\[\text{Growth rate of output: } g_Y = (Y - Y_{-1})/Y_{-1} \tag{16}\]

\[\text{Leverage ratio: } lev = L / K \tag{17}\]

Symbols and Values

Parameters

Symbol	Description	Value/Calibration
c₁	Propensity to consume out of wage income	0.9
c₂	Propensity to consume out of capital income	0.75
c₃	Propensity to consume out of deposits	0.474
g_K	Growth rate of capital	US 1960-2010 mean value of g_Y
int_D	Interest rate on deposits	US 1960-2010 mean value
int_L	Interest rate on loans	US 1960-2010 mean value
s_F	Retention rate of firms	0.179
s_W	Wage share	US 1960-2010 mean value
v	Capital productivity	Calculated using equations (14) and (15)

Endogenous Variables

Symbol	Description	Calculation
W	Wage income of households	Equation (1)
Y_C	Capital income of households	Equation (2)
C	Consumption expenditures	Equation (3)
D	Deposits	Equation (4)
Y	Output	US 1960 value (trillion 2009 US$)
TP	Total profits of firms	Equation (6)
RP	Retained profits	Equation (7)
DP	Distributed profits	Equation (8)
I	Investment	Equation (9)
K	Capital stock	US 1960 value (trillion 2009 US$)
L	Loans	US 1960 value (trillion 2009 US$)
BP	Profits of banks	Equation (12)
D_red	Deposits (redundant)	Equation (13)
Y*	Potential output	Equation (14)
u	Capacity utilisation	US 1960 value
g_Y	Growth rate of output	US 1960-2010 mean value
lev	Leverage ratio	Equation (17)

R Implementation

Step 1: Setup

Clear the workspace and set the number of time periods:

rm(list=ls(all=TRUE))
T <- 51  # 51 periods for 1960-2010

Step 2: Load Data

Download US data for 1960-2010 (from FRED and BIS):

Data <- read.csv("Data.csv")

# To estimate the mean of a variable:
mean(Data[,c("g_Y")])

Step 3: Initialize Variables

Create vectors for all endogenous variables:

# Endogenous variables 
W <- vector(length=T)    
Y_C <- vector(length=T)    
CO <- vector(length=T)    
D <- vector(length=T)      
Y <- vector(length=T)      
TP <- vector(length=T)    
RP <- vector(length=T)    
DP <- vector(length=T)    
I <- vector(length=T)    
K <- vector(length=T)    
L <- vector(length=T)    
BP <- vector(length=T)    
D_red <- vector(length=T)    
Y_star <- vector(length=T)  # auxiliary variable   
u <- vector(length=T)       # auxiliary variable   
g_Y <- vector(length=T)     # auxiliary variable   
lev <- vector(length=T)     # auxiliary variable

Step 4: Set Parameters and Initial Values

# Parameters 
for (i in 1:T) { 
  if (i == 1) { 
    for (iterations in 1:10){ 
      c_1 <- 0.9  
      c_2 <- 0.75  
      c_3 <- 0.473755074
      g_K <- mean(Data[,c("g_Y")])  
      int_D <- mean(Data[,c("int_D")])  
      int_L <- mean(Data[,c("int_L")])  
      s_F <- 0.17863783
      s_W <- mean(Data[,c("s_W")])  
      v <- Y[i]/(K[i]*u[i])  
      
      # Initial values 
      W[i] <- s_W*Y[i]  
      Y_C[i] <- DP[i]+BP[i]+int_D*(D[i]/(1+g_K))  
      CO[i] <- Y[i]-I[i]  
      D[i] <- L[i]   
      Y[i] <- Data[1,c("Y")]  
      TP[i] <- Y[i]-W[i]-int_L*(L[i]/(1+g_K))  
      RP[i] <- s_F*TP[i]/(1+g_K) 
      DP[i] <- TP[i]-RP[i]  
      I[i] <- (g_K/(1+g_K))*K[i]  
      K[i] <- Data[1,c("K")]              
      L[i] <- Data[1,c("L")]  
      BP[i] <- int_L*(L[i]/(1+g_K))-int_D*(D[i]/(1+g_K))  
      D_red[i] <- L[i]  
      Y_star[i] <- v*K[i]  
      u[i] <- Data[1,c("u")]  
      g_Y[i] <- g_K  
      lev[i] <- L[i]/K[i]  
    } 
  }

Step 5: Run the Model

  # Equations 
  else { 
    for (iterations in 1:10){ 
      # Households 
      W[i] <- s_W*Y[i] 
      Y_C[i] <- DP[i]+BP[i]+int_D*D[i-1] 
      CO[i] <- c_1*W[i-1]+c_2*Y_C[i-1]+c_3*D[i-1] 
      D[i] <- D[i-1]+W[i]+Y_C[i]-CO[i] 
      
      # Firms 
      Y[i] <- CO[i]+I[i] 
      TP[i] <- Y[i]-W[i]-int_L*L[i-1] 
      RP[i] <- s_F*TP[i-1] 
      DP[i] <- TP[i]-RP[i] 
      I[i] <- g_K*K[i-1] 
      K[i] <- K[i-1]+I[i] 
      L[i] <- L[i-1]+I[i]-RP[i] 
      
      # Banks 
      BP[i] <- int_L*L[i-1]-int_D*D[i-1] 
      D_red[i] <- L[i] 
      
      # Auxiliary equations 
      Y_star[i] <- v*K[i] 
      u[i] <- Y[i]/Y_star[i] 
      g_Y[i] <- (Y[i]-Y[i-1])/Y[i-1] 
      lev[i] <- L[i]/K[i] 
    } 
  } 
}

Key Model Insights

Endogenous Money Creation

In this model, money (deposits) is created endogenously when banks extend loans to firms:

Firms need external financing: I - RP (investment minus retained profits)
Banks create deposits when they grant loans
The redundant identity D_red = L confirms balance sheet consistency

Stock-Flow Consistency

The model maintains perfect accounting consistency:

Horizontal consistency: Every financial asset has a corresponding liability
Vertical consistency: All flows are tracked from source to destination
Quadruple entry: Each transaction affects four cells in the matrices

Steady State Properties

In the baseline scenario, the model achieves a steady state where:

Economic growth equals the mean US growth rate (1960-2010)
All ratios (leverage, capacity utilisation) remain constant
The system is fully balanced

References

This model and documentation are based on teaching materials developed for SFC modeling courses. For the theoretical foundations, see:

Godley, W. & Lavoie, M. (2007) Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth
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”

This page presents the simple SFC model developed by Yannis Dafermos and Maria Nikolaidi for educational purposes.

--- title: "A Simple SFC Model" subtitle: "Three-Sector Stock-Flow Consistent Model" author: - name: "Yannis Dafermos" affiliation: "University of the West of England" - name: "Maria Nikolaidi" affiliation: "University of Greenwich" date: "August 2019" toc: true --- ## Brief Description This is a simple SFC model that consists of three sectors: **firms**, **households**, and **banks**. - **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 The balance sheet matrix shows the stock positions of all sectors: | | Households | Firms | Commercial banks | Total | |---|:---:|:---:|:---:|:---:| | Deposits | +D | | -D | 0 | | Loans | | -L | +L | 0 | | Capital | | +K | | +K | | **Total (net worth)** | **+D** | **+V~F~** | **0** | **+K** | **Key observations:** - Household wealth consists entirely of deposits - Firm net worth (V~F~) equals capital minus loans: V~F~ = K - L - Banks have zero net worth (assets = liabilities) - Total economy net worth equals physical capital stock ## Transactions Flow Matrix The transactions flow matrix shows all flows between sectors: | | Households | Firms | | Commercial 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 | +int~D~D~-1~ | | | -int~D~D~-1~ | | 0 | | Interest on loans | | -int~L~L~-1~ | | +int~L~L~-1~ | | 0 | | Change in deposits | -ΔD | | | | +ΔD | 0 | | Change in loans | | | +ΔL | | -ΔL | 0 | | **Total** | **0** | **0** | **0** | **0** | **0** | **0** | **Key observations:** - Every row sums to zero (stock-flow consistency) - Every column sums to zero (budget constraints satisfied) - Current and capital accounts are separated for firms and banks ## Model Equations ### Households $$\text{Wage income: } W = s_w Y \tag{1}$$ $$\text{Capital income: } Y_C = DP + BP + int_D D_{-1} \tag{2}$$ $$\text{Consumption: } C = c_1 W_{-1} + c_2 Y_{C-1} + c_3 D_{-1} \tag{3}$$ $$\text{Deposits (identity): } D = D_{-1} + W + Y_C - C \tag{4}$$ ### Firms $$\text{Output: } Y = C + I \tag{5}$$ $$\text{Total profits (identity): } TP = Y - W - int_L L_{-1} \tag{6}$$ $$\text{Retained profits: } RP = s_F TP_{-1} \tag{7}$$ $$\text{Distributed profits (identity): } DP = TP - RP \tag{8}$$ $$\text{Investment: } I = s_K K_{-1} \tag{9}$$ $$\text{Capital stock: } K = K_{-1} + I \tag{10}$$ $$\text{Loans (identity): } L = L_{-1} + I - RP \tag{11}$$ ### Banks $$\text{Profits (identity): } BP = int_L L_{-1} - int_D D_{-1} \tag{12}$$ $$\text{Deposits (redundant identity): } D_{red} = L \tag{13}$$ ### Auxiliary Equations $$\text{Potential output: } Y^* = vK \tag{14}$$ $$\text{Capacity utilisation: } u = Y / Y^* \tag{15}$$ $$\text{Growth rate of output: } g_Y = (Y - Y_{-1})/Y_{-1} \tag{16}$$ $$\text{Leverage ratio: } lev = L / K \tag{17}$$ ## Symbols and Values ### Parameters | Symbol | Description | Value/Calibration | |--------|-------------|-------------------| | c~1~ | Propensity to consume out of wage income | 0.9 | | c~2~ | Propensity to consume out of capital income | 0.75 | | c~3~ | Propensity to consume out of deposits | 0.474 | | g~K~ | Growth rate of capital | US 1960-2010 mean value of g~Y~ | | int~D~ | Interest rate on deposits | US 1960-2010 mean value | | int~L~ | Interest rate on loans | US 1960-2010 mean value | | s~F~ | Retention rate of firms | 0.179 | | s~W~ | Wage share | US 1960-2010 mean value | | v | Capital productivity | Calculated using equations (14) and (15) | ### Endogenous Variables | Symbol | Description | Calculation | |--------|-------------|-------------| | W | Wage income of households | Equation (1) | | Y~C~ | Capital income of households | Equation (2) | | C | Consumption expenditures | Equation (3) | | D | Deposits | Equation (4) | | Y | Output | US 1960 value (trillion 2009 US\$) | | TP | Total profits of firms | Equation (6) | | RP | Retained profits | Equation (7) | | DP | Distributed profits | Equation (8) | | I | Investment | Equation (9) | | K | Capital stock | US 1960 value (trillion 2009 US\$) | | L | Loans | US 1960 value (trillion 2009 US\$) | | BP | Profits of banks | Equation (12) | | D~red~ | Deposits (redundant) | Equation (13) | | Y* | Potential output | Equation (14) | | u | Capacity utilisation | US 1960 value | | g~Y~ | Growth rate of output | US 1960-2010 mean value | | lev | Leverage ratio | Equation (17) | ## R Implementation ### Step 1: Setup Clear the workspace and set the number of time periods: ```r rm(list=ls(all=TRUE)) T <- 51 # 51 periods for 1960-2010 ``` ### Step 2: Load Data Download US data for 1960-2010 (from FRED and BIS): ```r Data <- read.csv("Data.csv") # To estimate the mean of a variable: mean(Data[,c("g_Y")]) ``` ### Step 3: Initialize Variables Create vectors for all endogenous variables: ```r # Endogenous variables W <- vector(length=T) Y_C <- vector(length=T) CO <- vector(length=T) D <- vector(length=T) Y <- vector(length=T) TP <- vector(length=T) RP <- vector(length=T) DP <- vector(length=T) I <- vector(length=T) K <- vector(length=T) L <- vector(length=T) BP <- vector(length=T) D_red <- vector(length=T) Y_star <- vector(length=T) # auxiliary variable u <- vector(length=T) # auxiliary variable g_Y <- vector(length=T) # auxiliary variable lev <- vector(length=T) # auxiliary variable ``` ### Step 4: Set Parameters and Initial Values ```r # Parameters for (i in 1:T) { if (i == 1) { for (iterations in 1:10){ c_1 <- 0.9 c_2 <- 0.75 c_3 <- 0.473755074 g_K <- mean(Data[,c("g_Y")]) int_D <- mean(Data[,c("int_D")]) int_L <- mean(Data[,c("int_L")]) s_F <- 0.17863783 s_W <- mean(Data[,c("s_W")]) v <- Y[i]/(K[i]*u[i]) # Initial values W[i] <- s_W*Y[i] Y_C[i] <- DP[i]+BP[i]+int_D*(D[i]/(1+g_K)) CO[i] <- Y[i]-I[i] D[i] <- L[i] Y[i] <- Data[1,c("Y")] TP[i] <- Y[i]-W[i]-int_L*(L[i]/(1+g_K)) RP[i] <- s_F*TP[i]/(1+g_K) DP[i] <- TP[i]-RP[i] I[i] <- (g_K/(1+g_K))*K[i] K[i] <- Data[1,c("K")] L[i] <- Data[1,c("L")] BP[i] <- int_L*(L[i]/(1+g_K))-int_D*(D[i]/(1+g_K)) D_red[i] <- L[i] Y_star[i] <- v*K[i] u[i] <- Data[1,c("u")] g_Y[i] <- g_K lev[i] <- L[i]/K[i] } } ``` ### Step 5: Run the Model ```r # Equations else { for (iterations in 1:10){ # Households W[i] <- s_W*Y[i] Y_C[i] <- DP[i]+BP[i]+int_D*D[i-1] CO[i] <- c_1*W[i-1]+c_2*Y_C[i-1]+c_3*D[i-1] D[i] <- D[i-1]+W[i]+Y_C[i]-CO[i] # Firms Y[i] <- CO[i]+I[i] TP[i] <- Y[i]-W[i]-int_L*L[i-1] RP[i] <- s_F*TP[i-1] DP[i] <- TP[i]-RP[i] I[i] <- g_K*K[i-1] K[i] <- K[i-1]+I[i] L[i] <- L[i-1]+I[i]-RP[i] # Banks BP[i] <- int_L*L[i-1]-int_D*D[i-1] D_red[i] <- L[i] # Auxiliary equations Y_star[i] <- v*K[i] u[i] <- Y[i]/Y_star[i] g_Y[i] <- (Y[i]-Y[i-1])/Y[i-1] lev[i] <- L[i]/K[i] } } } ``` ## Key Model Insights ### Endogenous Money Creation In this model, money (deposits) is created endogenously when banks extend loans to firms: 1. Firms need external financing: I - RP (investment minus retained profits) 2. Banks create deposits when they grant loans 3. The redundant identity D~red~ = L confirms balance sheet consistency ### Stock-Flow Consistency The model maintains perfect accounting consistency: - **Horizontal consistency**: Every financial asset has a corresponding liability - **Vertical consistency**: All flows are tracked from source to destination - **Quadruple entry**: Each transaction affects four cells in the matrices ### Steady State Properties In the baseline scenario, the model achieves a steady state where: - Economic growth equals the mean US growth rate (1960-2010) - All ratios (leverage, capacity utilisation) remain constant - The system is fully balanced ## Related Resources - [SFC Modeling Overview](sfc-modeling.qmd) - Introduction to Stock-Flow Consistent modeling - [Economics Tutorials](economics-tutorials.qmd) - Hands-on tutorials with CircuitJS1 - [Examples](examples.qmd#economics--system-dynamics-models) - BOMD and other economic models ## References This model and documentation are based on teaching materials developed for SFC modeling courses. For the theoretical foundations, see: - Godley, W. & Lavoie, M. (2007) *Monetary Economics: An Integrated Approach to Credit, Money, Income, Production and Wealth* - 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" --- *This page presents the simple SFC model developed by Yannis Dafermos and Maria Nikolaidi for educational purposes.*