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Chapter 17: Problem 2

Assume that an individual expects to work for 40 years and then retire with a life expectancy of an additional 20 years. Suppose also that the individual's earnings increase at a rate of 3 percent per year and that the interest rate is also 3 percent (the overall) price level is constant in this problem). What (constant) fraction of income must the individual save in each working year to be able to finance a level of retirement income equal to 60 percent of earnings in the year just prior to retirement?

Short Answer

Expert verified
Answer: The individual should save approximately 9.87% of their income annually during their working years.

Step by step solution

01

Calculate the total amount saved during working years

To find the total amount saved during working years, we will use the formula for Future Value of an Increasing Annuity (as earnings grow at 3% per year) FV = Y0 * s * [((1+0.03)^40 - 1) / 0.03] Here, FV is the future value of the saved amount at the end of 40 years, and s is the constant fraction of income that the individual must save annually.
02

Calculate the present value of retirement income

Calculate the present value of the retirement income, which is the money needed for 20 years of retirement life receiving a fixed annual retirement income equal to 60% of Y40. PV = (0.6 * Y0 * (1+0.03)^40) * (1 - (1+0.03)^(-20)) / 0.03
03

Find the constant fraction of income (s)

At the start of the retirement, the total amount saved during the working years must be equal to the present value of required retirement income. Thus, we set FV equal to PV, and solve for s. Y0 * s * [((1+0.03)^40 - 1) / 0.03] = (0.6 * Y0 * (1+0.03)^40) * (1 - (1+0.03)^(-20)) / 0.03 Divide both sides by Y0: s * [((1+0.03)^40 - 1) / 0.03] = 0.6 * (1+0.03)^40 * (1 - (1+0.03)^(-20)) / 0.03 Now solve for s: s = (0.6 * (1+0.03)^40 * (1 - (1+0.03)^(-20))) / [((1+0.03)^40 - 1)] s ≈ 0.0987 or 9.87%
04

Interpret the result

To finance their desired retirement income (60% of their final-year earnings), the individual should save approximately 9.87% of their income annually, for each of their 40 working years.

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Key Concepts

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

Future Value of an Increasing Annuity
Understanding the future value of an increasing annuity is crucial for retirement planning. An annuity is a series of equal payments made at regular intervals, and when it's increasing, each payment grows at a steady rate. This concept helps individuals determine how much money they will have in the future if they save a growing amount each year, considering a constant interest rate.

When planning for retirement, one may expect their earnings to increase over time due to promotions, raises, or inflation adjustments. Using the future value of an increasing annuity formula, we can calculate the final amount of savings accumulated after a set number of years. Here, it's important to factor in the growth rate of the earnings as well as the interest rate to find the total savings that will be available at retirement.
Present Value of Retirement Income
The present value of retirement income is a critical figure for individuals planning for their non-working years. It represents the amount of money you would need today to fund your desired income in retirement, taking into account the time value of money.

With constant interest rates and life expectancy estimates, it's possible to calculate the lump sum needed at the start of retirement to ensure a stable income throughout. The present value calculation discounts the future income back to today's dollars, showing what that future money is 'worth' now. This helps in establishing a saving target during the working years to ensure there are sufficient funds to enjoy a comfortable retirement.
Constant Fraction of Income Saving
The strategy of saving a constant fraction of income each year is a disciplined approach to building a retirement nest egg. Instead of saving random amounts, you determine a specific percentage of your yearly income to set aside for the future.

This approach leverages the power of habit and compound interest. By saving a fixed percentage, your contributions naturally increase as your income grows. This method simplifies financial planning and can potentially lead to more significant retirement savings compared to sporadic or inconsistent savings. When you compute the right fraction to save, using the formulas for future value and present value, you tailor your savings rate precisely to meet your retirement goals.
Financial Planning for Retirement
Financial planning for retirement involves a series of careful calculations and foresighted decisions. The aim is to secure sufficient resources to maintain your desired lifestyle after you stop working.

It requires balancing present needs with future ones, investing wisely, and regularly reviewing your plans to accommodate life changes. Future value and present value calculations are instrumental in this process, helping to set clear savings goals based on anticipated retirement needs. By understanding these concepts and committing to a constant fraction of income saving strategy, individuals can create a robust financial foundation that will support them through their retirement years.

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Most popular questions from this chapter

This problem focuses on the interaction of the corporate profits tax with firms' investment decisions. a. Suppose (contrary to fact) that profits were defined 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 defined for tax purposes as \(\pi^{\prime}=p q-w l-\) 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 \(\pi^{\prime}\) affect the firm's choice of capital inputs? c. Given the conditions of part (b), describe how capital usage would be affected by adoption of "accelerated depreciation" policies, which 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?

An individual has a fixed wealth \((W)\) to allocate between consumption in two periods \(\left(c_{1} \text { and } c_{2}\right) .\) The individual's utility function is given by \\[ U\left(c_{1}, c_{2}\right) \\] and the budget constraint is \\[ W=c_{1}+\frac{c_{2}}{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 \(c_{1}\) and \(c_{2}\) such that the \(M R S\left(\text { of } c_{1} \text { for } c_{2}\right)\) is equal to \(1+r\) b. Show that \(\partial c_{2} / \partial r \geq 0\) but that the sign of \(\partial c_{1} / \partial r\) is ambiguous. If \(\partial c_{1} / \partial r\) is negative, what can you conclude about the price elasticity of demand for \(c_{2} ?\) c. How would your conclusions from part (b) be amended if the individual received income in each period ( \(y_{1}\) and \(y_{2}\) ) such that the budget constraint is given by \\[ y_{1}-c_{1}+\frac{y_{2}-c_{2}}{1+r}=0 ? \\]

The notion that people might be "shortsighted" was formalized by David Laibson in "Golden Eggs and Hyperbolic Discounting" (Quarterly Journal of Economics, May \(1997,\) pp. \(443-77\) ). In this paper the author hypothesizes that individuals maximize an intertemporal utility function of the form \\[ \text { utility }=U\left(c_{t}\right)+\beta \sum_{\tau=1}^{\tau=T} \delta^{\tau} U\left(c_{t+\tau}\right) \\] where \(0<\beta<1\) and \(0<\delta<1 .\) The particular time pattern of these discount factors leads to the possibility of shortsightedness. a. Laibson suggests hypothetical values of \(\beta=0.6\) and \(\delta=0.99 .\) Show that, for these values, the factors by which future consumption is discounted follow a general hyperbolic pattern. That is, show that the factors decrease significantly for period \(t+1\) and then follow a steady geometric rate of decrease for subsequent periods. b. Describe intuitively why this pattern of discount rates might lead to shortsighted behavior. c. More formally, calculate the \(M R S\) between \(c_{t+1}\) and \(c_{t+2}\) at time \(t .\) Compare this to the \(M R S\) between \(c_{t+1}\) and \(c_{t+2}\) at time \(t+1 .\) Explain why, with a constant real interest rate, this would imply "dynamically inconsistent" choices over time. Specifically, how would the relationship between optimal \(c_{t+1}\) and \(c_{t+2}\) differ from these two perspectives? d. Laibson explains that the pattern described in part (c) will lead "early selves" to find ways to constrain "future selves" and so achieve full utility maximization. Explain why such constraints are necessary. e. Describe a few of the ways in which people seek to constrain their future choices in the real world.

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 \(\$ 14,000(=\$ 400 \cdot 35)\) 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.

The calculations in Problem 17.4 assume there is no difference between the decisions to cut a single tree and to manage a woodlot. But managing a woodlot also involves replanting, which should be explicitly modeled. To do so, assume a lot owner is considering planting a single tree at a cost \(w,\) harvesting the tree at \(t^{*},\) planting another, and so forth forever. The discounted stream of profits from this activity is then \\[ V=-w+e^{-r t}[f(t)-w]+e^{-r 2 t}[f(t)-w]+\cdots+e^{-r n t}[f(t)-w]+\cdots \\] a. Show that the total value of this planned harvesting activity is given by \\[ V=\frac{f(t)-w}{e^{-r t}-1}-w \\] b. Find the value of \(t\) that maximizes \(V\). Show that this value solves the equation \\[ f^{\prime}\left(t^{*}\right)=r f\left(t^{*}\right)+r V\left(t^{*}\right) \\] c. Interpret the results of part (b): How do they reflect optimal usage of the "input" time? Why is the value of \(t^{*}\) specified in part (b) different from that in Example \(17.2 ?\) d. Suppose tree growth (measured in constant dollars) follows the logistic function \\[ f(t)=50 /\left(1+e^{10-0.1 t}\right) \\] What is the maximum value of the timber available from this tree? e. If tree growth is characterized by the equation given in part (d), what is the optimal rotation period if \(r=0.05\) and \(w=0 ?\) Does this period produce a "maximum sustainable" yield? f. How would the optimal period change if \(r\) decreased to 0.04 ? Note: The equation derived in part (b) is known in forestry economics as Faustmann's equation.
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