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Chapter 23: Problem 8

Suppose an individual has \(W\) dollars to allocate between consumption this period (Co) and consumption next period \((Q)\) 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 \(\left(W-\mathrm{C}_{0}\right)\) b. Suppose that after the individual makes his or her savings decision (by purchasing oneperiod bonds), the interest rate falls tor', How will this alter the individual's budget con straint? 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 capi tal gains. If all such gains are valued in terms of \(\mathrm{C}_{\mathrm{o}}\) as they are "accrued," show how those gains should be measured. Call this value \(G_{L}\) 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 addi tional \(C_{Q}\). 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 fall in \(r\), measured in terms of \(C_{o .}\) Call this "true" capital gain \(\mathrm{G}_{3}\). Show that \(\mathrm{G}_{3}<\mathrm{G}_{2}<\mathrm{G}^{\wedge}\) 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.)

Short Answer

Expert verified
In this economics problem, an individual allocates their wealth, W, between consumption in the current period, Co, and consumption in the next period, CQ, with given interest rate r. The solution involves (a) graphing the individual's initial equilibrium and indicating the total value of current-period savings (W-Co); (b) examining the effects of a falling interest rate on the budget constraint and capital gain; (c) calculating capital gains (GL) based on accrued value for income tax; (d) calculating realized capital gains, G2, based on cashing in bonds for additional CQ consumption; and (e) developing a new measure of true capital gain, G3, and comparing the different measures of capital gains. The analysis shows that different measures of capital gains (realized, accrued, and true capital gain) do not fully capture the individual's full change in well-being. By solely taxing realized gains (current policy), the true capital gains are potentially overestimated. This indicates that the current policy of taxing only realized gains might not be the most accurate measure of an individual's financial well-being.

Step by step solution

01

Budget Constraint

The budget constraint for this problem is given by \(C_0 + \frac{C_Q}{1+r} = W\) where \(C_0\) is the consumption in the current period, \(C_Q\) is the consumption in the next period, r is the interest rate, and W is the total wealth.
02

Total Value of Savings

To find the total value of current-period savings, subtract \(C_0\) from the total wealth, W: \(W - C_0\). This is represented as a point on the budget constraint line in the graph. #b. Suppose that after the individual makes his or her savings decision (by purchasing one-period 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.#
03

New Budget Constraint Line

When the interest rate falls to r', the new budget constraint becomes \(C_0 + \frac{C_Q}{1+r'} = W\). We can plot this new budget constraint line on the graph, along with the old budget constraint, to visualize the effect on individual's decision-making.
04

New Utility-maximizing Position

The individual will move to a new utility-maximizing position, which will be the tangency point between the new budget constraint line with the utility curve. This will result in a change in both \(C_0\) and \(C_Q\).
05

Capital Gain

The capital gain resulting from the fall in the interest rate will be the difference between the original utility-maximizing position (with the higher interest rate) and the new utility-maximizing position (with the lower interest rate). This can be interpreted as an improvement in the individual's financial situation due to the "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 \(G_{L}\)#
06

Calculation of Accrued Capital Gains

To measure the capital gains (accrued) in terms of \(C_0\), calculate the difference in \(C_0\) values between the original utility-maximizing position and the new utility-maximizing position after the interest rate change: \(G_L = C_{0, new} - C_{0, old}\). #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 \(C_Q\). Show how these realized gains can be measured. Call this amount \(G_{2}\)#
07

Calculation of Realized Capital Gains

To find the realized capital gains, we look at the difference of the change in the consumption of \(C_Q\) between the original and new utility-maximizing positions: \(G_2 = C_{Q,new} - C_{Q,old}\). #e. Develop a measure of the true increase in utility that results from the fall in \(r\), measured in terms of \(C_{0}\). Call this "true" capital gain \(G_{3}\). Show that \(G_{3}<G_{2}<G_1\) What do you conclude about the current policy that taxes only realized gains?#
08

Calculation of True Capital Gain

To find the true capital gain (\(G_3\)), we must find a method to compare the utilities associated with the original and new utility maximizing positions in terms of \(C_0\). We can look at changes in the amount of consumption bundle that can achieve the same level of utility under the new interest rate.
09

Comparison of Capital Gains Measures

Based on the scenario, it is evident that \(G_{3}<G_{2}<G_1\). This implies that not all capital gain measures capture the individual's full change in well-being. When comparing the two methods of taxation, taxing only realized gains (as the current policy does) can lead to potentially overestimating the true capital gains (\(G_3\)) and hence might not be the most accurate measure of an individual's financial well-being. (Note: This problem is adapted from J. Whalley, "Capital Gains Taxation and Interest Rate Changes," National Tax Journal (March 1979]: 87-91.)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Budget Constraint
The budget constraint is a fundamental aspect of economic choices and affects how individuals allocate their wealth over different periods of time. In this scenario, an individual must decide how much of their wealth, denoted as \( W \), to consume now (\( C_0 \)) and how much to save for future consumption (\( C_Q \)). The budget constraint is expressed by the equation: \[ C_0 + \frac{C_Q}{1+r} = W \]Here, \( r \) is the interest rate, which determines how much the saved money will grow over time. The budget constraint represents all possible combinations of current and future consumption that the individual can afford.
This provides a framework for understanding their consumption choices. When graphed, the slope of the line represents the trade-off between current and future consumption, which is deeply influenced by the interest rate. As the interest rate changes, so does the budget constraint, affecting the individual's decision-making process.
Capital Gains Tax
Capital gains tax is a levy on the profit from an investment that is incurred when the investment is sold. In the context of intertemporal choice, it becomes crucial when considering changes in interest rates. When interest rates decrease, the value of bonds or savings invested at higher rates becomes more attractive. This can lead to a perceived increase in wealth, termed as a 'capital gain'. There are different ways to measure these gains:
  • Accrued gains (\(G_L\)): This refers to gains measured while still holding the investment, reflecting potential changes in value.
  • Realized gains (\(G_2\)): These gains are measured only when the investment is sold or liquidity is extracted through consumption.
The decision of taxing accrued versus realized capital gains can significantly impact financial outcomes. Current policies typically tax realized gains, affecting how individuals manage their investments and savings.
Interest Rate Changes
Interest rates have a profound influence on the decisions made by individuals regarding savings and consumption. When the interest rate decreases from \( r \) to \( r' \), the attractiveness of saving diminishes because the future value of saved money will be lower. This change shifts the budget constraint: \[ C_0 + \frac{C_Q}{1+r'} = W \]The impact of a lower interest rate results in a steeper slope, altering the trade-off between current and future consumption. With a reduced cost of future consumption, individuals might adjust their behavior by opting for more current consumption, thus maximizing their utility under the new conditions.
Ultimately, these adjustments illustrate how sensitive financial decisions are to changes in interest rates and demonstrate the dynamic interplay between economic policy and individual financial choices.
Utility Maximization
Utility maximization involves making choices that lead to the highest possible satisfaction or utility. In economic terms, this means selecting a combination of current and future consumption that lies on the highest possible indifference curve given the budget constraint.With the shift in interest rates, the utility-maximizing position also changes. Under the new interest rate \( r' \), a higher utility can be achieved, indicating a better financial position despite the same level of wealth \( W \). This is due to the change in the slope of the budget constraint, allowing for more favorable consumption trade-offs.
Capital gains resulting from such adjustments are perceived as the difference in utility levels between the old and new budget constraints. Measuring "true" capital gains (\( G_3 \)) provides a clearer picture of improved well-being, factoring in both the realized and potential gains. Understanding these concepts is crucial for evaluating economic policies and personal financial strategies aimed at utility maximization.

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Most popular questions from this chapter

An individual has a fixed wealth \((W)\) to allocate between consumption in two periods \((Q\) and \(\mathrm{C}_{2}\) ). The individual's utility function is given by \\[ T J(C \quad C | \\] and the budget constraint is \\[ w=c+\frac{c_{*}}{1+r} \\] where \(r\) is the one-period interest rate. a. Show that in order to maximize utility given this budget constraint, the individual should choose \(C,\) and \(C_{2}\) so that the \(M R S\) (of \(Q\) for \(C_{2}\) ) is equal to \(1+r\) b. Show that \(d C / d r>0\) but that the sign of \(d Q / d r\) is ambiguous. If \(d C_{x} / d r\) is negative, what can you conclude about the price elasticity of demand for \(\mathrm{C}_{2} ?\) c. How might your analysis of this problem be amended if the individual received income in each period \(\left(Y_{f} \text { and } Y_{2}\right)\) such that the budget constraint is given by

As in Example \(23.2,\) suppose trees are produced by applying one unit of labor at time 0. The value of the wood contained in a tree is given at any time \((t)\) by \(f(t)\). If the market wage rate is \(w\) and the instantaneous rate is \(\mathrm{r}\), what is the \(P D V\) of this production process and how should \(t\) be chosen to maximize this \(P D V ?\) a. If the optimal value of \(t\) is denoted by \(t^{*}\), show that the no-pure- profit condition of perfect competition will necessitate that Can you explain the meaning of this expression? b. A tree sold before \(t^{*}\) will not be cut down immediately. Rather, it still will make sense for the new owner to let the tree continue to mature until \(t^{*}\). Show that the price of a M-year-old tree will be \(w e^{m \prime}\) and that this price will exceed the value of the wood in the tree \([/(\text { ( } x)]\) for every value of \(u\) except \(u=t^{*}\) when these two values are equal. c. Suppose a landowner has a "balanced" woodlot with one tree of "each" age from 0 to \(t^{*} .\) What is the value of this woodlot? (Hint: It is the sum of the values of all trees in the lot.) d. If the value of the woodlot is \(V\), show that the instantaneous interest on \(V\) (that is, \(r \cdot V\) ) is equal to the "profits" earned at each instant by the landowner, where by profits we mean the difference between the revenue obtained from selling a fully matured tree \(\left[/\left(f^{*}\right)\right]\) and the cost of planting a new one \((w) .\) This result shows there is no pure profit in borrowing to buy a woodlot, because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree.

This problem focuses on the interaction of the corporate profits tax with firms' investment decisions. a. Suppose (contrary to fact) that profits were denned for tax purposes as what we have called pure economic profits. How would a tax on such profits affect investment decisions? b. In fact, profits are denned for tax purposes as \\[ m^{\prime}=P Q-w L-\text { depreciation } \\] where depreciation is determined by governmental and industry guidelines that seek to allocate a machine's costs over its "useful" lifetime. If depreciation were equal to actual physical deterioration and if a firm were in long-run competitive equilibrium, how would a tax on \(\pi\) 'affect the firm's choice of capital inputs? c. Under the conditions of part (b), how would capital usage be affected by adoption of "accelerated depreciation" policies that specify depreciation rates in excess of physical deterioration early in a machine's life, but much lower depreciation rates as the machine ages? d. Under the conditions of part (c), how might a decrease in the corporate profits tax affect capital usage?

Optimal control theory can be used to generalize the model of intertemporal consumption choice contained in Example 23.1 . Consider the following simple life cycle model: An individual receives wages (w) each period and a return on his or her invested capital. Let \(k=\) capital, \(r=\) market interest rate at which the individual can borrow or lend. During each period, the individual chooses consumption (c) to maximize \\[ f(c) e-p^{\prime} d t \\] where \(p\) is the individual's rate of time preference. Given these assumptions, the intertemporal budget constraint for this problem is \\[ k=w+r k-c \\] with constraints on initial and final \(k\) of the form \(k(0)=k(T)=0\) a. What are the necessary conditions for a maximum for this problem? b. Under what conditions would optimal consumption rise over time? When would con sumption fall over time? c. Suppose \(U(c)=\ln (c),\) what is the optimal pattern of consumption? d. More generally, suppose \\[ U(c)=j \quad 8<1 \\] What is the optimal time pattern for consumption? How does this compare to the special case in part (c)? e. How does the optimal time pattern for consumption in this problem determine this in dividual's measured wealth at various points in the life cycle?

A high-pressure life insurance salesman was heard to make the following argument: "At your age a \(\$ 100,000\) whole life policy is a much better buy than a similar term policy. Under a whole life policy you'll have to pay \(\$ 2,000\) per year for the first four years, but nothing more for the rest of your life. A term policy will cost you \(\$ 400\) per year, essentially forever. If you live 35 years, you'll pay only \(\$ 8,000\) for the whole life policy, but \(\$ 14,000\) for the term policy. Surely, the whole life is a better deal." Assuming the salesman's life expectancy assumption is correct, how would you evaluate this argument? Specifically, calculate the present discounted value of the premium costs of the two policies assuming the interest rate is 10 percent.
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