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 \(M R S\) 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

**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

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.

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**

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**

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} \\]

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} \\]

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**

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**

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.

Key concepts

Intertemporal Utility

Intertemporal utility is a concept that describes how individuals make decisions about consumption over different periods. The idea is to consider not just the immediate benefits, but also the long-term satisfaction derived from consumption. To mathematically express this, economists use an intertemporal utility function, which takes future consumption into account based on certain discount factors.

This calculation allows individuals to prioritize different time periods when considering consumption. Normally, future utility is discounted to reflect a time preference, meaning that immediate benefits often have a higher perceived value than future benefits.

When analyzing intertemporal utility, scholars use parameters such as \( \beta \) and \( \delta \) to adjust how future utilities are factored into present decision-making.

• \( \beta \) typically modifies the importance of immediate versus future consumption, where a smaller \( \beta \) increases the present bias.
• \( \delta \) represents a standard geometric discount factor applied to consumption in each future period.

These parameters create a model that tries to mimic actual human behavior when it comes to decisions spread out over time. By understanding intertemporal utility, individuals and policymakers can better design systems that help people optimize long-term happiness without sacrificing present satisfaction.

Dynamic Inconsistency

Dynamic inconsistency occurs when an individual's preferences change over time in such a way that what was planned or expected at one time does not align with choices made later. This often happens due to hyperbolic discounting, where the perceived value of a later reward changes as the time of receipt approaches.

People who suffer from dynamic inconsistency may make plans for future actions that they do not follow through on because their preference shifts over time.

• For example, you may plan to start saving next month but when the time arrives, you decide to spend instead.
• This inconsistency matters because it leads to making decisions that might not maximize long-term utility, despite plans stating otherwise.

Calculative adjustments like the marginal rate of substitution (MRS) can show how this inconsistency emerges between different time periods.

Hyperbolic discounting can lead to an overvaluation of current consumption and undervaluation of future consumption as periods approach. This inconsistent handling of consumption over time is what economists try to address by understanding and theorizing about dynamic inconsistency.

Marginal Rate of Substitution

The marginal rate of substitution (MRS) is a key concept in understanding how individuals make choices regarding consumption across different time periods. It expresses the rate at which a person is willing to give up a certain amount of one good in exchange for another while maintaining the same level of overall satisfaction or utility.

In the context of hyperbolic discounting and intertemporal utility, the MRS helps to identify how the discounting of future utilities affects decisions over time.

• For instance, calculating MRS between consumption in two consecutive future periods allows us to see the preference shifts from one time period to the next.
• The formula \( MRS_t = \frac{\beta \delta U'(c_{t+1})}{\beta \delta^2 U'(c_{t+2})} \) simplifies to \( \frac{1}{\delta} \), showing how each future consumption period is discounted in relation to another.

By examining the MRS at different points in time, we can assess whether individual decision-making exhibits consistency over time. If preferences do not change, the MRS remains constant. However, varying values can suggest dynamic inconsistency and highlight potential areas where people may deviate from their originally intended consumption paths.

Shortsighted Behavior

Shortsighted behavior refers to the tendency of individuals to focus on immediate consumption at the expense of long-term benefits. This behavior is often a result of hyperbolic discounting, where people disproportionately value rewards available sooner compared to those available later.

The immediate future is greatly prioritized, which leads to decisions like overconsumption and poor savings for the future.

• For example, choosing to eat a dessert now rather than sticking to a diet plan can be an instance of shortsighted behavior.
• In financial terms, overspending today instead of saving for retirement is another manifestation.

This behavior can have significant consequences on personal well-being, affecting financial stability, health, and overall quality of life. Thus, understanding shortsighted tendencies helps in designing strategies where individuals can find ways to self-regulate.

Conferences such as setting up constraints and limitations become essential to counteract such tendencies. By knowing this, individuals can implement measures like savings plans, automatic deductions, and setting penalties, which help in aligning immediate actions with long-term goals.