IF lecture 4

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Questions and Answers

What does the Social Discount Rate (SDR) represent?

The rate at which benefits and losses are traded off across time (correct)

The annual growth rate of consumption

The total utility gained from consumption

The level of inequality in a society

In the formula for Social Discount Rate (SDR), what does 'g' signify?

The growth rate of consumption (correct)

The rate of inflation affecting consumption

The discount rate applied to future consumption

The level of future utility expected

What are the two main components that influence the Social Discount Rate (SDR)?

g and U(Ct)

δ and g (correct)

δ and η

η and U(Ct)

Why is there disagreement regarding the parameters used in the Social Discount Rate?

Different contexts require different interpretations of the parameters Signup and view all the answers

What does the variable δ represent in the Social Discount Rate formula?

The time preference rate Signup and view all the answers

Which value of 'η' is the minimum required in the utility function U(Ct) = $\frac{C_t^{1-η}}{1-η}$?

Greater than or equal to 0 Signup and view all the answers

What does the term 'intergenerational fairness' imply in the context of discounting?

Valuing future generations' benefits similarly to current generations Signup and view all the answers

Which of the following best describes the standard exponentially discounted utilitarian criterion?

A criterion that decreases the weight of future consumption exponentially Signup and view all the answers

What does the author suggest about the interests of posterity in relation to a Utilitarian's perspective?

They are equally important as the interests of contemporaries. Signup and view all the answers

According to Ramsey (1928), what is ethically indefensible?

Discounting future enjoyments compared to present ones. Signup and view all the answers

What was Arrow's key consideration regarding output in his changed view on discounting?

Output consists of completely perishable goods. Signup and view all the answers

What is the general stance of the author towards discounting future enjoyments?

It arises from a lack of imagination. Signup and view all the answers

What aspect of humanity's existence affects a man's happiness, according to the content?

His actions toward future generations. Signup and view all the answers

Which of the following describes Sidgwick's stance as interpreted by Ramsey?

Enjoyments should not be discounted ethically. Signup and view all the answers

What general topic do the phrases 'Discounting: ethically defensible?' and 'Alternative discounting functions' address?

The moral implications of valuing future enjoyment. Signup and view all the answers

What issue does the ongoing debate about discounting primarily revolve around?

The ethical justification for valuing future well-being. Signup and view all the answers

What does time inconsistency in social welfare imply?

It can arise from non-exponential discounting functions. Signup and view all the answers

What is a characteristic of a 'smart' planner regarding time inconsistency?

They anticipate changes in social welfare preferences. Signup and view all the answers

What does a 'modular' theory of intergenerational justice allow for?

It separates attitudes towards discounting from inequality. Signup and view all the answers

Which of the following statements is true about discounting in relation to inequality?

Attitudes towards discounting and inequality can be chosen separately. Signup and view all the answers

Which approach helps clarify policy implications concerning social welfare?

Disentangling discounting from inequality. Signup and view all the answers

What might be a consequence of using a non-exponential discounting function?

It may violate the principle of time consistency. Signup and view all the answers

What do specific conditions in intergenerational justice allow for?

Diverse criteria that better reflect individual views. Signup and view all the answers

In the context of intergenerational fairness, what is an effect of discounting?

It can lead to biased resource allocation. Signup and view all the answers

What is the primary purpose of the aggregator function V?

To aggregate consumption streams in equivalent time. Signup and view all the answers

How is the aggregator function V defined mathematically?

V(y) = W(y(ω)) Signup and view all the answers

What characteristic does the aggregator function V possess according to equivalent time definition?

It is anonymous. Signup and view all the answers

Which of the following is NOT an example of a standard social welfare function mentioned?

Utility-maximizing Signup and view all the answers

What does the function W represent in the context of the aggregator function?

The social welfare function. Signup and view all the answers

In which context is the aggregator function V introduced?

Intergenerational fairness and consumption. Signup and view all the answers

What is implied about the relationship between time-discounting and equivalent time representation?

Time-discounting plays a critical role in how consumption streams are evaluated. Signup and view all the answers

Which term best describes the nature of the aggregator function V?

Anonymous and atemporal. Signup and view all the answers

What phenomenon is used to model individuals' behavior regarding future rewards?

Hyperbolic discounting Signup and view all the answers

According to Thaler's findings, which amount is preferred immediately over a delayed reward?

$15 immediately Signup and view all the answers

What is the trend observed in annual discount rates as delays increase, according to Thaler's findings?

Rates decline over time Signup and view all the answers

What is the discount rate when an individual is indifferent between $60 after 1 year?

139 Signup and view all the answers

How much does an individual weigh an immediate reward compared to a reward delayed by 3 years?

$15 over $100 Signup and view all the answers

What type of discounting shows a declining rate as the time for reward increases?

Hyperbolic discounting Signup and view all the answers

What reward amount is presented after 1 year in Thaler's analysis?

$60 Signup and view all the answers

What is the annual discount rate when an individual considers $30 after 3 months?

277 Signup and view all the answers

What conclusion does Arrow reach regarding the optimal saving rate in the absence of impatience?

It could become arbitrarily close to 100%. Signup and view all the answers

What does Koopmans argue about preference ordering over infinite-dimensional consumption streams?

It must be continuous and sensitive. Signup and view all the answers

What ethical concern does Arrow bring into the debate on discounting?

Discounting may be ethically defensible. Signup and view all the answers

Which aspect of investment does Arrow highlight when discussing the first generation's opportunity?

Investment sacrifices create a perpetual stream of income. Signup and view all the answers

According to Arrow, what happens when time preference is absent?

Greater investments are always favored. Signup and view all the answers

What position did Arrow change regarding discounting due to his analysis?

He now considers discounting necessary. Signup and view all the answers

What does Arrow imply regarding the relationship between impatience and investment?

Impatience makes investments less appealing. Signup and view all the answers

What is indicated by the concept of intergenerational fairness in Arrow's discussion?

Future generations must have equal access to resources. Signup and view all the answers

Study Notes

Intergenerational Fairness - Lecture 4

Lecture was on October 16, 2024
Topics included discounting, inequality, and intergenerational fairness.
Recap of discounting was given.
The Social Discount Rate (SDR) is the rate at which benefits and losses are traded off across time.
The standard exponentially discounted utilitarian criterion is used;
W(Co, C1, ...) = ∑ exp {-t} U(Ct) where U(Ct) = C¹⁻ⁿ / (1-η) for η ≥ 0.
The SDR is given by the Ramsey rule: SDR = δ + ng, where g is the growth rate of consumption.
δ is about discounting; and ng is about inequality.
There is disagreement on how to define and use these parameters.
Several open questions are posed: Why exponential discounting? What are the alternatives? Why the weighted sum of CES utilities? What are the alternatives? How restrictive is thinking in terms of discounting and inequality?

Discounting - Ethical and Defensible?

Most philosophers oppose discounting, but economists often use it.
Sidgwick argued that the time a person exists doesn't affect the value of their happiness
Ramsey agreed with Sidgwick but later included a rate of discount in his analyses.
Arrow (1999) changed his view, concluding that without impatience, the optimal saving rate could approach 100%.
Koopmans considered an infinite world and concluded that ethically-based choice requires a preference ordering and needs impatience in infinite consumption streams.

Discounting: A Justification

The strongest justifications for discounting in general and exponentially-discounted utilitarianism are Koopmans (1960) and Diamond (1965).
Diamond (1965) showed there is no social welfare function that satisfies efficiency, more consumption is better, and finite anonymity.

Koopmans (1960)

Koopmans used an axiomatic approach to rank consumption streams.
Koopmans derived utility functions for each generation and the constant discount factor from a planner's preferences.
He proposed five postulates to establish the social welfare function.

Koopmans' Postulates

Continuity: Small variations in consumption streams do not drastically change social welfare.
Sensitivity: The level of consumption in the first period matters.
Non-complementarity: Future assessments are independent of the present, and vice versa.
Stationarity: Ranking of consumption streams is independent of the starting point in time.
Boundedness: Growing consumption streams cannot be unambiguously better/worse than any constant stream.

Koopmans' Result

A social welfare function that satisfies these postulates can be expressed as: W(C) = ∑ β³ U(Ct)
β∈ (0,1) and U is a continuous and strictly increasing function.

Discussion

The equivalence result of Koopmans' postulates is strong, meaning any disagreement with the result requires disagreement with some postulate(s).
Each postulate should be critically assessed, as some may seem restrictive.
Stationarity and non-complementarity are particularly controversial postulates.

Alternative Discounting Functions

Hyperbolic discounting is suggested to model individual behavior.
Thaler (1981) showed that individuals are indifferent between smaller immediate rewards and larger rewards further in the future.
This indicates that annual discount rates decrease.

Quasi-Hyperbolic Discounting (Laibson, 1997)

Quasi-hyperbolic discounting describes discounting with a different rate for immediate vs later periods.
When β=1 exponentially discounted utilitarianism emerges.
When β<1, there is first-period discounting, but discounting weakens as time extends.
Non-exponential discounting functions violate time consistency, which requires that a better option should remain better regardless of the time frame.

Inequality Versus Discounting

A recent work with Frikk Nesje shows how social welfare functions can disentangle discounting from inequality.
Under specific conditions, attitudes towards discounting are orthogonal to attitudes towards inequality.
This "modular" theory offers many new criteria and policy implications.

The Disentanglement

Transforming consumption streams from calendar time to equivalent time is crucial.
Duration of each generation is adjusted to create a common moral value for consumption at each point in time.
This allows use of any standard aggregation function.

Consumption Streams

Consumption streams are functions from time into the positive reals.
The set of possible consumption streams is denoted by C.
A theory of intergenerational justice is described by a social welfare function W: C → R⁺.

Time-Discounting Function

Time is expressed in calendar time, but can be represented on the unit interval.
A continuous and strictly increasing function transforms calendar time into a comparable scale.

The Aggregator Function

Given a social welfare function and a time discounting function , a complete aggregator function can be defined.
This aggregator function can use different standard social welfare functions depending on the desired criteria.

A modular social welfare function is one in which consumption streams can be represented in equivalent time, with a resulting aggregator function (v) dependent on both a time discounting function and another independent welfare function (W).
All common welfare criteria are modular.

Some Modular Theories

There is a table of various representations of common modular models in economics for intergenerational justice.

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Description

This quiz explores key concepts regarding the Social Discount Rate (SDR), including its formula, components, and ethical implications. Analyze questions related to intergenerational fairness and the perspectives of notable economists such as Ramsey and Arrow. Test your understanding of the SDR and its significance in economics.