532 P A R T V I Monetary Theory
demand for money accurately) implies that velocity is predictable as well. If we can predict what velocity will be in the next period, a change in the quantity of money will produce a predictable change in aggregate spending. Even though velocity is no longer assumed to be constant, the money supply continues to be the primary determinant of nominal income as in the quantity theory of money. Therefore, FriedmanÕs theory of money demand is indeed a restatement of the quantity theory, because it leads to the same conclusion about the importance of money to aggregate spending.
You may recall that we said that the Keynesian liquidity preference function (in which interest rates are an important determinant of the demand for money) is able to explain the procyclical movements of velocity that we find in the data. Can FriedmanÕs money demand formulation explain this procyclical velocity phenomenon as well?
The key clue to answering this question is the presence of permanent income rather than measured income in the money demand function. What happens to permanent income in a business cycle expansion? Because much of the increase in income will be transitory, permanent income rises much less than income. FriedmanÕs money demand function then indicates that the demand for money rises only a small amount relative to the rise in measured income, and as Equation 8 indicates, velocity rises. Similarly, in a recession, the demand for money falls less than income, because the decline in permanent income is small relative to income, and velocity falls. In this way, we have the procyclical movement in velocity.
To summarize, FriedmanÕs theory of the demand for money used a similar approach to that of Keynes but did not go into detail about the motives for holding money. Instead, Friedman made use of the theory of asset demand to indicate that the demand for money will be a function of permanent income and the expected returns on alternative assets relative to the expected return on money. There are two major differences between FriedmanÕs theory and KeynesÕs. Friedman believed that changes in interest rates have little effect on the expected returns on other assets relative to money. Thus, in contrast to Keynes, he viewed the demand for money as insensitive to interest rates. In addition, he differed from Keynes in stressing that the money demand function does not undergo substantial shifts and is therefore stable. These two differences also indicate that velocity is predictable, yielding a quantity theory conclusion that money is the primary determinant of aggregate spending.
Empirical Evidence on the Demand for Money
As we have seen, the alternative theories of the demand for money can have very different implications for our view of the role of money in the economy. Which of these theories is an accurate description of the real world is an important question, and it is the reason why evidence on the demand for money has been at the center of many debates on the effects of monetary policy on aggregate economic activity. Here we examine the empirical evidence on the two primary issues that distinguish the different theories of money demand and affect their conclusions about whether the quantity of money is the primary determinant of aggregate spending: Is the demand for money sensitive to changes in interest rates, and is the demand for money function stable over time?16
16If you are interested in a more detailed discussion of the empirical research on the demand for money, you can find it in an appendix to this chapter on this bookÕs web site at www.aw.com/mishkin.
534 P A R T V I Monetary Theory
economics profession after the sharp drop in velocity during the years of the Great Depression.
3.John Maynard Keynes suggested three motives for holding money: the transactions motive, the precautionary motive, and the speculative motive. His resulting liquidity preference theory views the transactions and precautionary components of money demand as proportional to income. However, the speculative component of money demand is viewed as sensitive to interest rates as well as to expectations about the future movements of interest rates. This theory, then, implies that velocity is unstable and cannot be treated as a constant.
4.Further developments in the Keynesian approach provided a better rationale for the three Keynesian motives for holding money. Interest rates were found to be important to the transactions and precautionary components of money demand as well as to the speculative component.
5.Milton FriedmanÕs theory of money demand used a similar approach to that of Keynes. Treating money like any other asset, Friedman used the theory of asset demand to derive a demand for money that is a function of the expected returns on other assets relative to the expected return on money and permanent income. In contrast to Keynes, Friedman believed that the demand for money is stable and insensitive to interest-rate movements. His belief that velocity is predictable (though not constant) in turn leads to the quantity theory conclusion that money is the primary determinant of aggregate spending.
6.There are two main conclusions from the research on the demand for money: The demand for money is sensitive to interest rates, but there is little evidence that the liquidity trap has ever existed; and since 1973, money demand has been found to be unstable, with the most likely source of the instability being the rapid pace of financial innovation.
Key Terms
equation of exchange, p. 518 |
monetary theory, p. 517 |
real money balances, p. 523 |
liquidity preference theory, p. 521 |
quantity theory of money, p. 519 |
velocity of money, p. 518 |
QUIZ Questions and Problems
Questions marked with an asterisk are answered at the end of the book in an appendix, ÒAnswers to Selected Questions and Problems.Ó
*1. The money supply M has been growing at 10% per year, and nominal GDP PY has been growing at 20% per year. The data are as follows (in billions of dollars):
2001 2002 2003
PY 1,000 1,200 1,440
Calculate the velocity in each year. At what rate is velocity growing?
2.Calculate what happens to nominal GDP if velocity remains constant at 5 and the money supply increases from $200 billion to $300 billion.
*3. What happens to nominal GDP if the money supply grows by 20% but velocity declines by 30%?
4.If credit cards were made illegal by congressional legislation, what would happen to velocity? Explain your answer.
*5. If velocity and aggregate output are reasonably constant (as the classical economists believed), what happens to the price level when the money supply increases from $1 trillion to $4 trillion?
6.If velocity and aggregate output remain constant at 5 and 1,000, respectively, what happens to the price level if the money supply declines from $400 billion to $300 billion?
*7. Looking at Figure 1 in the chapter, when were the two largest falls in velocity? What do declines like this sug-
to chapter |
A Mathematical Treatment of |
appendix 1 |
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22 |
the Baumol-Tobin and Tobin |
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Mean-Variance Models |
Baumol-Tobin Model of Transactions Demand for Money
The basic idea behind the Baumol-Tobin model was laid out in the chapter. Here we explore the mathematics that underlie the model. The assumptions of the model are as follows:
1.An individual receives income of T0 at the beginning of every period.
2.An individual spends this income at a constant rate, so at the end of the period, all income T0 has been spent.
3.There are only two assetsÑcash and bonds. Cash earns a nominal return of zero, and bonds earn an interest rate i.
4.Every time an individual buys or sells bonds to raise cash, a fixed brokerage fee of b is incurred.
Let us denote the amount of cash that the individual raises for each purchase or sale of bonds as C, and n the number of times the individual conducts a transaction in bonds. As we saw in Figure 3 in the chapter, where T0 1,000, C 500, and n 2:
n T0 C
Because the brokerage cost of each bond transaction is b, the total brokerage costs for a period are:
nb bTC0
Not only are there brokerage costs, but there is also an opportunity cost to holding cash rather than bonds. This opportunity cost is the bond interest rate i times average cash balances held during the period, which, from the discussion in the chapter, we know is equal to C/2. The opportunity cost is then:
iC
2
Combining these two costs, we have the total costs for an individual equal to:
bT0 iC COSTS C 2
A Mathematical Treatment of the Baumol-Tobin and Tobin Mean-Variance Models 2
The individual wants to minimize costs by choosing the appropriate level of C. This is accomplished by taking the derivative of costs with respect to C and setting it to zero.1 That is:
Solving for C yields the optimal level of C:
C 2bTi 0
Because money demand Md is the average desired holding of cash balances C/2,
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Md |
1 |
2bT |
0 |
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bT0 |
(1) |
2 |
i |
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2i |
This is the famous square root rule.2 It has these implications for the demand for money:
1.The transactions demand for money is negatively related to the interest rate i.
2.The transactions demand for money is positively related to income, but there are economies of scale in money holdingsÑthat is, the demand for money rises less
than proportionally with income. For example, if T0 quadruples in Equation 1, the demand for money only doubles.
3.A lowering of the brokerage costs due to technological improvements would decrease the demand for money.
4.There is no money illusion in the demand for money. If the price level doubles,
T0 and b will double. Equation 1 then indicates that M will double as well. Thus the demand for real money balances remains unchanged, which makes sense because neither the interest rate nor real income has changed.
1To minimize costs, the second derivative must be greater than zero. We find that it is, because:
d2COSTS |
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2 |
( bT0 ) |
2bT0 |
0 |
dC |
2 |
C |
3 |
C |
3 |
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2An alternative way to get Equation 1 is to have the individual maximize profits, which equal the interest on bonds minus the brokerage costs. The average holding of bonds over a period is just:
T20 C2
Thus profits are:
PROFITS 2i (T0 C ) bTC0
Then:
d PROFITS |
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i |
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bT0 |
0 |
dC |
2 |
2 |
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C |
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This equation yields the same square root rule as Equation 1.
3 Appendix 1 to Chapter 22
Tobin Mean-Variance Model
TobinÕs mean-variance analysis of money demand is just an application of the basic ideas in the theory of portfolio choice. Tobin assumes that the utility that people derive from their assets is positively related to the expected return on their portfolio of assets and is negatively related to the riskiness of this portfolio as represented by the variance (or standard deviation) of its returns. This framework implies that an individual has indifference curves that can be drawn as in Figure 1. Notice that these indifference curves slope upward because an individual is willing to accept more risk if offered a higher expected return. In addition, as we go to higher indifference curves, utility is higher, because for the same level of risk, the expected return is higher.
Tobin looks at the choice of holding money, which earns a certain zero return, or bonds, whose return can be stated as:
RB i g
where i interest rate on the bond g capital gain
Tobin also assumes that the expected capital gain is zero3 and its variance is g2. That is,
E(g) 0 and so E(RB) i 0 i
Var(g) E[g E(g)]2 E(g2) g2
F I G U R E 1 Indifference Curves
in a Mean-Variace Model
The indifference curves are
upward-sloping, and higher indifExpected Return ference curves indicate that utility
is higher. In other words,
U3 U2 U1.
Standard Deviation of Returns
3 This assumption is not critical to the results. If E(g) ≠ 0, it can be added to the interest term i, and the analysis proceeds as indicated.
A Mathematical Treatment of The Baumol-Tobin and Tobin Mean-Variance Models 4
where E expectation of the variable inside the parentheses Var variance of the variable inside the parentheses
If A is the fraction of the portfolio put into bonds (0 ≤ A ≤ 1) and 1 A is the fraction of the portfolio held as money, the return R on the portfolio can be written as:
R ARB (1 A)(0) ARB A(i g)
Then the mean and variance of the return on the portfolio, denoted respectively as and 2, can be calculated as follows:
E(R) E(ARB) AE(RB) Ai
2 E(R )2 E[A(i g) Ai]2 E(Ag)2 A2E(g2) A2 g2
Taking the square root of both sides of the equation directly above and solving for A yields:
Substituting for A in the equation Ai using the preceding equation gives us:
Equation 3 is known as the opportunity locus because it tells us the combinations of and that are feasible for the individual. This equation is written in a form in which the variable corresponds to the Y axis and the variable to the X axis. The opportunity locus is a straight line going through the origin with a slope of i/ g. It is drawn in the top half of Figure 2 along with the indifference curves from Figure 1.
The highest indifference curve is reached at point B, the tangency of the indifference curve and the opportunity locus. This point determines the optimal level of risk* in the figure. As Equation 2 indicates, the optimal level of A, A*, is:
A* *
g
This equation is solved in the bottom half of Figure 2. Equation 2 for A is a straight line through the origin with a slope of 1/ g. Given *, the value of A read off this line is the optimal value A*. Notice that the bottom part of the figure is drawn so that as we move down, A is increasing.
Now letÕs ask ourselves what happens when the interest rate increases from i1 to i2. This situation is shown in Figure 3. Because g is unchanged, the Equation 2 line in the bottom half of the figure does not change. However, the slope of the opportunity locus does increase as i increases. Thus the opportunity locus rotates up and we move to point C at the tangency of the new opportunity locus and the indifference curve. As you can see, the optimal level of risk increases from *1 and *2 the optimal fraction of the portfolio in bonds rises from A*1 to A*2. The result is that as the interest
5 Appendix 1 to Chapter 22
F I G U R E 2 Optimal Choice of
the Fraction of the Portfolio in Bonds
The highest indifference curve is reached at a point B, the tangency of the indifference curve with the opportunity locus. This point determines the optimal risk *, and using Equation 2 in the bottom half of the figure, we solve for the optimal fraction of the portfolio in bonds A*.
Slope = i/ g
Eq. 3
Opportunity
Locus
B
*
A*
Slope = 1/ g
rate on bonds rises, the demand for money falls; that is, 1 A, the fraction of the portfolio held as money, declines.4
TobinÕs model then yields the same result as KeynesÕs analysis of the speculative demand for money: It is negatively related to the level of interest rates. This model, however, makes two important points that KeynesÕs model does not:
1.Individuals diversify their portfolios and hold money and bonds at the same time.
2.Even if the expected return on bonds is greater than the expected return on money, individuals will still hold money as a store of wealth because its return is more certain.
4The indifference curves have been drawn so that the usual result is obtained that as i goes up, A* goes up as well. However, there is a subtle issue of income versus substitution effects. If, as people get wealthier, they are willing to bear less risk, and if this income effect is larger than the substitution effect, then it is possible to get the opposite result that as i increases, A* declines. This set of conditions is unlikely, which is why the figure is drawn so that the usual result is obtained. For a discussion of income versus substitution effects, see David Laidler, The Demand for Money: Theories and Evidence, 4th ed. (New York: HarperCollins, 1993).
A Mathematical Treatment of The Baumol-Tobin and Tobin Mean-Variance Models 6
F I G U R E 3 Optimal Choice of
the Fraction of the Portfolio in Bonds
as the Interest Rate Rises
The interest rate on bonds rises from i1 to i2, rotating the opportunity locus upward. The highest indifference curve is now at point C, where it is tangent to the new opportunity locus. The optimal level of risk rises from *1 to 2*, and then Equation 2, in the bottom haf of the figure, shows that the optimal fraction of the portfolio in bonds rises from A1* to A2*.
Slope = i2/ g
B
* *
1 2
A*
1
A*
2
A
Slope = 1/ g
