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.
Wealthy

Borrowers

Banks


Loans

 L

L


Deposits

D

 D


Land

p. A^{w}

p. A^{b}

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

W^{w}

W^{b}


Income from land

rra . A^{w}_{t1}

rra . A^{b}_{t1}


Loan interest

 r_{ t1} . L_{t1}

r_{ t1} . L_{t1}


Deposit interest

r_{ t1} . D_{t1}

 r_{ t1} . D_{t1}


Consumption

C^{w}

C^{b}


Purchase of land

 p . ΔA^{w}

 p . ΔA^{b}


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}_{t1 }. p + D_{t1 })
C^{b} = α_{by} . Yd^{b} + α_{bv}
. ( A^{b}_{ t1} . p  L_{ t1} )
GDP is equal to total consumption
Y = C^{w} + C^{b}
Disposable income of each type of household is equal to
their share of nonproperty income, plus their respective net property income.
Yd^{w} = β_{w} . ( 1  β_{a} ) . Y +
rra . A^{w}_{t1 }+ r_{t1 }. D_{t1}
Yd^{b} = ( 1  β_{w} ) . ( 1  β_{a}
) . Y + rra . A^{b}_{t1 } r_{t1 }. L_{t1}
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_{t1} )
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}_{t1}
Interest rates are adjusted based on the difference between
GDP and a target level.
r = r_{t1 }+ ε_{r} . ( Y_{t1 } Y*
)
Variables
Name

Description

Opening
Value

A^{b}

Land held by borrowers

200

A^{t}

Total land

400

A^{w}

Land held by wealthy

200

C^{b}

Consumption of borrowers

47

C^{w}

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

Yd^{b}

Disposable income of borrowers

47

Yd^{w}

Disposable income of wealthy

53

r

Interest rate

3.00%

ra^{e}

Expected return on land

5.00%

rra

Rental rate on land

5.00%

p

Price of land

1.000

p^{e}

Expected price of land

1.000

Parameters
Name

Description

Opening
Value

α_{bv}

Borrower marginal propensity to consume out of income

0.89362

α_{by}

Borrower marginal propensity to consume out of net equity

0.05

α_{wy}

Wealthy marginal propensity to consume out of income

0.71698

α_{wv}

Wealthy marginal propensity to consume out of wealth

0.05

β_{a}

Share of GDP attributable to land

0.2

β_{w}

Share of nonland GDP going to wealthy

0.5

ε_{e}

Adjustment rate of land price expectation

0.5

ε_{L}

Adjustment rate of loans

0.1

ε_{r}

Adjustment rate of interest rate

0.001

λ_{L0}

Loan demand parameter

0.80

λ_{L1}

Loan demand parameter

10.0

λ_{w0}

Wealthy portfolio allocation parameter

0.46667

λ_{w1}

Wealthy portfolio allocation parameter

10.0

The simulation involved an increase in λ_{L0} to 1.00.
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 1exp(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?
ReplyDeleteSorry  that's too complicated for me. May be right  I don't know.
DeleteYes, I needed an iteration at each time step.
Wouldn't wealthy MPC out of wealth be lower then non wealthy? Are non wealthy "borrowers" in this model?
ReplyDeleteInteresting 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.
DeleteOut 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 reran the numbers. The results gave a fairly similar pattern.