CLASS #21: INFLATION AND SEIGNORAGE

Inflation is always and everywhere a monetary phenomenon.
Milton Friedman (Nobel Prize Winner)

• Inflation.
• Does the government gain from printing money? How much? seignorage.
• Seignorage and the inflation tax.

More on Money Demand

Remember that the demand for real money balances is:

$$m=\frac{Y2-G2}{i}$$

Taking logs (and forgetting about G for the moment) we obtain:

$$\ln{m}=\ln{Y}-\ln{i}$$

The relationship that has been estimated is more like:

$$\ln{m}=\alpha_1 \ln{Y}-\alpha_2 \ln{i}$$

Notice that $\alpha_1$ and $\alpha_2$ are elasticities: the income elasticity of money demand $\alpha_1$ measures how much real money demand grows when output grows by 1%. The interest elasticity $\alpha_2$ measures how much real money demand grows when the nominal interest grows by 1%.
Empirical estimate (Goldfeld 1973) are $\alpha_1=\frac{2}{3}$, $\alpha_1=\frac1{10}$.

The Quantity Theory of Money

Define the velocity of money v as:

$$v=\frac{PY}{M}$$

that is: v measures the speed at which money circulates, the extent to which M is able to support a volume of transactions PY.
The quantity theory of money says that velocity is constant.
This implies that real money balances are only a function of output, and not of the interest rate:

$$\frac{M}{P}=vY$$

Empirically, is velocity constant?

• Certainly not across different inflationary regimes.
  Think of hyperinflation: money is ``hot" (example: 1 day deposits in Mexico).
• Not constant across time: financial innovation.

What are the implications of the "Quantity Theory of Money"?

• For given output, ``inflation is a monetary phenomenon'':
  $m=\frac{M}{P}$ is constant.
  
• A one shot increase in the money supply produces a one-to-one increase in the price level.
• The price level grows (Inflation) at the same rate of growth of the money supply:
  
  $$\ln{P_{t+1}}-\ln{P_t}=\ln{M_{t+1}}-\ln{M_t}.$$
  
  

Our two-period model is ill suited to distinguish between a one-shot increase in the money supply, and a change in the growth rate of the money supply. However with the help of intuition we will try to discuss the effects of both.

Inflation and changes in the growth rate of the Money Supply

At what rate do prices grow (how large is inflation) if the money supply grows at a constant rate? If the demand for real balances m is (as in our two period model and as in the book) a function of output and next period nominal interest rate, that is:

m=L(Y,i)

and if output is exogenously given and constant, the following is a equilibrium:

• The money supply grows at a constant rate $\mu$ ( $\ln{M_{t+1}}-\ln{M_t}=\mu$).
• The price level grows at a constant rate $\mu$ ( $\ln{P_{t+1}}-\ln{P_t}=\mu$).
• The nominal interest rate is constant and equal to $r+\mu$.

provided that the government budget constraint is satisfied. Why is it an equilibrium?

• If prices grow at a constant rate $\mu$, then expected inflation is constant and equal to $\mu$.
• But then the nominal interest rate is constant and equal to $r+\mu$.
• Consequently, real money demand m is also constant over time because i and Y are constant.
• This implies that the initial conjecture, i.e., that the money supply and prices grow at the rate $\mu$, is true:
  
  $$\ln{P_{t+1}}-\ln{P_t}=\ln{M_{t+1}}-\ln{M_t}=\mu$$
  
  

• ... provided that the government budget constraint is satisfied. Why? We will see that in a minute.

so far it appears that Friedman is right: inflation is a monetary phenomenon, but...

The Forward-Looking Nature of Prices

An -expected- increase in the rate of growth of the money supply from 0% to 10% ten years from now triggers inflation today.

• Let us say that in 1998 the central bank is keeping the money supply constant (rate of growth of 0%, M=M₀), and that the public expects it to do so forever. Inflation is 0%, i=r, and real money demand is equal to m₀=L(Y,r) (in all this analysis we assume Y does not change).
• Suddenly, in December 1998, the Central Bank announces that in 2008 it will change its policy, and that money supply from 2008 onward will grow at the constant rate of 10% (until 2007 M is still constant at M₀).
• For the reasoning just made, if money supply is expected to grow at 10% in year 2008, and to grow at that rate forever since, inflation in 2008 will be 10%.
• Since people know this, expected inflation for 2008 will be 10%, so the nominal interest rate in year 2007 will be equal to $r+10\%$.
• Consequently, real money demand m in 2007 will be lower than m₀, because the nominal interest rates have gone up.
• But in 2007 money supply has not changed yet! If M is still M₀, and m has gone down, it must be that prices go up in 2007, before money supply actually increases.
• But then in 2006 people will expect some inflation from 2006 to 2007. This means that the interest rate will be higher than r, and that real money demand m in 2006 will be lower than m₀.
• For the same reasoning, if M is still M₀, and m has gone down, it must be that we observe an increase in prices in 2006.
• Applying this reasoning again and again, we see that prices must increase now, in December 1998, even if money supply will increase only ten years from now.

Conclusion: prices react now to future increases in the money supply.

A One-shot increase in the Money Supply

• let us say that today the government surprises everyone by doubling the money supply (and that everyone is informed that it has doubled).
• However, let us also say that the government promises to keep the money supply constant from now on, and that people believe it.
• Then expected inflation is still zero, and the nominal interest rate and the real money demand do not change.
• Therefore, it must be that prices double:
  one shot increase in the money supply produces a one-to-one increase in the price level.
• Things are a little different if people do not realize immediately that the money supply has doubled, and think that relative prices have changed (Lucas: islands model).

Seignorage

If the Central Bank prints money to buy government paper (or equivalently to finance government expenditures) this means that the government can finance part of its deficit without having to pay interest on it. Let us recall the first and second period government budget constraints:

$$G1-T1=\frac{M2-M1+B}{P1}$$

$$G2-T2=\frac{(1+i)B+M2}{P2}$$

Notice that that the government is "monetizing'', that is, financing by printing money, an amount equal to $\Delta M=M2-M1$ of its deficit.
How much is the government gaining by doing so? If in period 1 the government did not print $\Delta M$ money, it would have to finance the deficit by issuing debt, and in the next period it would have to pay an amount equal to $i \Delta M$.
In real terms, the gain that the government derives from financing its deficit by printing money, called seignorage, is equal to

$$i \frac{\Delta M}{P2}$$

.

Is inflation always a monetary phenomenon?

• Let us say, for example, that the Mexican government decides to keep money supply constant.
• If money supply is constant, no deficit is financed by printing money, and seignorage is zero.
• Let us also say that the government is unable -for political reasons- to raise taxes beyond the current level.
• Assuming that the government budget constraint is now balanced, the loss of seignorage coupled with the impossibility of raising taxes implies that the government is accumulating debt.
• But the government cannot accumulate debt forever...soon or later the government will have to either raise taxes, or restart printing money, and if it cannot do the former, it will do the latter.
• But prices are forward looking...people know that X years in the future the money supply will grow again: prices will continue to grow.
• In conclusion, to stop printing money is not enough to stop inflation:
  a fiscal adjustment is required: inflation is not only a monetary phenomenon - but a fiscal one as well!
• Sargent: The end of Four Big Inflations
• A currency board in Mexico?