Mini Minsky Model

This is a little model inspired by Minsky's Financial Instability Hypothesis.  It is intended to illustrate the role of debt in asset price bubbles and business cycles, within the context of a simple stock-flow framework.

There are three sectors: households, firms and banks, and four asset classes: bank deposits, productive loans made to firms, speculative loans made to households, and equity of firms.  Banks are assumed to have no equity so that total loans always equals deposits.  Firm equity is represented by the number of securities multiplied by a price.  The financial balance sheet matrix for the model is shown below:

	Households	Firms	Banks	Net
Deposits	D		-D	0
Productive Loans		-LK	LK	0
Speculative Loans	-LS		LS	0
Equity	e.p	-e.p		0
Net	V	- ( LK + e.p )	0	0

If speculative loans are set at zero, then the model shows a steady growth path for the economy.  Once speculative loans are included, however, the economy exhibits a repeated cycle of boom and bust.  Debt levels and equity prices show a similar pattern to that of output.

GDP (log scale)

What is happening in the model is as follows: increased demand for equity, financed by speculative loans, pushes up the equity price.  As the equity price rises, so does realised return on equity.  This leads to more demand for speculative loans and further equity price rises.

In the absence of any check, this would continue indefinitely.  However, there is assumed to be a cap on speculative loans, a maximum amount that banks are prepared to lend relative to income.  Once this cap is reached, no more speculative loans can be made which slows the effective demand for equity.  This in turn slows the rise in equity prices and without the price gains, equity returns fall.  The fall in the return leads to falling demand for speculative loans, which further reduces the equity demand.  The process continues until speculative loans are reduced back to zero.

The whole process then starts up again.

The cycle of speculation has implications for national income.  Wealth is included in the model as a determining factor of consumer spending.  As equity values rise, spending increases and when equities collapse, spending drops back.  Debt drives asset prices which drive spending.  Spending goes up, even though the loans are not directly used to finance spending.  There is an interesting dynamic here, arising from the stock-flow consistency, that I intend to discuss further in a later post.

Model Specification

The following is intended to be a very brief, but complete description of the model.  The variables are as follows: 

Variable	Description
C	Consumer spending
D	Balance of deposits
DP	Distributed profit
I	Firms' expenditure on investment
LK	Balance of loans to firms
LS	Balance of speculative loans to households
LST	Target level of speculative loans
RP	Retained profit
V	Household net wealth
W	Wages
Y	GDP
e	Number of equities in issuance
P	Equity price
re	Equity return

Income is consumption plus investment.

Y_t = C_t + I_t

Wages are a fixed share of income.  Distributed profit is a fixed share of the surplus.

W_t = µ_w . Y_t

DP_t = µ_d . ( Y_t - W_t )

RP_t = Y_t - W_t - DP_t

The budget constraint of firms determines the change in productive loans.

ΔLK_t = I_t - RP_t

The budget constraint of households determines the change in deposits.

ΔD_t = W_t + DP_t - C_t + ΔLS_t

Household wealth is net assets:

V_t = D_t + e . p_t - LS_t

Consumption is assumed to be a function of current income and previous period wealth.

C_t = α₀ . (W_t + DP_t ) + α₁ . V_t-1

Investment is assumed to be a fixed multiple of retained profit.

I_t = β . RP_t

Households are assumed to hold equities either as a fixed share of net wealth, or fully funded by borrowing.   This relationship can be arranged as:

e . p_t =  θ₁ . D_t + LS_t

The number of equities in issue is assumed to be fixed, so this relationship effectively determines the price of equity.

The return on equity is given by:

re_t = [ ( DP_t / e ) + p_t ] / p_t-1 - 1                                         

The demand for speculative loans is a function of the return on equity.

LS^T_t = ( λ₀ + λ₁ . re_t-1 ) . Y_t

Speculative loans adjust incrementally towards this target.

ΔLS_t = ε . ( LS^T_t - LS_t-1 )

subject to:

θ₂ . Y_t - LS_t-1 <= ΔLS_t => - LS_t-1

The first part of this limits the level of speculative loans to some proportion of national income.  This cap (perhaps a limit on how much banks are prepared to lend) provides the trigger for the slump.  The second part prevents the measure of speculative loans becoming zero.  This prevents the downturn continuing indefinitely.

Note on parameters

The only fully exogenous variable in the model is the number of equities.  The opening values of stocks are exogenous, but over time all stock values are determined within the model.  This means there are no anchors to tie the long run values down, such as would typically be included in a more comprehensive model.

Partly because of this, the parameters need to be chosen appropriately in order for the model to fall into the cyclical path.  The values also reflect the fact that fairly short time periods are used so as to get a smooth plot on the chart.

The following values were used for parameters and opening values.

Parameter	Value
α1	0.2
α2	0.02
β	2.0
ε	0.05
λ0	-2.0
λ1	200.0
θ1	1.0
θ2	5.0
µw	0.8
µd	0.5

Variable	Opening value
D	100
LK	100
LS	0
V	200
e	100
p	1.00
re	1.00%