## Monday, 24 February 2014

### 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)  At = (YDt-1 - Cwt-1) / ( pt / pt-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 1 0.8 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 1 1 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.)

1. Hi Nick,

Maybe you should specify the complete model. If I am correct:
- the worker households save half of their disposable income;
- the retired households consume all of the savings of the worker households from the previous period;

The model (lower case letters: real variables):
a = ( yd(-1) – cw(-1) ) / (p/p(-1))
cw = 0.5*yd
cr = a(-1)
g = exogenous
y = c + g
yd = y - t
p = ?

What is the equation for p?

Anton

1. This comment has been removed by a blog administrator.

2. Forgot t:
t = g

3. Correction: cr = a ?

4. Yes. Most of that's right, except for cw, which is actually given by cw = cr(+1). This is saying that each household aims to spend the same amount in their working period as they will in retirement.

In addition we have p(1) = p(2) = 1, which is the sticky prices in periods 1 and 2.

And we have y(t) = y* = 100, for all t >= 3, i.e. once prices are fully flexible we have full employment output.

So we solve for y in periods 1 and 2, and p for all later periods. Obviously, because the equation for cw references future values, you can't solve this by determining the period 1 values, then moving to period 2, etc.

5. OK, but assuming that working households always save half of their income, that is analogous to cw = 0.5*yd, it seems to me.

But what is the equation for p after t=2? Is it: p = p0 * y/y0

Anton

1. Workers don't always save half their income, e.g. period 2 in the first scenario. I think I might have confused it, by saying they will be saving half their income in equilibrium - I meant long term equilibrium, when prices are allowed to adjust and we have full employment. I might change that.

After period 2, we have two equations in y: y = c + g and y = 100, so you find the p that satisfies both equations. Or you re-arrange some of the equations to give an equation in p.