Modeling of Monetary Financial Flows in System Dynamics

The model describes a small open economy with six sectors. They are the central bank, the government, banks, households, firms and the rest of the world. Stocks are the items of the consolidated sector balance sheets. Flows are the accounting operations that change those stocks. One time unit equals one quarter.

The model follows the stock-flow consistent framework of Godley and Lavoie (2007) and is written as a system dynamics model in continuous time. Behavioural equations govern spending, credit, borrowing and taxation. The interest rate closes the main balancing loop of the model. It reacts to the inflation gap, the output gap and the balance of credit demand and supply.

Every financial asset is someone else’s liability, so each instrument summed across all sectors gives zero. For a sector balance sheet with assets \(A\), liabilities \(L\) and equity \(E\) the same rule holds.

\[ \sum_{s=1}^{n} B_{j,s} = 0 \quad \forall j, \qquad A - L - E = 0 \]

Here \(B_{j,s}\) is the holding of instrument \(j\) by sector \(s\). An asset enters with a plus sign and a liability with a minus sign.

Balance sheet and transactions

Table 1 shows the balance sheet matrix and Table 2 the transaction flow matrix. Point at a row to see that it sums to zero.

Table 1. Balance sheet matrix of the model.

Instrument	HH	F	B	CB	G	RoW
Household deposits, H_H	+H_H		−H_H
Firm deposits H_F		+H_F	−H_F
Gov. deposits at CB H_G				−H_G	+H_G
Bank reserves R			+R	−R
Loans to firms L_F		−L_F	+L_F
Loans to households L_HH	−L_HH		+L_HH
Government bonds B			+B		−B
Nostro accounts H_BROW			+H_BROW			−H_BROW
CB FX reserves FX_CB				+FX_CB		−FX_CB
Σ	0	0	0	0	0	0

Source: Author.

Table 2. Transaction flow matrix of the model.

Flow	HH	F	B	CB	G	RoW
Wages, W	+W	−W
Consumption, C	−C	+C
Gov. spending, G		+G			−G
Bank spending, C_B		+C_B	−C_B
Exports, M_F		+M_F				−M_F
Firm imports, M^imp_F		−M^imp_F				+M^imp_F
HH imports, M_HH	−M_HH					+M_HH
HH taxes, T_H	−T_H				+T_H
Firm taxes, T_F		−T_F			+T_F
Interest on HH loans, ΦL_HH	−ΦL_HH		+ΦL_HH
Interest on firm loans, ΦL_F		−ΦL_F	+ΦL_F
Interest on bonds, ΦB			+ΦB		−ΦB
Firm dividends, D_F		−D_F				+D_F
Bank dividends, D_B			−D_B			+D_B
Saving, S	S_HH	S_F	S_B	0	S_G	S_ROW
HH deposits, ΔH_H	−ΔH_H		+ΔH_H
Firm deposits, ΔH_F		−ΔH_F	+ΔH_F
TGA, ΔH_G				+ΔH_G	−ΔH_G
Reserves, ΔR			−ΔR	+ΔR
Firm loans, ΔL_F		+ΔL_F	−ΔL_F
HH loans, ΔL_HH	+ΔL_HH		−ΔL_HH
Gov. bonds, ΔB			−ΔB		+ΔB
Nostro, ΔH_BROW			−ΔH_BROW			+ΔH_BROW
Σ	0	0	0	0	0	0

Source: Author.

Stocks

Each stock changes only through its inflows and outflows. Figure 1 shows the stocks and flows of the model. The stock equations below follow the Godley tables, sector by sector.

Click a stock to see what fills and drains it. Click a flow to see where it leaves one stock and enters another.

Figure 1. Forrester diagram of the model. A rectangle is a stock, a variable that accumulates through its own differential equation. An arrow with a solid dot is a flow, the rate at which a stock fills or drains. A circle marks the boundary of the flow. Source: Author

Central bank and government

Bank reserves \(R\) rise when the government spends or pays interest on its bonds. They fall when the government collects taxes or sells bonds. Government deposits \(H_G\) move the opposite way. The government finances its deficit by selling bonds \(B\).

\[ \frac{dR}{dt} = G + I_G - (T + \Delta B) \tag{1} \]

\[ \frac{dH_G}{dt} = T + \Delta B - (G + I_G) \tag{2} \]

\[ \Delta B = G + I_G - T \tag{3} \]

\[ \frac{dB}{dt} = \Delta B \tag{4} \]

\[ \frac{dG_E}{dt} = T - (G + I_G) \tag{5} \]

Households

Household deposits \(H_H\) grow with wages \(W\) and new loans \(\Delta L_{\mathrm{HH}}\). They fall with consumption \(C\), taxes, interest \(I_{\mathrm{HH}}\), imports \(M_{\mathrm{HH}}\) and loan repayments \(R_{\mathrm{HH}}\).

\[ \frac{dH_H}{dt} = W + \Delta L_{\mathrm{HH}} - (C + T + I_{\mathrm{HH}} + M_{\mathrm{HH}} + R_{\mathrm{HH}}) \tag{6} \]

\[ \frac{dL_{\mathrm{HH}}}{dt} = \Delta L_{\mathrm{HH}} - R_{\mathrm{HH}} \tag{7} \]

\[ \frac{d\mathit{HH}_E}{dt} = W - (C + T + I_{\mathrm{HH}} + M_{\mathrm{HH}}) \tag{8} \]

Firms

Firm deposits \(H_F\) grow with new loans \(\Delta L_F\), consumption, government spending, bank spending \(C_B\) and exports \(M_F\). They fall with wages, interest \(I_F\), dividends \(D_F\), imports and loan repayments \(R_F\).

\[ \frac{dH_F}{dt} = \Delta L_F + C + G + C_B + M_F - (W + I_F + D_F + M_F^{\mathrm{imp}} + R_F) \tag{9} \]

\[ \frac{dL_F}{dt} = \Delta L_F - R_F \tag{10} \]

\[ \frac{dF_E}{dt} = C + G + C_B + M_F - (W + I_F + D_F + M_F^{\mathrm{imp}}) \tag{11} \]

Banks

Nostro accounts \(H_{\mathrm{BROW}}\) are the claims of domestic banks on the foreign banking system. They grow with exports and fall with imports and with dividends paid abroad. Net financial assets of banks \(B_E\) grow with interest on loans and fall with bank spending and dividends.

\[ \frac{dH_{\mathrm{BROW}}}{dt} = M_F - (M_F^{\mathrm{imp}} + M_{\mathrm{HH}} + D_F + D_B) \tag{12} \]

\[ \frac{dB_E}{dt} = I_F + I_{\mathrm{HH}} - (C_B + D_B) \tag{13} \]

Rest of the world

\[ \frac{dH_{\mathrm{ROW}}}{dt} = M_F^{\mathrm{imp}} + M_{\mathrm{HH}} + D_F + D_B - M_F \tag{14} \]

\[ \frac{dE_{\mathrm{ROW}}}{dt} = M_F^{\mathrm{imp}} + M_{\mathrm{HH}} + D_F + D_B - M_F \tag{15} \]

Output, credit and the interest rate

GDP is the turnover of the firms’ deposits. Wages, government spending and taxes are fixed shares of GDP.

\[ \mathit{GDP} = \frac{H_F}{V_F} \tag{16} \]

\[ W = W_s \cdot \mathit{GDP} \tag{17} \]

\[ G = g_s \cdot \mathit{GDP} \tag{18} \]

\[ T = \tau \cdot \mathit{GDP} \tag{19} \]

Firms borrow the smaller of their loan demand \(\mathit{DL}_F\) and the loan supply of banks \(\mathit{SL}_B\). Banks lend up to a leverage multiple of their net financial assets. Households borrow a fixed share of GDP. Both sectors repay a fixed share of their debt in each period.

\[ \mathit{DL}_F = H_F \cdot (\beta^{\max} - \beta_S \cdot r) \tag{20} \]

\[ \mathit{SL}_B = B_E \cdot \mathit{Lev}_r \tag{21} \]

\[ \Delta L_F = \min(\mathit{DL}_F,\ \mathit{SL}_B) \tag{22} \]

\[ \Delta L_{\mathrm{HH}} = \beta_{\mathrm{HH}} \cdot \mathit{GDP} \tag{23} \]

\[ R_F = \rho_F \cdot L_F \tag{24} \]

\[ R_{\mathrm{HH}} = \rho_{\mathrm{HH}} \cdot L_{\mathrm{HH}} \tag{25} \]

Interest is paid on government bonds and on new loans.

\[ I_G = r \cdot B \tag{26} \]

\[ I_F = r \cdot \Delta L_F \tag{27} \]

\[ I_{\mathrm{HH}} = r \cdot \Delta L_{\mathrm{HH}} \tag{28} \]

Growth comes from net new credit. The output gap is the deviation of growth from its steady state. Inflation follows a simple Phillips curve.

\[ \mathit{GR} = \frac{\Delta L_F + \Delta L_{\mathrm{HH}} - R_F - R_{\mathrm{HH}}}{H_F} \tag{29} \]

\[ \mathit{Out}_{\mathrm{GAP}} = \mathit{GR} - \mathit{GR}_{ss} \tag{30} \]

\[ \pi = \pi_{ss} + \Phi_p \cdot \mathit{Out}_{\mathrm{GAP}} \tag{31} \]

\[ \pi_{\mathrm{GAP}} = \pi - \pi_{ss} \tag{32} \]

The effective interest rate is the central bank rate plus a market component. The market component accumulates two terms. The Taylor gap weighs the inflation gap and the output gap. The credit gap \(\Delta r\) compares the loan demand of firms and new government bonds with the loan supply of banks.

\[ r = r^{\mathrm{CB}} + r_{\mathrm{market}} \tag{33} \]

\[ \mathit{Taylor}_{\mathrm{GAP}} = a_p \cdot \pi_{\mathrm{GAP}} + a_y \cdot \mathit{Out}_{\mathrm{GAP}} \tag{34} \]

\[ \Delta r = \frac{\kappa_r \cdot (\mathit{DL}_F + \Delta B - \mathit{SL}_B)}{\mathit{GDP}} \tag{35} \]

\[ \frac{dr_{\mathrm{market}}}{dt} = \Delta r + \mathit{Taylor}_{\mathrm{GAP}} \tag{36} \]

The interest rate closes the balancing loops B1 and B4. Figure 2 shows all feedback loops of the model. Switch them on and off, or follow one through the diagram.

Click a loop to follow it through the diagram. The switches turn loops on and off.

Figure 2. Causal loop diagram of the model. + marks a positive and − a negative causal link. R marks reinforcing and B balancing loops. Source: Author.

Spending, trade and profits

Households spend a share of their deposits that moves with the inflation gap. Banks spend a fixed share of their net financial assets.

\[ S_{\mathrm{HH}}^{\mathrm{eff}} = S_{\mathrm{HH}} \cdot (1 - \lambda_{\mathrm{inf}}) \cdot \pi_{\mathrm{GAP}} \tag{37} \]

\[ C = S_{\mathrm{HH}}^{\mathrm{eff}} \cdot H_H \tag{38} \]

\[ C_B = S_B \cdot B_E \tag{39} \]

Exports and imports are shares of GDP.

\[ M_F = E_{\mathrm{rat}} \cdot \mathit{GDP} \tag{40} \]

\[ M_{\mathrm{HH}} = I_{\mathrm{rat}}^{\mathrm{HH}} \cdot \mathit{GDP} \tag{41} \]

\[ M_F^{\mathrm{imp}} = I_{\mathrm{rat}}^{F} \cdot \mathit{GDP} \tag{42} \]

Profits of firms \(F_p\) and of banks \(B_p\) set the dividends paid to foreign owners.

\[ F_p = C + G + C_B + M_F - (W + I_F + M_F^{\mathrm{imp}}) \tag{43} \]

\[ D_F = \mathit{FD}_{\mathrm{rat}} \cdot F_p \tag{44} \]

\[ B_p = I_F + I_G + I_{\mathrm{HH}} - C_B \tag{45} \]

\[ D_B = \mathit{BD}_{\mathrm{rat}} \cdot B_p \tag{46} \]

Parameters and initial values

All stocks are in millions of Czech crowns. Flows and ratios are unitless proportions. Table 3 lists the values of the baseline scenario on the scale of the paper. The Ravel file holds the same values multiplied by 1,000.

Table 3. Initial stocks and parameters of the baseline scenario.

Symbol	Meaning	Type	Value
\(H_H\)	Household deposits	Stock	5000
\(H_F\)	Firm deposits	Stock	1500
\(H_G\)	Government deposits at the central bank	Stock	300
\(R\)	Bank reserves	Stock	1000
\(H_{\mathrm{BROW}}\)	Nostro accounts	Stock	800
\(\mathit{FX}_{\mathrm{CB}}\)	Central bank FX reserves	Stock	3000
\(L_F\)	Loans to firms	Stock	3500
\(L_{\mathrm{HH}}\)	Loans to households	Stock	2000
\(B\)	Government bonds	Stock	3300
\(H_{\mathrm{ROW}}\)	Assets of the rest of the world	Stock	8000
\(L_{\mathrm{ROW}}\)	Liabilities of the rest of the world	Stock	3000
\(E_{\mathrm{ROW}}\)	Net financial assets of the rest of the world	Stock	5000
\(B_E\)	Net financial assets of banks	Stock	4100
\(F_E\)	Net financial assets of firms	Stock	−2000
\(\mathit{HH}_E\)	Net financial assets of households	Stock	3000
\(G_E\)	Net financial assets of the government	Stock	−3000
\(\mathit{CB}_E\)	Net financial assets of the central bank	Stock	1000
\(\mathit{BD}_{\mathrm{rat}}\)	Dividend ratio of banks	Parameter	0.05
\(\beta_{\mathrm{HH}}\)	Borrowing ratio of households	Parameter	0.003
\(\beta^{\max}\)	Maximum borrowing ratio of firms	Parameter	0.06
\(\beta_S\)	Sensitivity of firm borrowing to the interest rate	Parameter	0.015
\(\rho_F\)	Repayment rate of firms	Parameter	0.04
\(\rho_{\mathrm{HH}}\)	Repayment rate of households	Parameter	0.02
\(E_{\mathrm{rat}}\)	Exports to GDP	Parameter	0.6
\(\mathit{FD}_{\mathrm{rat}}\)	Dividend ratio of firms	Parameter	0.6
\(\mathit{GR}_{ss}\)	Steady-state growth rate	Parameter	0.06
\(I_{\mathrm{rat}}^{F}\)	Imports of firms to GDP	Parameter	0.07
\(I_{\mathrm{rat}}^{\mathrm{HH}}\)	Imports of households to GDP	Parameter	0.05
\(\pi_{ss}\)	Inflation target	Parameter	0.004
\(r_{\mathrm{market}}\)	Market component of the interest rate, initial value	Parameter	0.01
\(r^{\mathrm{CB}}\)	Central bank policy rate	Parameter	0.01
\(\kappa_r\)	Sensitivity of the market rate to the credit gap	Parameter	0
\(\mathit{Lev}_r\)	Leverage ratio of banks	Parameter	8
\(\Phi_p\)	Price sensitivity to the output gap	Parameter	0.03
\(S_B\)	Spending rate of banks	Parameter	0.005
\(g_s\)	Government spending to GDP	Parameter	0.18
\(S_{\mathrm{HH}}\)	Spending rate of households	Parameter	0.035
\(\tau\)	Tax rate	Parameter	0.15
\(V_F\)	Turnover ratio of firm deposits	Parameter	0.3
\(W_s\)	Wage share of GDP	Parameter	0.6
\(a_p\)	Central bank response to the inflation gap	Parameter	0.03
\(a_y\)	Central bank response to the output gap	Parameter	0.01
\(\lambda_{\mathrm{inf}}\)	Sensitivity of consumption to the inflation gap	Parameter	0.15

Source: Author.

Scenarios and simulation

One time unit equals one quarter and the baseline runs for 30 quarters. The relative error is 0.01 and the absolute error 0.001. The solver is a fourth-order Runge–Kutta method with no implicit solver.

The boom scenario raises the export-to-GDP ratio from 0.6 to 0.8 and the borrowing capacity of firms from 0.06 to 0.14. It also lowers the repayment rate of firms slightly.

The negative shock scenario cuts the export ratio to 0.43 and the turnover of firms to 0.3. The wage share rises to 0.6. Credit is tightened through a lower maximum borrowing fraction of 0.08, and the tax rate is 0.21.

Results

The three scenarios suggest that public debt may positively influence the financial position of the private sector in a small open economy. In the baseline, rising government debt goes together with rising net financial assets of firms and a gradual fall in their borrowing. The boom scenario strengthens these patterns. In the negative shock scenario, lower government debt and weaker external demand reduce the net financial assets of firms and their borrowing.

The results are highly sensitive to the foreign sector. Even a modest external shock reshapes the sector balance sheets of a small open economy.

Model code and data

Ravel model fileFull system dynamics model with all stocks, flows and parameters Coming soon

References

Bykov, P. (2025). Modeling of monetary financial flows in system dynamics. ACTA VŠFS, 19(2), 177–203. https://doi.org/10.37355/acta-2025/2-05
Godley, W., & Lavoie, M. (2007). Monetary economics: An integrated approach to credit, money, income, production and wealth. Palgrave Macmillan.

Keywords: stock flow consistent modeling, system dynamics, public debt, monetary economics
JEL codes: E51, C63, E17

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