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?

<p>Different contexts require different interpretations of the parameters (D)</p> Signup and view all the answers

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

<p>The time preference rate (C)</p> Signup and view all the answers

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

<p>Greater than or equal to 0 (C)</p> Signup and view all the answers

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

<p>Valuing future generations' benefits similarly to current generations (C)</p> Signup and view all the answers

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

<p>A criterion that decreases the weight of future consumption exponentially (B)</p> Signup and view all the answers

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

<p>They are equally important as the interests of contemporaries. (A)</p> Signup and view all the answers

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

<p>Discounting future enjoyments compared to present ones. (C)</p> Signup and view all the answers

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

<p>Output consists of completely perishable goods. (C)</p> Signup and view all the answers

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

<p>It arises from a lack of imagination. (D)</p> Signup and view all the answers

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

<p>His actions toward future generations. (C)</p> Signup and view all the answers

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

<p>Enjoyments should not be discounted ethically. (D)</p> Signup and view all the answers

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

<p>The moral implications of valuing future enjoyment. (C)</p> Signup and view all the answers

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

<p>The ethical justification for valuing future well-being. (A)</p> Signup and view all the answers

What does time inconsistency in social welfare imply?

<p>It can arise from non-exponential discounting functions. (B)</p> Signup and view all the answers

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

<p>They anticipate changes in social welfare preferences. (C)</p> Signup and view all the answers

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

<p>It separates attitudes towards discounting from inequality. (D)</p> Signup and view all the answers

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

<p>Attitudes towards discounting and inequality can be chosen separately. (B)</p> Signup and view all the answers

Which approach helps clarify policy implications concerning social welfare?

<p>Disentangling discounting from inequality. (D)</p> Signup and view all the answers

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

<p>It may violate the principle of time consistency. (D)</p> Signup and view all the answers

What do specific conditions in intergenerational justice allow for?

<p>Diverse criteria that better reflect individual views. (C)</p> Signup and view all the answers

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

<p>It can lead to biased resource allocation. (A)</p> Signup and view all the answers

What is the primary purpose of the aggregator function V?

<p>To aggregate consumption streams in equivalent time. (D)</p> Signup and view all the answers

How is the aggregator function V defined mathematically?

<p>V(y) = W(y(ω)) (D)</p> Signup and view all the answers

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

<p>It is anonymous. (A)</p> Signup and view all the answers

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

<p>Utility-maximizing (C)</p> Signup and view all the answers

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

<p>The social welfare function. (B)</p> Signup and view all the answers

In which context is the aggregator function V introduced?

<p>Intergenerational fairness and consumption. (B)</p> Signup and view all the answers

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

<p>Time-discounting plays a critical role in how consumption streams are evaluated. (B)</p> Signup and view all the answers

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

<p>Anonymous and atemporal. (C)</p> Signup and view all the answers

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

<p>Hyperbolic discounting (B)</p> Signup and view all the answers

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

<p>$15 immediately (D)</p> Signup and view all the answers

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

<p>Rates decline over time (A)</p> Signup and view all the answers

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

<p>139 (B)</p> Signup and view all the answers

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

<p>$15 over $100 (D)</p> Signup and view all the answers

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

<p>Hyperbolic discounting (D)</p> Signup and view all the answers

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

<p>$60 (D)</p> Signup and view all the answers

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

<p>277 (B)</p> Signup and view all the answers

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

<p>It could become arbitrarily close to 100%. (B)</p> Signup and view all the answers

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

<p>It must be continuous and sensitive. (D)</p> Signup and view all the answers

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

<p>Discounting may be ethically defensible. (B)</p> Signup and view all the answers

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

<p>Investment sacrifices create a perpetual stream of income. (D)</p> Signup and view all the answers

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

<p>Greater investments are always favored. (B)</p> Signup and view all the answers

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

<p>He now considers discounting necessary. (A)</p> Signup and view all the answers

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

<p>Impatience makes investments less appealing. (B)</p> Signup and view all the answers

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

<p>Future generations must have equal access to resources. (C)</p> Signup and view all the answers

Flashcards

Universal Happiness

The concept that the timing of someone's existence shouldn't affect the value of their happiness from a universal standpoint.

Intergenerational Fairness

The principle that future generations deserve our concern just as much as individuals living today, considering the potential impact of our actions on them.

Discounting

The process of reducing the value of future benefits or costs compared to present ones.

Ethical Indefensibility of Discounting

The idea that discounting future benefits or costs is ethically wrong because it undermines fairness across generations, arising from our limited ability to imagine the future.

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Uncertain Future

The argument against discounting suggests that if we can't predict the future perfectly, we should not discount it, as future generations might be more important than the present.

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Pragmatic Justification for Discounting

The argument that discounting is necessary for economic decision-making, even though it's ethically questionable.

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Perishable Output

A scenario where economic output consists of perishable goods only, meaning that investment is impossible because any gains cannot be saved.

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Ethical and Practical Discounting

The idea that discounting in economic models is used not only for practical reasons but also for ethical reasons.

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Social Discount Rate (SDR)

The rate at which we value benefits and losses at different points in time. It reflects our preference for present over future outcomes.

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Exponentially Discounted Utilitarian Criterion

A model used to represent how much value we place on future consumption compared to present consumption. It involves a parameter for discounting and a parameter for inequality aversion.

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Growth Rate of Consumption (g)

The rate at which consumption grows over time. It influences the weight given to future generations in the decision-making process.

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Discount Rate (δ)

The parameter in the discounted utilitarian criterion that captures our preference for present over future benefits. It reflects our impatience or desire for immediate gratification.

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Inequality Aversion (η)

The parameter in the discounted utilitarian criterion that reflects how much we dislike inequality in consumption across time. It measures our concern for fairness between generations.

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Ramsey Rule

The formula that links the Social Discount Rate (SDR) to the discount rate (δ), the inequality aversion (η), and the growth rate of consumption (g). It helps us understand how these factors influence our future-oriented decisions.

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Parameter Selection in Intergenerational Fairness

A significant area of debate in intergenerational fairness, focusing on the selection of appropriate values for the discount rate (δ) and the inequality aversion (η).

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Current Debates in Intergenerational Fairness

The ongoing discussion about the validity, applicability, and limitations of the exponentially discounted utilitarian criterion and the Ramsey rule in the context of intergenerational fairness.

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Impatience in Economics

The idea that without a preference for immediate consumption (impatience), individuals would always prefer to save more, even if it means sacrificing current consumption.

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Investment with Perpetual Benefits

A situation where individuals are willing to sacrifice some current consumption to invest in a project that yields a perpetual stream of benefits for future generations.

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Discount Rate

The rate at which future consumption is valued relative to present consumption. A lower discount rate suggests that future generations' well-being is valued more highly.

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Preference Ordering

A preference ordering that ranks infinite consumption streams based on their desirability. A higher-ranked stream is preferred.

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Sensitive Preference Ordering

A preference ordering that is sensitive to changes in consumption at any point in time, meaning even a small increase in consumption at any point would result in a higher ranking.

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Continuous Preference Ordering

A preference ordering that is continuous, meaning small changes in consumption streams should lead to small changes in the ranking, ensuring smoothness in the ordering.

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Koopmans' Argument for Impatience

The idea that if a preference ordering is both continuous and sensitive (as Koopmans argues), then individuals must exhibit some degree of impatience.

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Arrow's Conclusion on Discounting

Arrow's conclusion based on Koopmans' argument that discounting is necessary for making ethical and consistent choices between generations.

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Hyperbolic Discounting

A model of individual behavior that suggests people value immediate rewards more highly than future rewards, leading to declining discount rates over time.

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Indifference Point

A situation where individuals are indifferent between receiving a smaller amount immediately and a larger amount at a later date, demonstrating their diminishing discount rate over time.

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Thaler's Finding

The observation that individuals value immediate gratification more than future rewards, leading to steeply declining discount rates over time.

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Present Value Analysis

The process of determining the current value of a future benefit, based on its perceived value at a specific point in time.

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Cost-Benefit Analysis

The evaluation of the benefits and drawbacks of a particular choice, taking into account the time value of money.

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Technological Optimism

The assumption that future generations will have access to resources and technologies that are not currently available, leading to a lower discount rate.

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Time Consistency

A concept that says if one consumption plan is better than another today, it should still be better in the future, even as time passes.

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Time Inconsistency Problem

An issue that arises when a decision made today about future events might not be consistent with future preferences when those events actually occur.

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Modular Social Welfare Function

Social welfare that takes into account both discounting and inequality.

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Disentangling Discounting and Inequality

The idea that social attitudes towards discounting (valuing present over future) can be considered separately from attitudes towards inequality (fairness between individuals).

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Modular Theory of Intergenerational Justice

A framework that distinguishes between how we value future generations (discounting) and how we address fairness between individuals (inequality).

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Key Advantage of Modular Theory

This theory highlights that views on discounting and inequality can be selected independently, leading to more options and clarity in dealing with intergenerational issues.

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Social Discount Rate

Measures how much we value future well-being, factoring in both the time it occurs and the level of inequality it affects.

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Aggregator function

A function that aggregates consumption streams over time, accounting for both the social welfare function and time discounting.

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Time-discounting function

A mathematical function that reflects our preference for present benefits over future benefits. It captures the idea that a benefit received today is worth more than the same benefit received in the future.

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Equivalent time representation

A way to convert consumption streams across different points in time into equivalent values, enabling fair comparisons of social welfare across generations.

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Additive social welfare function

A social welfare function that assumes the value of a society's welfare is simply the sum of the individual welfare of each member.

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Rank-weighted social welfare function

A social welfare function that weights the welfare of individuals based on their position in the distribution of welfare, giving more weight to those with lower welfare.

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Rank-dependent social welfare function

A social welfare function that is not defined by a simple sum or weighted sum of individual welfares but instead considers the entire distribution of welfare and its impact on social well-being.

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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.

The Modular Social Welfare Function

  • 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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