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.

We Don't Need Banks To Get Us Out Of A Loanable Funds Constraint

I wanted to say something about how the existence of a bank intermediary relates to the idea of loanable funds, as I often see it suggested that the ability of banks to create money is what removes the loanable funds constraint.

Consider an economy with only two private agents - Patient and Impatient.  Impatient always spends all his income plus anything he can borrow.  There is also a government which occasionally spends and occasionally taxes.  Any difference it funds by printing money.  Right now it is spending, but not taxing, so it is increasing the supply of money.

Patient has three possible uses for his income:

1. Consumption spending

2. Making loans to Impatient

3. Increasing his holding of money

The total of these three must equal his income.  He therefore has two and only two independent choices.  What happens in the economy depends critically on these choices.  Aggregate spending will increase if he increases either consumption spending or loans to Impatient (the latter because, by assumption, Impatient will immediately spend the amount loaned).  The only choice that will reduce aggregate spending is one that involves trying to increase his holding of money.

Making loans funds Impatient and accumulating money funds the government.  However, a decision to lend to Impatient dictates Impatient's spending; a decision to accumulate money has no impact on the government's spending.

Patient's two choices are independent.  It might be that Patient decides that if he makes a loan to Impatient he will cut his own spending by an equivalent amount.  In that case, lending to Impatient would have no impact on aggregate demand.  We would have what looked like a loanable funds model.

However, there is no reason Patient has to behave this way.  He could equally decide to fund a loan to Impatient by reducing his holding of money.  In that case, the increased lending would lead to greater aggregate expenditure.

So greater debt can lead to greater aggregate demand, but it depends on the various choices made by the lender.  This has nothing to do with the endogeneity or otherwise of money.  So how might banks and financial intermediaries matter here?

Let's now suppose that the government's money and the loans to Impatient are held by a bank and all Patient holds is bank deposits.  Now Patient only has one choice to make - whether to spend or accumulate bank deposits.  The other decision - how much to lend to Impatient - is now taken by the bank.  So, whereas before there might possibly have been some connection between how much Patient spent and how much was loaned to Impatient, there are now likely to be unrelated.  Intermediation splits the decision taking.

Financial intermediation (or "endogenous money" or whatever) is not at all necessary for the loanable funds constraint not to apply.  What it does do is distance saving and lending decisions, making it less likely that the two are correlated.

"Neo-Fisherites" and Fiscal Policy

I wanted to make an observation on what Noah Smith calls the "Neo-Fisherite" idea that raising interest rates raises inflation.  I don't intend to go over all aspects of it; suffice it to say, I think the idea is flawed.  I just wanted to look at one particular point which I think is interesting in that it says quite a lot about how economists of different schools frame questions differently.

The Fisher equation says that the expected real rate of interest is (approximately) equal to the nominal rate of interest less expected inflation.  The Neo-Fisherite idea is that if the central bank increases its nominal rate and holds it there, then the rate of inflation will adjust to restore the natural real rate of interest.  One way this might be achieved in theory would be through a sudden sharp fall in the price level as the immediate response to the change in interest.  From this point, the price level could then drift upwards at the increased inflation rate.  Needless to say, the sharp fall in prices is somewhat implausible.  What is rather more realistic in this scenario is that prices would edge down very slowly with depressed output in the mean time.

If this were the analysis, it would be fairly unremarkable.  What we would really be saying is that high interest rates lead to a prolonged period of deflation, which may eventually reach the point at which it can turn around.  For the Neo-Fisherite case to be interesting, it must involve no significant deflationary period.

There are all sorts of things to be said here, but one thing that intrigues me is what happens to government debt in this scenario.  If the central bank were to permanently raise its rate, then the price of long-dated bonds must fall.  So, without a sharp drop in the general price level, the immediate result of this policy would be a fall in the real value of government debt held by the private sector.  So we can't claim monetary neutrality - we have to accept there will be real effects.

You could deal with this by simply assuming all debt is short term or assuming that the revaluation has no effect on private sector behaviour.  However, to me this seems rather pointless, because you're then just assuming away most of the most important reasons why the Neo-Fisherite position shouldn't apply.

In Stephen Williamson's QE paper (which provides a detailed model in which the Neo-Fisherite case holds), this problem is avoided a different way.  With regards to fiscal policy, Wlliamson makes the assumption that taxes and transfers respond passively to central bank policy, so as to maintain a constant real value of government debt (unnumbered equation appearing between (25) and (26)).  So, because an increase in the central bank rate reduces the real value of debt, as described above, we are saying here that the government responds with expansionary fiscal policy.

So the argument appears to be that if the central bank raises rates and the government reacts by cutting taxes, then we will get inflation. 

Which is not really a very surprising result.  You could put that into an IS-LM / AS-AD model and get the result that raising interest rates would raise inflation.  You just need to assume that the fiscal policy is sufficiently expansionary to validate the central bank's inflation target, which is what Williamson's assumption achieves.

It may well be that this assumption is not essential for Williamson's result.  As I said, there are other ways of avoiding the problem, although to me all they do is make it more questionable whether that the result wouldn't apply in the real world.  It just seems really odd to me to make an assumption like this about fiscal policy and pay such little regard to it, when the same assumption would have massive consequences in a differently framed model.  

The Impact of Flight to Quality on Bank Lending

John Cochrane's paper on run-free banking, which I mentioned in my last post, includes a section discussing why financial crises have real effects.  It's an important question.  Understanding why a fall in the value of securities or a bank run might lead to a reduction in output should shape how we think about financial regulation.[1]

One of the main features of the recent crisis was the flight to quality, in which investors sought to sell privately issued asset-backed securities in favour of government paper.  The simple story here then might be that the fall in demand for private paper reduces the price and raised the yield. This raised the cost of borrowing for new borrowers wanting to access this market.  Faced with this higher effective interest rate, private agents looked to defer spending.

It's a reasonable story and no doubt has an element of truth.  However, I think it misses some important features of the way things work in practice.

Much of the flight to quality of the recent crisis involved short-term investors refusing to take private long-term securities as collateral for short term investment.  This had two important implications for banks.  First, banks were effectively underwriting the liquidity of these securities, so that when the collateral based funding dried up, the banks had to take them onto their balance sheet.  Secondly, the problems with financing the securities caused holders to try to offload them leading to big falls in the market value.  Many banks were heavily exposed to this and took significant losses, eating into their capital base.

So banks suffered a fall in capital at the same time as they were being forced to increase their holdings of assets.  Banks were also having to reassign positions from the trading book to the banking book, due to the increasing illiquidity of those positions, putting further pressure on capital.

In the short run, capital represents an important constraint on bank lending.  In theory, it should be just a question of price.  If the bank needs more capital to lend, it simply charges the borrower the amount needed to cover the cost of the additional capital.  In practice, capital is not something that can be turned on like a tap.  Banks normally rely heavily on retained earnings to provide capital growth.  Although they can do equity issues if needed, capital is inherently long term and paying a high long term price may not be appropriate to deal with a shorter term problem.  So one implication to the flight to quality was a big reduction in bank lending capacity.

Another issue is that a willingness to hold loans at a particular price is not indicative of a willingness to advance new loans at that price.  Banks may be prepared to hang on to distressed loans, rather than sell them at the prevailing market price, because they expect the effective return at that price to outweigh realised losses.  This implies that were they to make new loans at a comparable rate, the profit on the good loans would cover the expected level of defaults. 

However, simply because a strategy might appear to be profitable does not mean a bank will want to pursue it.  Banks may be concerned that undertaking what is apparently riskier lending will cause concern with the shareholders, even if such a tactic could be very profitable.  Things like strategy and focus play an important role in shaping behaviour, not just profit maximisation.  Of course, this may leave the market open to new lenders who are prepared to run higher loss levels, but it can take a long time for this to happen.

The point about factors such as these is that the flight to quality is not just about an increase in the rate at which the private sector can borrow.  In fact, for many people, it means that they can no longer borrow at any price.  The fall in lending played an important role in transmitting the effects of the financial crisis to the real economy, but the quantity response needs to be analysed independently of the price response.  

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[1] The analysis here is not intended to be the same as Cochrane's, nor to necessarily contradict what he is saying.