# Tobin Mean Variance Model

Tobin's 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 That is,

E(g) = 0 and so E(RB) = i + 0 = i Var(g) = E[g - E(g)]2 = E(g2) = ag2

FIGURE 1 Indifference Curves In a Mean-Variace Model The indifference curves are upward-sloping, and higher indifference curves indicate that utility is higher. In other words,

Expected Return |

Higher Utility

Higher Utility Standard Deviation of Returns a

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.

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:

Then the mean and variance of the return on the portfolio, denoted respectively as |i and a2, can be calculated as follows:

I = E(R) = E(ARb) = AE(Rb) = Ai a2 = E(R — |)2 = E[A(i + g) — Ai]2 = E(Ag)2 = A2E(g2) = A2a\

Taking the square root of both sides of the equation directly above and solving for A yields:

Substituting for A in the equation |i = Ai using the preceding equation gives us:

Equation 3 is known as the opportunity locus because it tells us the combinations of | and a that are feasible for the individual. This equation is written in a form in which the | variable corresponds to the Y axis and the a variable to the X axis. The opportunity locus is a straight line going through the origin with a slope of i/ag. 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 a * in the figure. As Equation 2 indicates, the optimal level of A, A*, is:

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/ag. Given a*, 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 let's ask ourselves what happens when the interest rate increases from ix to i2. This situation is shown in Figure 3. Because ag 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 a^ and a2 the optimal fraction of the portfolio in bonds rises from A*1 to A*2. The result is that as the interest

FIGURE 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 ct*, and using Equation 2 in the bottom half of the figure, we solve for the optimal fraction of the portfolio in bonds A*. rate on bonds rises, the demand for money falls; that is, 1 — A, the fraction of the portfolio held as money, declines.4

Tobin's model then yields the same result as Keynes's 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 Keynes's 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).

FIGURE 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 ct* to ct2, and then Equation 2, in the bottom haf of the figure, shows that the optimal fraction of the portfolio in bonds rises from A* to A*. appendix 2 to chapter 