23.8 Suppose an individual has W dollars to allocate between consumption this period (C0) and consumption next period (C1) and that the interest rate is given by r.

a. Graph the individual’s initial equilibrium and indicate the total value of current-period savings (W
C0).

b. Suppose that after the individual makes his or her savings decision (by purchasing oneperiod bonds), the interest rate falls to r. How will this alter the individual’s budget constraint? Show the new utility-maximizing position. Discuss how the individual’s improved position can be interpreted as resulting from a “capital gain” on his or her initial bond purchases.

c. Suppose the tax authorities wish to impose an “income” tax based on the value of capital gains. If all such gains are valued in terms of C0 as they are “accrued,” show how those gains should be measured. Call this value G1.

d. Suppose instead that capital gains are measured as they are “realized”—that is, capital gains are defined to include only that portion of bonds that is cashed in to buy additional C0. Show how these realized gains can be measured. Call this amount G2.

e. Develop a measure of the true increase in utility that results from the fall in r, measured in terms of C0. Call this “true” capital gain G3. Show that G3 G2 G1. What do you conclude about the current policy that taxes only realized gains?

(Note: This problem is adapted from J. Whalley, “Capital Gains Taxation and Interest Rate Changes,” National Tax Journal (March 1979]: 87–91.)