文档介绍：3 Appendix 1 to Chapter 22
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:
ϭϩ
RB i g
where i ϭ interest rate on the bond
g ϭ capital gain
3 2
Tobin also assumes that the expected capital gain is zero and its variance is g. That is,
ϭϭϩϭ
E(g) 0 and so E(RB) i 0 i
ϭϪ 2 ϭ 2 ϭ2
Var(g) E[g E(g)] E(g ) g
FIGURE 1 Indifference Curves
in a Mean-Variace Model
The indifference curves are
upward-sloping, and higher indif- Expected Return
ference curves indicate that utility U3
is higher. In other words, U
ϾϾ 2
U3 U2 U1.
U1
Higher
Utility
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 analy-
sis proceeds as indicated.
A Mathematical Treatment of The Baumol-Tobin and Tobin Mean-Variance Models 4
where E ϭ expectation of the variable inside the parentheses
Var ϭ variance of the variable inside the parentheses
If A is the fraction of the portfolio put into bonds (0 ≤ A ≤ 1) and 1 Ϫ A is the
fraction of the portfolio held as money, the return R on the