Question

Suppose an individual has \(W\) dollars to allocate between consumption this period \(\left(c_{0}\right)\) and consumption next period \(\left(c_{1}\right)\) 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 \(\left(W-c_{0}\right)\) b. Suppose that, after the individual makes his or her savings decision (by purchasing one-period bonds), the interest rate decreases to \(r^{\prime} .\) How will this alter the individual's budget constraint? Show the new utilitymaximizing 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 \(c_{0}\) as they are "accrued," show how those gains should be measured. Call this value \(G_{1}\) 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 \(c_{0}\). Show how these realized gains can be measured. Call this amount \(G_{2}\) e. Develop a measure of the true increase in utility that results from the decrease in \(r,\) measured in terms of \(c_{0}\) Call this "true" capital gain \(G_{3}\). Show that \(G_{3}

Short answer

Answer: The decrease in interest rate flattens the budget constraint, making it easier for the individual to borrow and consume more today. Under an income tax system, capital gains are valued as they are "accrued." Under a realized gains tax system, capital gains include only the portion of bonds cashed in to buy additional current-period consumption. A tax policy that taxes only realized gains would overestimate the true increase in utility resulting from the decrease in interest rate, leading to a higher tax burden for the individual compared to taxing the true capital gain.

Step-by-step solution

a. Initial Equilibrium and Savings

The individual has \(W\) dollars to allocate between consumption in period 0 (\(c_0\)) and consumption in period 1 (\(c_1\)). Given the interest rate \(r\), the budget constraint can be represented as: $$(1+r) * (W - c_{0})=c_{1}$$ At the initial equilibrium, the individual maximizes their utility subject to this budget constraint. Graphically, this occurs where the indifference curve is tangent to the budget constraint. The slope of the budget constraint is given by \(-(1+r)\). The total value of current-period savings can be calculated as \(W - c_{0}\).

b. Decrease in Interest Rate and New Utility-Maximizing Position

When the interest rate decreases to \(r'\), the new budget constraint is given by: $$(1+r')*(W - c_{0}) = c_{1}$$ The decrease in the interest rate essentially flattens the budget constraint, making it easier for the individual to borrow and consume more today. In the new equilibrium, the individual's utility-maximizing position changes. Graphically, this occurs where the new indifference curve is tangent to the new budget constraint. The initial bond purchases are now worth more due to the decrease in the interest rate, which can be interpreted as a "capital gain."

c. Capital Gains under Income Tax

Under an income tax system, capital gains \(G_{1}\) are valued in terms of \(c_{0}\) as they are "accrued." To find the value of \(G_{1}\), we need to measure the increase in consumption possibilities due to the decrease in the interest rate: $$G_{1} = (1+r')(W - c'_{0}) - (1+r)(W - c_{0})$$

d. Capital Gains under Realized Gains Tax

Under a realized gains tax system, capital gains \(G_{2}\) are defined to include only the portion of bonds that is cashed in to buy additional \(c_{0}\). To find the value of \(G_{2}\), we need to measure the increase in \(c_{0}\) due to the decrease in the interest rate: $$G_{2} = c_{0}'-c_{0}$$

e. True Capital Gain and Comparison

To find the true increase in utility resulting from the decrease in the interest rate (\(G_{3}\)), we need to derive a measure of it in terms of \(c_{0}\). The true capital gain is the difference between the individual's unconstrained optimal consumption in period 0 in the two scenarios (\(c_{0}\) and \(c_{0}'\)): $$G_{3} = c_{0}' - c_{0}$$ From the formulas derived earlier, we can see that: $$G_{3} < G_{2} < G_{1}$$ This implies that a tax policy that taxes only realized gains (i.e., \(G_{2}\)) would overestimate the true increase in utility resulting from the decrease in interest rate. It would lead to a higher tax burden for the individual compared to taxing the true capital gain (\(G_{3}\)).