Question

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 MRS between \(c_{t+1}\) and \(c_{t+2}\) at time \(t .\) Compare this to the \(M R S\) between \(c_{l+1}\) and \(c_{l+2}\) 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 seck to constrain their future choices in the real world.

Short answer

Answer: The hyperbolic discount pattern represents shortsighted behavior because individuals heavily discount future consumption in the short term, prioritizing immediate gratification over long-term welfare. This inconsistency in time preferences may lead to suboptimal decision-making. Practical examples of people constraining their future choices include setting up automatic retirement savings contributions, making long-term investments, and using commitment devices to pre-commit to a savings plan or debt repayment goal. These strategies can help individuals align their actions with their long-term well-being and avoid shortsighted behavior due to hyperbolic discounting.

Step-by-step solution

a. Hyperbolic Discount Pattern

We are given the values of \(\beta=0.6\) and \(\delta=0.99\). The discount factors for future consumption in the given utility function are represented as \(\beta\delta^{\tau}\). Let's calculate the discount factors for \(\tau=1, 2, 3, ...\): \(\text{Discount factor for } t+1: \beta\delta^1 = 0.6 \times 0.99 = 0.594\) \(\text{Discount factor for } t+2: \beta\delta^2 = 0.6 \times (0.99)^2 = 0.58806\) \(\text{Discount factor for } t+3: \beta\delta^3 = 0.6 \times (0.99)^3 = 0.58217814\) As we see from the calculations, the discount factor decreases significantly from \(t\) to \(t+1\) and then follows a steady geometric rate of decrease for subsequent periods. This confirms that the discount factors follow a general hyperbolic pattern.

b. Intuition for Shortsighted Behavior

The given pattern of discount rates represents shortsighted behavior because individuals discount future consumption much more heavily in the short term, particularly between periods \(t\) and \(t+1\). This means that they may prioritize immediate gratification over long-term welfare. However, for consumption beyond period \(t+1\), the rate at which the discount factors decrease is less drastic, indicating a greater consideration for future welfare in distant periods. This inconsistency in time preferences can lead individuals to make shortsighted decisions that may not be optimal in the long run.

c. Marginal Rate of Substitution (MRS)

The marginal rate of substitution (MRS) is defined as the ratio of the marginal utility of consumption in two consecutive periods. Let \(MRS_{t+1,t+2}\) denote the MRS between \(c_{t+1}\) and \(c_{t+2}\) at time \(t\) and \(MRS_{l+1,l+2}\) denote the MRS between \(c_{l+1}\) and \(c_{l+2}\) at time \(t+1\). We will calculate both \(MRS_{t+1,t+2}\) and \(MRS_{l+1,l+2}\) as follows: \(MRS_{t+1,t+2} = \frac{U'(c_{t+1})}{U'(c_{t+2})} = \frac{\beta\delta^1}{\beta\delta^2} = \frac{\delta}{1} = 0.99\) \(MRS_{l+1,l+2} = \frac{U'(c_{l+1})}{U'(c_{l+2})} = \frac{\beta\delta^0}{\beta\delta^1} = \frac{1}{\delta} = \frac{1}{0.99}\) Note that \(MRS_{t+1,t+2} < MRS_{l+1,l+2}\). In the context of a constant real interest rate, this implies dynamically inconsistent choices over time because the individual's preference for current consumption over future consumption is greater at time \(t\) than at time \(t+1\). This would result in suboptimal allocation of consumption between periods \(t+1\) and \(t+2\).

d. Constraints for Full Utility Maximization

To achieve full utility maximization, early selves need to find ways to constrain future selves because dynamically inconsistent time preferences lead to shortsighted decisions, as the individual places excessive importance on immediate consumption over future welfare. By imposing constraints on future selves, early selves essentially force future selves to make more optimal choices that align better with the individual's long-term well-being. This can help the individual achieve a more balanced consumption allocation and maximize their overall utility.

e. Constraining Future Choices in the Real World

There are several ways individuals can constrain their future choices to avoid making shortsighted decisions. Some examples include: 1. Retirement savings: People may set up automatic contributions to retirement accounts such as 401(k) or IRA, essentially "locking away" a portion of their income for future use. 2. Long-term investments: Investing in assets with long-term growth potential, such as stocks or real estate, can help individuals prioritize their future financial well-being. 3. Commitment devices: Individuals can use various commitment devices, such as pre-committing to a savings plan or setting a goal to pay off a certain amount of debt by a specific date. This kind of commitment can help provide accountability and motivation to make better long-term choices. By adopting these strategies, individuals can better align their current actions with their long-term well-being and avoid the shortsighted behavior that might arise from hyperbolic discounting.