Some Peculiar Dynamics with Long Term Money Neutrality

Certain monetarists would assert that, given time, there is a fixed relationship between the quantity of base money and the price level.  For example, in this post Scott Sumner says the following:

"If the Fed wants to increase all nominal variables by 100 fold, it simply increases the base 100 fold."

The idea behind this is simple.  What matters to agents are real variables.  If the only policy variable fixed in nominal terms is the quantity of base money, then the nominal values of everything else must ultimately be pegged off that.  It may take a long time, but in the end that's where it will end up.

If you can step back from how policy is actually conducted and imagine that the quantity of base money is actually used a policy tool, then it is a persuasive argument.  Actually, I think it is wrong but I'm going to leave my reasons for thinking so until a later post.  For now, I just wanted to look at one particular interesting aspect about the claim which is that it says nothing about what happens as the economy moves from one set of nominal prices to another.

I want to consider a very simple scenario.  I'm going to assume an economy at an equilibrium level of output with stable prices.  I then want to imagine a one-off increase in the level of base money and see how the economy might adjust to a new equilibrium.

The first thing to note is that it's not just a question of adjusting all prices.  Aside from base money, there will be a variety of financial assets and liabilities denominated in nominal terms.  As a rise in prices will reduce the real value of these, additional flows will need to take place to restore all the real ratios.  

For example, if the private sector is holding a certain amount of government debt, then the erosion in the real value of this debt may require a period of budget deficit for the private sector to accumulate more nominal debt.  If government spending is fixed in real terms, then this may require a period of reduced real output in order to reduce real tax revenues.

To look at this I used a little model of an economy with three sectors: a private sector, government (including the central bank) and the rest of the world.  The model has only two financial assets: government bonds, held by the private sector and the rest of the world, and base money.  As usual, a full equation listing with parameter values is given at the end of the post.

I then ran a simulation for a 10% increase in base money (achieved by a repurchase of government bonds).  The resulting path of nominal GDP and real GDP are shown below:

What is happening here is as follows.  The increase in base money and repurchase of debt causes a depreciation of the exchange rate.  Most of the exchange rate movement reflects a revision in the expected future exchange rate.  The construction of the model requires a future equilibrium exchange rate 10% lower reflecting a 10% higher price level which in turn reflects a 10% higher base money stock.  In fact, the exchange rate overshoots slightly due to a reduced domestic interest rate resulting from the base money increase.

The exchange rate depreciation initially boosts GDP due to improved trade.  However, the increase in the relative price of foreign goods, coupled with the increase in demand, immediately starts to push up domestic prices.  The erosion of the real value of private sector holdings of financial assets causes the private sector to cut spending in an attempt to accumulate more assets and restore the real ratio of assets to income.  This causes GDP to fall and to persist at a reduced level for some time, slowly returning to its original value.

I would stress that I am not suggesting that this is the inevitable path of GDP in response to monetary easing.  The particular pattern depends very much on the relative elasticities and response rates.  All I am interested in looking at here is the point that the simple intuitive result hides the possibility of a complex response which may be, at least temporarily, the opposite of what might be expected.

Equation listing

Real GDP is made up of private spending, public spending and net exports.

yr = pxr + gr + xr - mr

Nominal GDP is real GDP multiplied by the price level.

Y = yr . p

Private disposable income is a fixed proportion of nominal GDP based on the tax rate.

YD = ( 1 - τ ) . Y

The private sector is assumed to want to hold financial assets in some proportion (θ)to disposable income.  The acquisition of financial assets is based on the difference between this target amount and the outstanding amount.

ΔFA = ε . ( θ . YD - FA_-1 )

Private spending is equal to disposable income less acquisition of financial assets.

pxr = ( YD - ΔFA ) / p

Exports and imports are based on relative prices (the foreign price level is assumed to be unity).   Imports are also based on private expenditure.

xr = xrn . ( p . e )^σx

mr = µ . pxr . ( p . e )^σm

Demand for base money is based on nominal GDP and the domestic interest rate.  Demand for money is equal to supply.

Hd = ( λ₀ + λ₁ . rd ) . Y

Hd = Hs

The first of these equations is rearranged as an equation for the interest rate.

The exchange rate is based on the expected exchange rate for the next period and uncovered interest parity.

e = E[e₊₁] . ( 1 + rd ) / ( 1 + rf )

The expected exchange rate for the next period is a weighted average of the current period level and the long term equilibrium level. 

E[e₊₁] = e^β . E[e_LT]^(1-β)

In the long run, the domestic interest rate tends towards the foreign interest rate and purchasing power parity holds.  The equilibrium exchange rate can be determined from the money demand function as follows.

E[e_LT] = ( λ₀ + λ₁ . rf ) . yrn / Hs

Within this simple model, this rule of thumb mechanism produces expected exchange rates fairly close to their actual outcomes if the value of β is chosen appropriately.

Finally, prices are based on lagged prices, foreign prices and domestic output.

p = p_-1^α . ( 1 / e )^(1-α) . yr / yrn

Variables and Parameters

Variable	Definition	Opening Value
FA	Financial assets	150
Hd	Demand for base money balances	10
Hs	Supply of base money balances	10
Y	Nominal GDP	100
YD	Nominal disposable income	75
e	Exchange rate	1.00
eLT	The long term equilibrium exchange rate	1.00
gr	Real public expenditure	25
mr	Real imports	25
p	Domestic price level	1.00
pxr	Real private expenditure	75
rd	Domestic interest rate	3.00%
rf	Foreign interest rate	3.00%
xr	Real exports	25
xrn	Baseline exports	25
yr	Real GDP	100
yrn	Normal real GDP	100

E[z] denotes the expected value of z.

Parameter	Definition	Value
α	Price adjustment rate	0.75
β	Expected exchange rate weighting	0.75
ε	Adjustment rate of financial asset holdings	0.10
θ	Target ratio of financial asset holdings	2.00
λ0	Money demand parameter	0.25
λ1	Money demand interest rate elasticity	-5.00
µ	Basic propensity to import	0.33
σm	Price elasticity of imports	0.30
σx	Price elasticity of exports	-0.10
τ	Tax rate	0.25

[Edit - thanks to Anton van de Haar for drawing my attention to one or two incorrect values in the parameter listing - now hopefully all correct.]