Preference Functions and the Balanced Budget Multiplier

One of the things that fascinates me in economics is how different economists manage to come up with completely different conclusions about the results of particular policy measures.  Where this happens, I think it provides an excellent opportunity to examine how results flow from assumptions and to appreciate how fragile certain conclusions might be.

One case of this that struck me recently is the New Keynesian position that a balanced budget increase in public expenditure, which is expected to be permanent, has no impact on GDP.  This result, which contrasts with the old Keynesian result, is said to be down to the assumption of rational inter-temporally maximising agents.  However, what is perhaps not so well appreciated is that it is also contingent on some rather ad hoc assumptions about household preferences.

To look at this, I'm going to assume a closed economy with no investment, so GDP (Y) comprises simply public spending (G) and household consumption (C) (all in real terms, assuming a single good):

(1)  Y = G + C

I'm going to go with the standard assumption that public spending is funded with lump sum taxes (T).  I don't like this assumption very much, partly because I think it sometimes skews the results, but it makes it easier to be clear about what constitutes a balance budget change in spending.  So household disposable income (YD) is given:

(2)  YD = Y - T

Now, in the standard New Keynesian analysis, consumption is based on lifetime income over an infinite life.  So, if it is assumed that price flexibility will ensure full employment GDP in the long run, then the permanent increase in taxes will reduce permanent disposable income.  Expected future consumption will therefore fall and so households will also cut current consumption.

However, this result arises only because of our assumption about household preferences.  We could take a different approach. 

For example, we could assume that household maximise over a finite lifetime.  In its simplest form, assume each household lives for two periods and then dies.  It works and earns only in the first period but splits its income so as to consume the same amount in each period (so we are assuming, for simplicity, zero elasticity of substitution between consumption in the two periods.)  We will also assume an equal number of new households is "born" in every period.  Each period, we therefore have half our households spending their savings from the previous period and half spending some fraction of their current earnings.  In equilibrium, these households will be saving half their income, so the stock of savings at any time will be equal to 50% of disposable income. [Edit - to clarify, equilibrium here means a steady-state non-growth equilibrium when prices have been able to fully adjust to remove any output gap.]

It will help at this stage to put some numbers on this.  I am going to assume that full employment GDP is 100 and the current real stock of savings is 40.  The savings take the form of public debt, but I am going to assume that public spending and taxes are both currently zero.  Lastly, prices are assumed to be sticky in the current period and the next, but to be fully flexible thereafter.

So in this instance, we find that expenditure falls short of what is needed for full employment.  The table below shows the general price level for goods (p), and savings, consumption, public spending, GDP, taxes and disposable income, all in real terms.  In addition to identities (1) and (2), we have the real value of savings (A) is given as:

(3)  A_t = (YD_t-1 - Cw_t-1) / ( p_t / p_t-1 )

where Cw is workers' consumption.  Consumption by retireds is equal to their savings.  Workers in one period are retireds in the next, so inter-temporal optimising under rational expectations requires that workers' consumption in one period is equal to retireds' consumption in the next.

Period	1	2	3
Price level	1.00	1.00	0.80
Value of savings	40	40	50
Workers' consumption	40	50	50
Retireds' consumption	40	40	50
Public spending	0	0	0
GDP	80	90	100
Taxes	0	0	0
Household disposable income	80	90	100

Each subsequent period will be the same as period 3.  So, in this instance, the sticky prices are creating a temporary shortfall in GDP.  What is the effect of a permanent balanced budget increase in public spending?  This is shown in the table below.

Period	1	2	3
Price level	1.00	1.00	1.00
Value of savings	40	40	40
Workers' consumption	40	40	40
Retireds' consumption	40	40	40
Public spending	20	20	20
GDP	100	100	100
Taxes	20	20	20
Household disposable income	80	80	80

Again, each subsequent period will be the same as period 3.  The effect has been an elimination of the temporary output gap.  The longer run effect is that the general deflation that occurred in the first example does not occur here.

It helps to understand what's going on here if you realise that the assumptions about household preferences imply a steady state ratio between consumption and the real value of savings.  In our example, the output gap arises because the real value of savings is too low.  GDP is prevented from rising, because if disposable income were to rise, households would try to save more and with the balanced budget assumption they cannot achieve this.  For the same reason, consumption stays the same when there is a balanced budget increase in public spending.  However, the fact that there is no price deflation in later periods keeps consumption at its existing level.

I am not arguing here that one particular assumption about household preferences is better than the other.  I think the latter approach is more consistent with heterodox views, and I find it more plausible, but I suspect what actually happens is much more complicated.  All I wanted to do here was illustrate what I thought was an interesting case of where conflicting conclusions arise from different assumptions, even if both involve rational inter-temporal maximising agents. 

(I should point out that I am aware that the result in my overlapping generation case also depends on how the tax is levied.  I have assumed the tax to be paid out of workers' income, but if the retired bear all or some of the burden there is a different result.  This is another reason why I am wary of unrealistic assumptions about the structure of taxation.)

Driving Tobin's Golfcart

Simon Wren-Lewis's asks whether New Keynesians have made a Faustian pact with the New Classicals.  Paul Krugman responds invoking Tobin's general equilibrium approach to monetary theory (but see Ramanan's comments), which in turn prompts Stephen Williamson to chime in.

Even if he generally tends to end up defending the mainstream, Wren-Lewis is at least prepared to address some heterodox concerns, perhaps indicating some nagging doubts that maybe he has indeed entered into a Faustian pact.  Williamson has no such reservations.  To him, we don't need Tobin any more because we've moved on.

"Why did Tobin's work disappear from discussion? It was superceded. We now have better models, that are much richer in their implications. Tobin was fine for his time, but why drive on the freeway in a golfcart, when you can drive a Ferrari (as one of my friends once said)?"

Williamson's "better models" involve deriving macro results from their micro-foundations, by making various assumptions about how individual agents might behave and deducing the consequences.  Heterodox economists can be a bit dismissive about this approach, but I find it very useful.  For me, it is not sufficient to rely on the stylized facts; I want to understand what the underlying behaviour is.  That is not to say I think that the assumptions micro-founders typically make are really the best ones and I like it when I see work that explores the consequences of different micro assumptions.  But I would say it is true that investigating what micro behaviour lies behind the macro outcomes adds richness to our understanding.

However, to incorporate everything that is relevant into a micro-founded approach is a massive task, even more so if you want to be sure that your results are not overly contingent on your assumptions.  Even Williamson recognises that his own work in this area is only a "piece of the puzzle", as it focuses specifically on pledgability of collateral.  Each bit of research may give some insight into one particular issue, but I'm less convinced that they can be relied upon in isolation.  I don't think we're near having a complete micro-founded picture.  And even if we did, I would be wary about how much we were relying on our particular behavourial assumptions.

Tobin's approach may be crude, but it provides a useful way of capturing the overall picture.  We need to think carefully about whether the assumed behaviour is realistic given what we believe about how individuals behave, but so far I see no reason to reject it.

So this Ferrari may sound very exciting, but I think it's not even half built yet.  We've got some very skilfully designed components, but whether we'll be able to use them, we'll have to see.  I'm interested in how it goes, but for the time being, if I want to get from A to B, I'm going to be driving Tobin's golfcart.

The Dynamics of Debt

I mentioned in my recent post on Steve Keen's latest paper that the relationship between debt and spending is a complex one.  This post is intended to give a brief outline of what I think are the main connections there.  The problem with tackling this topic is that whatever you write, there are always caveats and exceptions.  Still, you have to start somewhere.

To look at this, I want to imagine a simple closed economy with no public sector.  I have however, divided the non-financial sector into different types of agents,  This is necessary to construct the debt relationships.  The assets in this economy are loans, deposits and land.  Land is in fixed supply.  The different agents in this economy are:

- Wealthy households, that hold deposits and land and have no borrowing

- Speculators, that borrow to buy land

- Spenders, that borrow to acquire produced goods but hold no assets

- Banks, which make loans and take deposits

The national balance sheet is shown below:

	Wealthy	Mortgagors	Spenders	Banks
Loans		-Lm	-Ls	L
Deposits	D			-D
Land	p.Aw	p.Am

The table below shows the flow of funds for this economy.  I haven't included anything to represent production, so the top two rows sum to total income and total consumption respectively.  These will be equal and opposite.  Otherwise, all rows and columns sum to zero.

	Wealthy	Mortgagors	Spenders	Banks
Income	Yw	Ym	Ys
Consumption	-Cw	-Cm	-Cs
Loans		ΔLm	ΔLs	-ΔL
Deposits	-ΔD			ΔD
Land	p. -ΔAw	p. -ΔAm

Lending activity then matters in different ways.

1. Net new lending to Spenders is equal, by accounting identity, to their additional spending.  As Spenders hold no assets, their spending must be equal to their income plus their change in debt.

The increase in lending may either be seen as a cause of the increased spending, or as a result.  Maybe Spenders want to borrow more, but face credit constraints.  If Banks then relax their lending criteria, that could be seen as the cause of increased lending and spending.  On the other hand, these increases may arise from a change in the inter-temporal preferences of Spenders, in which case we might not want to say that the lending caused the spending.

If the economy was only made up of Spenders, then we would find that aggregate demand was strictly equal to income plus change in debt.  But, obviously it's impossible to have only borrowers without any lenders.

2. Mortgagors are borrowing to acquire land.  As with Spenders, there is a direct link between the amount of their borrowing and their demand for land.  However, there is also a fundamental difference, which is the distinction between stocks and flows.

The demand of Spenders for goods is a flow.  Spending on goods is measured over a period of time.  The demand of Mortgagors for land is a stock[1].  We need to know how much land Mortgagors want to hold at any point in time.  In relation to Spenders, therefore, it is the change in debt that matters.  In relation to Mortgagors, it is the outstanding debt[2].

As with Spenders, we cannot say anything conclusive about cause and effect.  The increased lending and land purchase may arise as a result of changes in bank lending policies or as a result of changes in household preferences.

We are assuming land to be fixed in supply.  The increased demand for land must therefore translate wholly into an increase in price.  The value of the landholdings of the Wealthy and of Mortgagors therefore rises.

So far in this analysis, there is no impact on GDP.  The rise in the value of land may, however, have an impact on spending.  Certain life cycle spending patterns can generate steady state stock-flow ratios between wealth and income.  Rising wealth, in the form of higher land values may then lead to greater spending as agents re-align their wealth income ratios.  In contrast with what happens with Spenders, the extent and rate to which this happens here is an empirical matter.

It is useful to ask what happens to the funds lent to the Mortgagors.  There are two places this can go, as we can see from the flow of funds.  The first is into higher consumption spending, to the extent that the Mortgagors respond to increased wealth as described above.  The other is in net acquisition of land from the Wealthy.  There is nowhere else for the funds to go.

This has an interesting implication.  As the debt of Mortgagors rises, they must either increase their consumption or end up owning more and more of the land stock.  We might therefore expect a continually rising level of Mortgagor debt to eventually translate into higher consumption.

3. The growth of loans requires a growth in deposits.  Note that, although I have chosen here to talk about deposits, there is no requirement that the assets created are monetary, in the sense of being able to be used for making payments.  An expanding loan book matched by expanding non-monetary assets would have the same effect.

So as new loans are made the stock of loans rises and the stock of deposits rises.  We may take the view that there are certain steady state stock-flow ratios in play, in particular that the wealthy have a target ratio of liquid assets to income.  In that case, we might expect the rise in deposits to lead to greater spending by the wealthy.

Mortgagors and Spenders have growing debts, which place a rising debt service burden on them.  The effects of this are complicated.  They may be tempted to borrow more to maintain their previous consumption levels.  However, over the longer term this is likely to be swamped by the limiting effects of high debt income ratios.  In theory, the ratio of a person's assets to income can rise indefinitely.   The same is not true for debt to income ratios.

Overall, the increase in liquid assets is likely to increase spending, but again the extent and pace of this is an empirical matter.

The combined effect of these things is to produce a complex dynamic response to an increase in debt.  Introducing a public sector and an overseas sector complicate this further.  But although the patterns that emerge do depend on agent behaviour, thinking about the relationships this way can bring out some important insights into longer term dynamics.

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[1] We could derive a flow demand for land as the difference in the stock demand from period to period.

[2] With assets that are heterogenous and less liquid, like land, the flow of debt may also matter because the market may only react slowly to sudden changes in demand.  I am ignoring this here, in order to be able to focus on the contrast with loans to Spenders.