Saturday, 8 June 2013

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.

Yt = Ct + It

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

Wt = µw . Yt

DPt = µd . ( Yt - Wt )

RPt = Yt - Wt - DPt

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

ΔLKt = It - RPt

The budget constraint of households determines the change in deposits.

ΔDt = Wt + DPt - Ct + ΔLSt

Household wealth is net assets:

Vt = Dt + e . pt - LSt

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

Ct = α0 . (Wt + DPt ) + α1 . Vt-1

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

It = β . RPt

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 . pt =  θ1 . Dt + LSt

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:

ret = [ ( DPt / e ) + pt ] / pt-1 - 1                                         

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

LSTt = ( λ0 + λ1 . ret-1 ) . Yt

Speculative loans adjust incrementally towards this target.

ΔLSt = ε . ( LSTt - LSt-1 )

subject to:

θ2 . Yt - LSt-1 <= ΔLSt => - LSt-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%


7 comments:

  1. This is great. Have you considered building this Minsky model in (Keen's) Minsky?

    ReplyDelete
    Replies
    1. Thanks.

      I hadn't considered it specifically for this model, but I did want to try out Keen's Minsky at some point. I've had some problems downloading it, but I'll have to give it another go, because I'm interested to see the construction.

      Delete
  2. I agree, this is great and is very much related to some things I am working on. Thank you. Any chance you could give out the formula specifications for the QE model. By the way, I have been using Excel with VBA macros to run my simulations. I am not sure why Keen wants to create the engineering-like programme "Minsky"... Any thoughts?

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    Replies
    1. The QE model has many more equations and a fairly complicated process for solving the expectations. I'm afraid I don't want to post a full listing of it at this stage, partly because of the time invested in it and partly because I would feel the need to an extensive write-up to explain why I've done each bit as I have. I might do so in the future, however, and I'm happy to answer specific questions about it.

      I also use Excel with VBA macros. It takes a bit longer than some dedicated modeling software, but I find it pretty versatile. I still haven't managed to download Minsky.

      Delete
  3. Hi Nick,

    I programmed your interesting model in Vensim. I played a little bit with the parameter values you used and modified them somewhat, in order to let them resemble real life values a little bit better:

    Parameter Value Parameter Value
    α1 0,8 λ1 1
    α2 0,1 θ1 1
    β 1.2 θ2 1
    ε 0,5 µw 0,8
    λ0 0 µd 0,5

    I experimented a little bit with the model using Synthesim, a nice option in Vensim in which you can change the parameter values using sliders. With these settings, the size of the speculative loans never reaches the cap you specified, but I noticed that the model still behaves in a cyclical fashion.

    What seems to happen is this:
    - an increasing return on equity leads to an increasing volume of speculative loans;
    - the increasing volume of speculative loans drives up the price of equities;
    - the increasing price of equities drives down the return on equity, but it also drives up consumption and investment and thus GDP, as an increase of the value of the equities leads to an increase in the net wealth of households, which leads to an increase in their consumption;
    - just after the return on equity has peaked and starts to decrease, GDP still increases more than the return on equity decreases (in a relative sense), leading to a furhter increase of the volume of speculative loans;
    - only after the (increasing) speed of decrease of the return on equity has overtaken the (decreasing) speed of GDP growth, the volume of speculative loans starts to decrease;
    - at the bottom of the cycle, the opposite happens.

    Best wishes,

    Anton van de Haar

    ReplyDelete
    Replies
    1. Do you mean that you could take away the cap and still get the same effect? That's quite interesting. Without the cap, I suspect that many parameter values would result in explosive behaviour, although I can believe that that is not the case for all values.

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  4. Yes, thats what I mean. Try for yourself with the parametervalues that I used.

    Anton

    ReplyDelete