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Chapter 17: Problem 12

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.

Short Answer

Expert verified
**Possible short answer:** Hyperbolic discounting is a pattern that can lead to shortsighted behavior because individuals may heavily prioritize immediate gratification, causing them to make suboptimal choices for long-term utility maximization. By constraining their future choices, such as using retirement accounts with penalties for early withdrawal or setting up automatic savings plans, individuals can help mitigate shortsighted behavior and move toward achieving full utility maximization in the long run.

Step by step solution

01

Understand and list the discount factors for the first few periods

Using the given formula for utility: \\[ \text { utility }=U\left(c_{t}\right)+\beta \sum_{\tau=1}^{\tau=T} \delta^{\tau} U\left(c_{t+\tau}\right) \\] We know we have \(\beta = 0.6\) and \(\delta = 0.99\). We can calculate the first few discount factors \(\beta \delta^{\tau}\) as follows: \(\beta \delta^1 = 0.6 * 0.99\) \(\beta \delta^2 = 0.6 * 0.99^2\) \(\beta \delta^3 = 0.6 * 0.99^3\) ... and so on.
02

Find the ratios of the discount factors

Now, calculate the ratios between these discount factors: \(\frac{\beta \delta^2}{\beta \delta^1}\) \(\frac{\beta \delta^3}{\beta \delta^2}\) ... and so on. Notice that this ratio should be approximately constant after the first few terms, indicating a geometric rate of decrease for subsequent periods, which is the general behavior of hyperbolic functions. **Part b: Explain intuitively why this pattern of discount rates can lead to shortsighted behavior**
03

Explain the pattern and its connection to shortsightedness

Due to the hyperbolic pattern observed in Part a, individuals tend to heavily discount the utility of future consumption in the immediate period, but relatively less so for further periods. As a consequence, they might prioritize immediate gratification more than they rationally should, hence leading to shortsighted behavior. **Part c: Calculate the MRS between \(c_{t+1}\) and \(c_{t+2}\) and analyze its implications on dynamically inconsistent choices**
04

Calculate the MRS at time t

MRS (marginal rate of substitution) at time t between \(c_{t+1}\) and \(c_{t+2}\) is given by: \\[ MRS_{t}=\frac{\beta \delta U^{\prime}\left(c_{t+1}\right)}{\beta \delta^{2} U^{\prime}\left(c_{t+2}\right)}=\frac{\delta}{\delta^{2}}=\frac{1}{\delta} \\]
05

Calculate the MRS at time t+1

MRS at time t+1 between \(c_{t+1}\) and \(c_{t+2}\) is given by: \\[ MRS_{t+1}=\frac{U^{\prime}\left(c_{t+1}\right)}{\delta U^{\prime}\left(c_{t+2}\right)}=\frac{1}{\delta} \\]
06

Explain the implications of the MRS values

Since the MRS values at time t and time t+1 are equal (\(\frac{1}{\delta}\)), it suggests that the individual's preference for \(c_{t+1}\) and \(c_{t+2}\) consumption remains constant across t and t+1, which would not lead to dynamically inconsistent choices over time in the presence of a constant real interest rate. However, considering the observed hyperbolic pattern, this result is somewhat counterintuitive. **Part d: Explain why constraints on future selves are necessary**
07

Explain the need for constraints to achieve full utility maximization

As seen in the above analysis, individuals may succumb to shortsighted behavior due to the distinct discount factors with a hyperbolic pattern. Restraining one's future selves in some ways can help mitigate this shortsighted behavior, thus helping them maximize utility in the long run. **Part e: Real world examples of constraining future choices**
08

Provide real-world examples

1. Saving money in a retirement account with penalties for early withdrawal. 2. Establishing a savings plan that automatically deducts money from a paycheck and invests it in a long-term fund. 3. Setting up a "cooling-off" period after making large purchases to avoid impulsive spending and give time for rational evaluation. 4. Signing a contract that commits you to a goal, such as a gym membership or a weight loss program, to ensure long-term goals override short-term temptation.

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Most popular questions from this chapter

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 ? \\]

As in Example \(17.3,\) suppose trees are produced by applying 1 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 real interest rate is \(r,\) 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 \\[ w=e^{-r t} f\left(t^{*}\right) \\] 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 \(u\) -year-old tree will be \(w e^{r u}\) and that this price will exceed the value of the wood in the tree \([f(u)]\) 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 \(\left[f\left(t^{*}\right)\right]\) 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

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.

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?

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?
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