Tuesday, 20 May 2014

Debt Dynamics Model

I thought I'd do another little model looking at the dynamics of changes in debt.  It's similar to stuff I've done before, but with a few differences.  The level of debt in this model is determined endogenously, but the central bank adjusts the interest rate in response to fluctuations in nominal GDP.

Many stock flow models of this type rely on an outside sector (generally the public sector) to provide an anchor for nominal GDP.  There is no outside sector here, so the only anchor is the central bank rate setting.

There are two type of households - wealthy and borrowers and a bank.  Land is measured as a quantity multiplied by a price.  The national balance sheet is shown below.



Wealthy
Borrowers
Banks
Loans

- L
L
Deposits
D

- D
Land
p. Aw
p. Ab



Demand consists only of consumption spending.  Owners of land receive a share of the national income with the rest going to wages.  The flow of funds is shown below.  All rows and columns sum to zero.



Wealthy
Borrowers
Banks
Wages
Ww
Wb

Income from land
rra . Awt-1
rra . Abt-1

Loan interest

- r t-1 . Lt-1
r t-1 . Lt-1
Deposit interest
r t-1 . Dt-1

- r t-1 . Dt-1
Consumption
Cw
Cb

Purchase of land
- p . ΔAw
- p . ΔAb

Change in loans

ΔL
- ΔL
Change in deposits
- ΔD

ΔD


The charts below show the outcome for a change in borrowers' optimal debt to income ratio.  The charts show the deviation from the opening (steady state) value.







There is no borrowing for consumption in this model, so the effects all play out through the price of land.  As borrowers take on more debt, they bid up the price of land.  Rising land prices increases nominal wealth for everyone leading to increased consumption (the increase in consumption is small relative to the additional debt).  The central bank responds by raising interest rates.  This reduces demand for land (people switch out of land into deposits) which brings the price down again,in turn reducing consumption.

Interest rates in this model do not figure in the consumption functions for either class of household.  Their only role is in relation to portfolio decisons.  They determine how the wealthy allocate their wealth between land and deposits and how much leverage borrowers will take on.

Nevertheless, there is a natural rate of interest in this model.  That is, there is a rate of interest in each period that will keep nominal GDP constant.  Loans still increase, but they are funded by wealthy househlds switching out of land and into deposits.  This model assumes that the central bank cannot know what this natural rate is and therefore has to respond to past deviations.

The fluctuations before the results settle down reflect the way the central bank is having to operate.  In fact, the central bank reaction function has to be set quite carefully to avoid bigger fluctuations .  Inclusion of an outside sector would provide a stabilising force.



Equation Listing


The consumption of each type of household is based on their disposable income and previous net wealth, revalued to current prices.

Cw = αwy . Ydw + αwv . ( Awt-1 . p + Dt-1 )

Cb = αby . Ydb + αbv . ( Ab t-1 . p - L t-1 )

GDP is equal to total consumption

Y = Cw + Cb

Disposable income of each type of household is equal to their share of non-property income, plus their respective net property income.

Ydw = βw . ( 1 - βa ) . Y + rra . Awt-1 + rt-1 . Dt-1

Ydb = ( 1 - βw ) . ( 1 - βa ) . Y + rra . Abt-1 - rt-1 . Lt-1

Wealth of the wealthy and the net equity of borrowers are based on their net assets.

V = Aw . p + D

NE = Ab . p - L

Borrowers have a target loan to net equity ratio based on the expected return on land and the interest on loans.  Actual loans adjust incrementally towards this target.

ΔL = εL . ( ( λL0 + λL1 . ( rae - r ) ) . NE - Lt-1 )

Deposits are equal to loans.

D = L

Land acquisitions by borrowers are equal to their new borrowing plus their saving, all divided by the price of land.   The remaining land is held by the wealthy.

ΔAb = ( ΔL + Ydb - Cb ) / p

Aw = At - Ab

The wealthy hold land as a proportion of their total wealth, based on the relative expected returns on land and deposits.  (For solving, this equation is arranged as an equation for p - the price of land.)

Aw . p = ( λw0 + λw1 . ( rae - r ) ) . V

Total land is assumed to earn a fixed share of GDP, which then gives the rental return on land.

rra = βa . Y / At

The expected return on land is based on the rental return and expected capital gains.

rae = ( rra + pe ) / p - 1

The expected price of land is based on adaptive expectations.

pe = εe . p + ( 1 - εe ) . pet-1

Interest rates are adjusted based on the difference between GDP and a target level.

r = rt-1 + εr . ( Yt-1 - Y* )


Variables



Name
Description
Opening Value
Ab
Land held by borrowers
200
At
Total land
400
Aw
Land held by wealthy
200
Cb
Consumption of borrowers
47
Cw
Consumption of wealthy
53
D
Deposits
100
L
Loans
100
NE
Net equity of borrowers
100
V
Wealth of wealthy
300
Y
Nominal GDP
100
Y*
Target nominal GDP
100
Ydb
Disposable income of borrowers
47
Ydw
Disposable income of wealthy
53
r
Interest rate
3.00%
rae
Expected return on land
5.00%
rra
Rental rate on land
5.00%
p
Price of land
1.000
pe
Expected price of land
1.000



Parameters



Name
Description
Opening Value
αbv
Borrower marginal propensity to consume out of income
0.89362
αby
Borrower marginal propensity to consume out of net equity
0.05
αwy
Wealthy marginal propensity to consume out of income
0.71698
αwv
Wealthy marginal propensity to consume out of wealth
0.05
βa
Share of GDP attributable to land
0.2
βw
Share of non-land GDP going to wealthy
0.5
εe
Adjustment rate of land price expectation
0.5
εL
Adjustment rate of loans
0.1
εr
Adjustment rate of interest rate
0.001
λL0
Loan demand parameter
0.80
λL1
Loan demand parameter
10.0
λw0
Wealthy portfolio allocation parameter
0.46667
λw1
Wealthy portfolio allocation parameter
10.0


The simulation involved an increase in λL0 to 1.00.

4 comments:

  1. Nick, I haven't studied your model very closely yet but the plots all look like classic 2nd order LTI system transient plots... I guess in this case in response to an input step function: you've got damped oscillations, rising exponentials of the form 1-exp(-t*a), etc. Did you find you had to do iterations at each time step to find a solution, or did each solution follow from the previous time step with a single calculation?

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    Replies
    1. Sorry - that's too complicated for me. May be right - I don't know.

      Yes, I needed an iteration at each time step.

      Delete
  2. Wouldn't wealthy MPC out of wealth be lower then non wealthy? Are non wealthy "borrowers" in this model?

    ReplyDelete
    Replies
    1. Interesting question. It doesn't have to be. Either a lower mpc income or a lower mpc wealth will produce a higher steady state ratio of wealth to income, which is all I'm looking for here.

      Out of interest, I changed the mpc of borrowers to the same as that of wealthy and increased their mpc wealth until the steady state values were the same as I used and then re-ran the numbers. The results gave a fairly similar pattern.

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