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.

C^w = α_wy . Yd^w + α_wv . ( A^w_t-1 . p + D_t-1 )

C^b = α_by . Yd^b + α_bv . ( A^b _t-1 . p - L _t-1 )

GDP is equal to total consumption

Y = C^w + C^b

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

Yd^w = β_w . ( 1 - β_a ) . Y + rra . A^w_t-1 + r_t-1 . D_t-1

Yd^b = ( 1 - β_w ) . ( 1 - β_a ) . Y + rra . A^b_t-1 - r_t-1 . L_t-1

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

V = A^w . p + D

NE = A^b . 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 = ε_L . ( ( λ_L0 + λ_L1 . ( ra^e - r ) ) . NE - L_t-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.

ΔA^b = ( ΔL + Yd^b - C^b ) / p

A^w = A^t - A^b

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.)

A^w . p = ( λ_w0 + λ_w1 . ( ra^e - r ) ) . V

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

rra = β_a . Y / A^t

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

ra^e = ( rra + p^e ) / p - 1

The expected price of land is based on adaptive expectations.

p^e = ε_e . p + ( 1 - ε_e ) . p^e_t-1

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

r = r_t-1 + ε_r . ( Y_t-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 λ_L0 to 1.00.