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 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 \(c_{0}\) 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

In summary, an individual with W dollars to allocate between consumption in two periods, 𝑐₀ and 𝑐₁, may be affected by changes in the interest rate and tax policies on capital gains. A decrease in the interest rate can lead to an increase in potential consumption in both periods, resulting in an improvement in the individual's utility. Measuring capital gains, realized gains, and the true increase in utility can help to better understand and evaluate the implications of various tax policies. Taxing only realized gains might not be the most efficient or equitable approach, as it may underestimate the true increase in utility and not capture the full value of accrued capital gains.

Step-by-step solution

a. Initial Equilibrium and Savings

To find the initial equilibrium and identify the total value of current-period savings, we will first identify the individual's budget constraint equation in both periods, \(c_0\) and \(c_1\): \(c_0 + \frac{c_1}{1+r} \le W\) Now we can graph the individual's budget constraint as a straight line, with \(c_0\) on the horizontal axis and \(c_1\) on the vertical axis: At \(c_0 = W\), \(c_1 = 0\), and at \(c_0 = 0\), \(c_1 = W(1+r)\). So the intercepts on the axes are \((W,0)\) and \((0, W(1+r))\). The individual's initial savings can be calculated by the remaining amount of wealth after consuming in the current period: Savings: \(W - c_0\)

b. Decrease in Interest Rate

When the interest rate decreases from \(r\) to \(r'\), the individual's budget constraint will change. The new budget constraint equation is: \(c_0 + \frac{c_1}{1+r'} \le W\) The intercept on the 𝑐₁-axis will change to \((0, W(1+r'))\). Since \(r' < r\), the new constraint will have a flatter slope, so the individual has a higher potential for consumption in both periods due to capital gain from the decrease in the interest rate. The new utility-maximizing position can be found by determining the tangency point between the indifference curve (representing the individual's preferences) and the budget constraint. The capital gain from the decrease in interest rate allows the individual to move to a higher indifference curve, thus improving their utility.

c. Measuring Capital Gains \(G_1\)

To measure capital gains \(G_1\), we must account for the accrued value of the individual's initial bond purchases as a result of the decrease in the interest rate. The increase in the value of 𝑐₁ is given by: \(G_1 = \frac{c_1}{1+r} - \frac{c_1}{1+r'}\) which represents the difference in value of 𝑐₁ in terms of 𝑐₀ in both scenarios (initial and decreased interest rate).

d. Measuring Realized Gains \(G_2\)

To measure realized gains \(G_2\), we need to calculate only the portion of the bonds that is cashed in to buy additional 𝑐₀ due to the decrease in interest rate: \(G_2 = (\frac{c_1}{1+r} - \frac{c_1'}{1+r'})c_0\) where \(c_1'\) is the consumption in the next period after buying additional 𝑐₀ with the realized gains.

e. True Increase in Utility and Comparison

To measure the true increase in utility (in terms of \(c_0\)) that results from the decrease in \(r\), we can find the equivalent amount of additional \(c_0\) that would give the same increase in utility: \(G_3 = c_0' - c_0\) where \(c_0'\) is the consumption in the current period at the new utility-maximizing position (after the decrease in the interest rate). We can now compare \(G_1\), \(G_2\), and \(G_3\) to understand the implications of a tax policy that taxes only realized gains: \(G_3 < G_2 < G_1\) This implies that taxing only realized gains (\(G_2\)) would underestimate the true increase in utility (\(G_3\)) and would not capture the full value of accrued capital gains (\(G_1\)). Thus, a tax policy that taxes only realized gains may not be the most efficient or equitable approach.

Key concepts

Understanding Budget Constraints

When studying microeconomic theory, a fundamental concept that comes into play is the budget constraint. This constraint represents the combinations of goods and services a consumer can purchase, given their income and the prices of those goods and services.

In the context of the textbook exercise, the budget constraint equation is defined for an individual deciding between consumption in the current period, denoted as \(c_0\), and consumption in the future period, \(c_1\). The equation is set as follows:

\[c_0 + \frac{c_1}{1+r} \le W\]
where \(W\) is the total wealth, and \(r\) is the interest rate. To graph this, one plots \(c_0\) on the horizontal axis and \(c_1\) on the vertical axis. At one extreme, if the individual spends all their wealth now \((c_0=W)\), they save nothing and have zero future consumption \((c_1=0)\). Conversely, if they save everything \((c_0=0)\), their future consumption can grow to \(W(1+r)\), factoring in the interest earned on savings.

This budget line visually and numerically depicts the trade-off between present and future consumption. A point on this line indicates a feasible combination of \(c_0\) and \(c_1\) given the interest rate and wealth.

The Interest Rate Effect on Consumption

Interest rates have a significant impact on a consumer's budget constraint and overall economic behavior, known as the interest rate effect. This concept becomes vital when there's a change in the interest rate after the initial saving decision, as outlined in the textbook exercise.

An individual has initially purchased one-period bonds, choosing a mix of \(c_0\) and \(c_1\) based on the pre-existing interest rate \(r\). But suppose the interest rate falls to \(r'\) later. This change flattens the budget constraint due to the increased present value of future consumption, meaning the same level of savings can now secure more future consumption, or conversely, the individual could consume more today without reducing future consumption as much.

Capital Gain from Interest Rate Fluctuations

The reduction in interest rate generates an unexpected capital gain on the individual's initial bond purchases. If the individual were to sell these bonds now, they would do so at a premium because new bonds reflecting the lower interest rate \(r'\) are less attractive. This capital gain enables the individual to reach a higher utility level, often illustrated by moving to a higher indifference curve on a graph representing their preferences.

Capital Gains Taxation Implications

The taxation of capital gains is a topical issue that's often discussed within microeconomic theory, especially in the context of different tax policies' fairness and efficiency. A capital gain occurs when the value of an asset increases.

The exercise prompts students to consider a scenario in which tax authorities impose an income tax on the value of capital gains as they are accrued or realized. Mathematically, one can measure these gains with the following formulas:

• Accrued gains (\(G_1\)): \(G_1 = \frac{c_1}{1+r} - \frac{c_1}{1+r'}\)
• Realized gains (\(G_2\)): \(G_2 = (\frac{c_1}{1+r} - \frac{c_1'}{1+r'})c_0\)

The exercise goes on to explore the concept of a 'true' capital gain (\(G_3\)), which reflects the actual utility increase obtained from the interest rate decrease. This measure is crucial because it shows that the actual economic benefit to the individual from the decrease in \(r\), measured in terms of the increased consumption now available to them, may be less than the potential increase in their wealth as calculated by realized gains (\(G_2\)) or accrued gains (\(G_1\)).

The key insight from comparing \(G_3\), \(G_2\), and \(G_1\) is that a taxation policy solely based on realized gains fails to capture the true economic benefit experienced by the individual and may incentivize behavioral responses to avoid taxation, leading to efficiency losses in the economy. Understanding these different measurements and their policy implications helps students grasp not just the mechanics of taxation, but also the broader economic impacts of tax policy.