A quantity equation attempts to link the money supplied to the number of transactions made. The quantity theory of money gives the *quantity equation*

where \(M\) is the quantity of money supplied, \(V\) is the *transaction velocity of money*, which is the number of times a rupee exchanges hands, \(P\) is the price of a typical transaction (the number of rupees exchanged), and \(T\) is the number of transactions made in a year. The right side of the equation looks from the perspective of transactions, whereas the left looks from the perspective of money used to make transactions.

However, it is very difficult to measure \(T\), so we replace the number of transactions \(T\) with the total output of the economy \(Y\) (real GDP). The more the economy produces, the more goods are bought and sold. They are technically not the same, for example because second-hand goods should be counted in \(T\) but are not in \(Y\).

\(V\) in this new equation \(MV = PY\) is called the *income velocity of money*. When using the quantity equation, we usually assume that \(V\) is constant.

Observe that since \(Y\) is the amount of output and \(P\) is the price of one unit of output, \(PY\) is just the nominal GDP.

\(M/P\) is referred to as the *real money balance*, which measures the purchasing power of a stock of money.

If the real GDP is constant, the inflation rate is equal to the money growth rate and if it is growing, the inflation rate is less than the money growth rate.

The building blocks that determine the level of prices are:

- The factors of production and production function, which determine \(Y\).
- The money supply, which determines the nominal value \(PY\) of output (This is due to the assumption that \(V\) is unchanging).

The price level \(P\) is then the ratio of \(PY\) to \(Y\).

If both \(V\) and \(Y\) are fixed, then the percentage change in \(M\) is equal to the percentage change in \(P\).

Thus, quantity theory yields that the central bank, which controls money supply, ultimately controls the inflation rate as well. If the money supply is kept stable, the price level is stable.

The money demand function is similar to the demand function for a good. Since the demand for money balance should be equal to the supply, we get \(M/P = kY\), so the income velocity is \(1/k\). This shows the link between demand and velocity.

If we hold a lot of money for each rupee of income (\(k\) is large), \(V\) is small and money changes hands frequently. Similarly, if \(k\) is small, then \(V\) is large.

The factors of production and production function determine the level of output \(Y\). The money supply determines the nominal value of output \(PY\). The price level \(P\) is thus the ratio of the nominal value of output \(PY\) to the real value of output \(Y\).

The revenue raised by printing of money is called *seigniorage*. A government can finance its spending in three ways - raising revenue through taxes, borrowing from public by selling bonds, and printing money.

The government itself does not print money, it asks the RBI to print it instead.

Seigniorage causes an increase in money supply, which causes inflation. The printing of money to increase revenue is like imposing an *inflation tax* which is levied on the holders of money (old money becomes less valuable).

Some results of inflation are:

- Shoeleather costs are resources wasted when inflation encourages people to reduce their money holdings. This includes the time and transactions costs of frequent bank withdrawals.
- Menu costs are the costs of changing prices, such as printing new menus and mailing new catalogs.
- Since firms don’t raise prices at the same time, relative prices can vary, which distorts resource allocation.
- It complicates long-range planning and comparison of rupee amounts over time.
- Nominal income grows faster than real income, and since taxes are based on nominal income and may not be adjusted for inflation, people pay more taxes even when their real incomes don’t rise.
- When inflation is high, another side effect is that debtors can repay their debt with rupees that aren’t worth much. Inflation lower than expected transfers purchasing power from debtors to creditors. A high inflation situation tends to be variable and less predictable than low inflation, and arbitrary redistributions of wealth are frequent.

*Hyperinflation* is said to occur when the inflation rate exceeds 50% per month. When it occurs, bartering or commodity money becomes prevalent.

Why do banks print money in countries facing hyperinflation? This is due to fiscal policy, when the government has inadequate tax revenue to pay for its spending. To cover this deficit, it prints money. Due to delayed tax collection, real tax revenue drops as inflation rises, and fiscal problems worsen. The reliance on seigniorage worsens the situation.

It can be fixed by reducing government spending and increasing taxes.

The interest rate paid by the banks is called the *nominal interest rate* \(i\) and the increase in purchasing power the *real interest rate* \(r\). If \(\pi\) is the rate of inflation, we have \(i = r + \pi\). This is known as the *Fisher effect*.

Quantity theory together with the Fisher equation tell us how money growth influences nominal interest rate.

Suppose a borrower and lender agree on some nominal interest rate (they do not know the inflation rate over the period of the loan). The *ex ante* real interest rate is that expected by the borrower and lender when a loan is made, and the *ex post* real interest rate is that which is actually realized.

Let \(\pi\) denote the actual future inflation and \(\pi^e\) the expectation of future inflation. Then the ex ante real interest rate is \(i - \pi^e\) and the ex post rate is \(i - \pi\). Since the nominal interest rate cannot adjust to the actual inflation (because it is not know when the nominal interest rate is set), the *modified Fisher effect* says \(i = r + \pi^e\). The ex ante rate \(r\) is determined by the equilibirium in the market as described by the GE model.

The nominal interest rate is essentially the opportunity cost of holding money - it is what is given up by holding money instead of bonds.

The Fisher effect just says that an increase in inflation causes an equal increase in the nominal interest rate, so the real interest rate (wealth) is unchanged.