Macroeconomics — Money Demand and the Interest Rate

Macroeconomics — The Interest Rate Is the Opportunity Cost of Holding Money

If Chapter 3 dealt with the supply of money (the monetary base and the multiplier), this chapter deals with demand. Money pays almost no interest, so why do people hold it? The answer shapes the LM curve and determines how effective monetary policy is.

1. Three motives for holding money

Keynes's liquidity preference
Motive	Description	Main variable it responds to
Transactions motive	Held for payments because spending and receipts occur at different times	Income (Y)
Precautionary motive	Held against unexpected spending	Uncertainty, income
Speculative (asset) motive	Held as money when bond prices are expected to fall	Interest rate (r)

Taken together, real money demand $L(Y,r)$ rises with income and falls with the interest rate. The interest rate is the return one would have earned by holding bonds instead of money — the opportunity cost of holding money.

2. The Baumol-Tobin model: transactions demand responds to interest rates too

This model shows that even money held for transactions responds to the interest rate. Suppose a person spends $Y$ won evenly over a year, keeps the money in a deposit account and withdraws it in $N$ instalments. Each withdrawal costs $F$ , and average cash holdings are $Y/(2N)$ .

Total cost is forgone interest $i\cdot Y/(2N)$ plus withdrawal costs $N\cdot F$ . Minimizing it with respect to $N$ gives

Baumol-Tobin money demand

M^* = \frac{Y}{2N^*} = \sqrt{\frac{Y\,F}{2\,i}}

Money demand is proportional to the square root of income and inversely proportional to the square root of the interest rate: an income elasticity of 0.5 and an interest elasticity of −0.5. Because of economies of scale, doubling income raises money holdings only about 1.41 times.

With annual spending of 24 million won and a withdrawal cost of 2,000 won:

At an interest rate of 2%: $M^*=\sqrt{24{,}000{,}000\times 2{,}000/0.04}\approx$ 1.1 million won, about 11 withdrawals a year
At an interest rate of 8%: $M^*\approx$ 550,000 won, about 22 withdrawals a year

When the interest rate quadrupled, average cash holdings halved. Technological change that lowers withdrawal costs (mobile banking, card payments) lowers $F$ and reduces money demand.

3. Velocity and the stability of money demand

Writing money demand as $M/P=L(Y,r)$ , velocity is $V=PY/M=Y/L(Y,r)$ . When the interest rate rises, people hold less money, so the same money circulates faster. With nominal GDP of 100 trillion won and a money stock of 20 trillion won, $V=5$ .

The quantity theory of money assumes that $V$ is stable and holds that the money supply determines nominal GDP. From the 1980s, as financial innovation and new payment technology made velocity swing sharply in many countries, central banks that had targeted the money supply switched to interest-rate targets. When money demand is unstable, fixing the money supply makes interest rates and output swing widely.

Money demand shocks and LM: if card payments spread and money demand falls at a given income, then with an unchanged money supply the LM curve shifts right and the interest rate falls. A central bank that targets the interest rate absorbs the extra liquidity to hold the rate, and the money supply adjusts on its own as a result.

4. The liquidity trap

When interest rates are very low, holding bonds yields almost nothing and leaves only the risk that rates will rise (bond prices will fall). People do not convert additional money into bonds but simply hold it. The interest elasticity of money demand approaches infinity and the LM curve becomes flat (Chapter 7). Expanding the monetary base no longer lowers the interest rate.

5. The inflation tax

When the government finances itself by printing money (seigniorage), the real purchasing power of money holders falls by the rate of inflation. In effect it is a tax on holding money.

The inflation tax

\text{Revenue} \approx \pi \times \frac{M}{P}

With real balances of 50 trillion won and inflation of 10%, revenue is about 5 trillion won. When inflation rises, people hold less money and the tax base shrinks. That is why revenue can actually fall in hyperinflation.

Check your understanding

In the Baumol-Tobin model, if income quadruples and the interest rate stays the same, by what factor does money demand change? By $\sqrt{4}=2$ . If the interest rate also quadruples over the same period, the factor is $\sqrt{4/4}=1$ and money demand does not change. With nominal GDP of 100 trillion won and money of 20 trillion won, if money rises to 24 trillion and velocity stays at 5, nominal GDP becomes 120 trillion won; if velocity falls to 4, it is only 96 trillion won.

References

William Baumol, “The Transactions Demand for Cash: An Inventory Theoretic Approach,” Quarterly Journal of Economics (1952)
James Tobin, “The Interest-Elasticity of Transactions Demand for Cash,” Review of Economics and Statistics (1956)
John Maynard Keynes, The General Theory, ch. 13–15
N. Gregory Mankiw, Macroeconomics, ch. 5, 19