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Questions and Answers
What is the total taxable income elasticity denoted as in the analysis by Piketty, Saez & Stantcheva (2014)?
What is the total taxable income elasticity denoted as in the analysis by Piketty, Saez & Stantcheva (2014)?
- z
- e (correct)
- Ï„
- dz
In the standard case discussed by Piketty et al (2014), if e2 = e3 = 0, what is the relationship between the elasticities?
In the standard case discussed by Piketty et al (2014), if e2 = e3 = 0, what is the relationship between the elasticities?
- e = e1 + e2 + e3
- e = e1 - e2
- e = e1 (correct)
- e = e1 - e2 - e3
According to the analysis, what is the optimal top labor income tax rate expressed as in terms of 'τ ∗' when considering the taxable income elasticity 'e1'?
According to the analysis, what is the optimal top labor income tax rate expressed as in terms of 'τ ∗' when considering the taxable income elasticity 'e1'?
- 1/(1 + a · e1) (correct)
- (1 + a · e1)/1
- (1−τ )/(1−τ )
- a · e1
What type of reduction in current cash compensation is provided as an example for tax avoidance in the analysis?
What type of reduction in current cash compensation is provided as an example for tax avoidance in the analysis?
Which channel does Piketty, Saez & Stantcheva (2014) identify as contributing to the increase in the top 1% income share with an elasticity denoted as 'e2'?
Which channel does Piketty, Saez & Stantcheva (2014) identify as contributing to the increase in the top 1% income share with an elasticity denoted as 'e2'?
What is the primary goal of financing the government purely through lump-sum taxation?
What is the primary goal of financing the government purely through lump-sum taxation?
In Mirrlees' model of optimal income taxation, what did practical tax policy lack until the late 1990s?
In Mirrlees' model of optimal income taxation, what did practical tax policy lack until the late 1990s?
What key concept did Diamond and Saez introduce in their 2011 publication that changed the approach to optimal taxation?
What key concept did Diamond and Saez introduce in their 2011 publication that changed the approach to optimal taxation?
Why is it necessary to tax economic outcomes like income or consumption if individuals' types cannot be observed?
Why is it necessary to tax economic outcomes like income or consumption if individuals' types cannot be observed?
How did Mirrlees' model impact contract theory in economics?
How did Mirrlees' model impact contract theory in economics?
What happens to tax revenue when the tax rate goes beyond the optimal tax rate?
What happens to tax revenue when the tax rate goes beyond the optimal tax rate?
What is the optimal top tax rate equation from the text?
What is the optimal top tax rate equation from the text?
What does the social welfare weighted net effect of changes in tax rates need to be at the optimal tax point?
What does the social welfare weighted net effect of changes in tax rates need to be at the optimal tax point?
What does the parameter 'a' represent in the context of optimal top income tax?
What does the parameter 'a' represent in the context of optimal top income tax?
In the context of optimal linear tax, what is the condition for the optimal tax rate?
In the context of optimal linear tax, what is the condition for the optimal tax rate?
Based on empirical estimates, what is the value of 'a' in the US tax return data?
Based on empirical estimates, what is the value of 'a' in the US tax return data?
What is the formula for the optimal linear tax rate, capturing the equity-efficiency trade-off?
What is the formula for the optimal linear tax rate, capturing the equity-efficiency trade-off?
How does the optimal tax rate behave in relation to 'g_{top}', according to the text?
How does the optimal tax rate behave in relation to 'g_{top}', according to the text?
What does e represent in the equation dB = −τ × e × Zdτ / (1−τ)?
What does e represent in the equation dB = −τ × e × Zdτ / (1−τ)?
What is the significance of 'etop' in determining the optimal top tax rate?
What is the significance of 'etop' in determining the optimal top tax rate?
Why is ν' considered to be the slope rate of the curve?
Why is ν' considered to be the slope rate of the curve?
What does the condition FOC (first-order condition) (1 − τ ) = v ′ (zi ) imply in the context of optimal linear tax?
What does the condition FOC (first-order condition) (1 − τ ) = v ′ (zi ) imply in the context of optimal linear tax?
Why does a steeper slope in the linear tax system imply individuals keep less of each additional dollar earned?
Why does a steeper slope in the linear tax system imply individuals keep less of each additional dollar earned?
What is the significance of Z(1 − τ ) = zi f (wi )dw in the context of optimal linear tax?
What is the significance of Z(1 − τ ) = zi f (wi )dw in the context of optimal linear tax?
In the context of optimal linear tax, what does the condition dU/dthing on x-cx_ij dzU/dthing on y-cx_ij slope ind, curveL -MRS= signify?
In the context of optimal linear tax, what does the condition dU/dthing on x-cx_ij dzU/dthing on y-cx_ij slope ind, curveL -MRS= signify?
What is the purpose of using a luck measure in the empirical analysis described?
What is the purpose of using a luck measure in the empirical analysis described?
In the context of the study, what does it mean when it is said that there is 'pay for luck'?
In the context of the study, what does it mean when it is said that there is 'pay for luck'?
What methodological approach is used to estimate the luck component in CEO pay in Piketty et al.'s empirical analysis?
What methodological approach is used to estimate the luck component in CEO pay in Piketty et al.'s empirical analysis?
How has 'pay for luck' changed since 1987 according to the analysis presented?
How has 'pay for luck' changed since 1987 according to the analysis presented?
Which factor has not been affected by the change in top tax rates according to the study findings?
Which factor has not been affected by the change in top tax rates according to the study findings?
What does a Linear Tax Rate of less than 1 reflect?
What does a Linear Tax Rate of less than 1 reflect?
What does the red line on the graph represent?
What does the red line on the graph represent?
What is the slope of the red line if the tax rate is 30%?
What is the slope of the red line if the tax rate is 30%?
What does the Optimal Linear Tax Revenue-Maximizing Linear Tax Rate maximize?
What does the Optimal Linear Tax Revenue-Maximizing Linear Tax Rate maximize?
What does the Laffer Curve illustrate?
What does the Laffer Curve illustrate?
What does the elasticity 'e' represent in the formula Ï„ max = 1 / (1+e)?
What does the elasticity 'e' represent in the formula Ï„ max = 1 / (1+e)?
What happens if Ï„ exceeds Ï„ max according to the text?
What happens if Ï„ exceeds Ï„ max according to the text?
'Z' in the formula R = Ï„ Z represents what?
'Z' in the formula R = Ï„ Z represents what?
'R′' in the formula for maximizing tax revenue represents what?
'R′' in the formula for maximizing tax revenue represents what?
Explain why ν' is considered the slope rate of the curve in the context of optimal linear tax.
Explain why ν' is considered the slope rate of the curve in the context of optimal linear tax.
What does the slope being 1 indicate in the context of the red curve?
What does the slope being 1 indicate in the context of the red curve?
Why does a steeper slope in the linear tax system imply individuals keep less of each additional dollar earned?
Why does a steeper slope in the linear tax system imply individuals keep less of each additional dollar earned?
What does the condition FOC (first-order condition) (1 − τ ) = v' (zi ) imply in the context of optimal linear tax?
What does the condition FOC (first-order condition) (1 − τ ) = v' (zi ) imply in the context of optimal linear tax?
What is the significance of Z(1 − τ ) = zi f (wi )dw in the context of optimal linear tax?
What is the significance of Z(1 − τ ) = zi f (wi )dw in the context of optimal linear tax?
What is the purpose of the tax function T(z) in the context of individual preferences?
What is the purpose of the tax function T(z) in the context of individual preferences?
Explain the significance of the social welfare function W = G(ui)f(wi)dw.
Explain the significance of the social welfare function W = G(ui)f(wi)dw.
What is the key characteristic of the function G(u) in transforming individual utilities into social welfare contributions?
What is the key characteristic of the function G(u) in transforming individual utilities into social welfare contributions?
In the context of optimal linear tax, what does the condition FOC (first-order condition) (1 - Ï„) = v'(zi) imply?
In the context of optimal linear tax, what does the condition FOC (first-order condition) (1 - Ï„) = v'(zi) imply?
What does the condition Z(1 - Ï„) = zi f(wi)dw represent in the context of optimal linear tax?
What does the condition Z(1 - Ï„) = zi f(wi)dw represent in the context of optimal linear tax?
What are the three channels analyzed by Piketty et al (2014) to explain the increase in the top 1% income share?
What are the three channels analyzed by Piketty et al (2014) to explain the increase in the top 1% income share?
In the context of total taxable income elasticity, what is the formula to calculate the total elasticity 'e'?
In the context of total taxable income elasticity, what is the formula to calculate the total elasticity 'e'?
What is the optimal tax rate formula in terms of 'τ ∗' when considering the taxable income elasticity 'e1'?
What is the optimal tax rate formula in terms of 'τ ∗' when considering the taxable income elasticity 'e1'?
How does the total taxable income elasticity 'e' relate to the tax avoidance ability of an agent?
How does the total taxable income elasticity 'e' relate to the tax avoidance ability of an agent?
What is the condition for optimal linear tax rate that captures the equity-efficiency trade-off?
What is the condition for optimal linear tax rate that captures the equity-efficiency trade-off?
What is the purpose of using individual-specific lump sum taxes in the context of optimal income taxation?
What is the purpose of using individual-specific lump sum taxes in the context of optimal income taxation?
According to Mirrlees (1971), what is the impact of the model on practical tax policy before the late 1990s?
According to Mirrlees (1971), what is the impact of the model on practical tax policy before the late 1990s?
What is the key contribution of Diamond (1998), Piketty (1997), and Saez (2001) to optimal taxation?
What is the key contribution of Diamond (1998), Piketty (1997), and Saez (2001) to optimal taxation?
In the context of optimal taxation, why must economic outcomes like income or consumption be taxed even when individuals' types cannot be observed?
In the context of optimal taxation, why must economic outcomes like income or consumption be taxed even when individuals' types cannot be observed?
What is the new approach introduced by Diamond and Saez in 2011 that changed the approach to optimal taxation?
What is the new approach introduced by Diamond and Saez in 2011 that changed the approach to optimal taxation?
Explain the perturbation argument used to derive the optimal tax rate in the context of the Optimal Linear Tax model.
Explain the perturbation argument used to derive the optimal tax rate in the context of the Optimal Linear Tax model.
What is the condition for the optimal tax rate in the Optimal Linear Tax model?
What is the condition for the optimal tax rate in the Optimal Linear Tax model?
How is the optimal tax rate derived in the Optimal Top Income Tax model?
How is the optimal tax rate derived in the Optimal Top Income Tax model?
What does the formula for the optimal linear tax rate reveal about the equity-efficiency trade-off?
What does the formula for the optimal linear tax rate reveal about the equity-efficiency trade-off?
Explain the significance of the three elasticities (e, e*, e1) in the context of A Tale of Three Elasticities.
Explain the significance of the three elasticities (e, e*, e1) in the context of A Tale of Three Elasticities.
Define the social marginal welfare weight gi in terms of public funds for an individual with earnings zi.
Define the social marginal welfare weight gi in terms of public funds for an individual with earnings zi.
What does gi = 1 for all individuals represent in the context of social preferences for redistribution?
What does gi = 1 for all individuals represent in the context of social preferences for redistribution?
Explain the significance of gi > 1 for individuals with low income and gi < 1 for individuals with high income in the context of Utilitarian preferences.
Explain the significance of gi > 1 for individuals with low income and gi < 1 for individuals with high income in the context of Utilitarian preferences.
Derive the formula for the marginal social welfare weight α in the case of quasi-linear utility.
Derive the formula for the marginal social welfare weight α in the case of quasi-linear utility.
What does the condition FOC (first-order condition) (1 − τ ) = v ′ (zi ) imply in the context of optimal linear tax?
What does the condition FOC (first-order condition) (1 − τ ) = v ′ (zi ) imply in the context of optimal linear tax?
What factors contribute to determining the optimal top tax rate according to the text?
What factors contribute to determining the optimal top tax rate according to the text?
How does the optimal top tax rate change with redistributive tastes ('g_top')?
How does the optimal top tax rate change with redistributive tastes ('g_top')?
What does the parameter 'a' represent in the context of optimal top income tax?
What does the parameter 'a' represent in the context of optimal top income tax?
What is the formula for the optimal top tax rate derived in the text?
What is the formula for the optimal top tax rate derived in the text?
How does the optimal top tax rate change with the thickness of the top tail ('a')?
How does the optimal top tax rate change with the thickness of the top tail ('a')?
What effect does a higher taxable income elasticity ('etop') have on the optimal top tax rate?
What effect does a higher taxable income elasticity ('etop') have on the optimal top tax rate?
Explain how redistributive tastes ('g_top') impact the optimal top tax rate.
Explain how redistributive tastes ('g_top') impact the optimal top tax rate.
How does the optimal top tax rate change with efficiency ('etop')?
How does the optimal top tax rate change with efficiency ('etop')?
Define the parameter 'a' in the context of top income taxation.
Define the parameter 'a' in the context of top income taxation.
What does the weighted private utility effect consider in the context of optimal top income tax?
What does the weighted private utility effect consider in the context of optimal top income tax?
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Study Notes
Here are the study notes for the provided text:
Overview of Optimal Taxation
- From an efficiency perspective, the government should be financed purely through lump-sum taxation.
- With redistributional concerns, the ideal is to levy individual-specific lump sum taxes, taxing higher-ability individuals more.
- However, this is not possible since individuals' types cannot be observed, so the government must tax economic outcomes (e.g., income or consumption), which leads to distortions.
Mirrlees (1971)
- Mirrlees' model pioneered optimal income taxation based on a structural approach.
- His model had a significant impact on information economics, particularly in contract theory.
- Until the late 1990s, the model had little influence on practical tax policy.
- Since then, researchers like Diamond (1998), Piketty (1997), and Saez (2001) have connected Mirrlees' model to practical tax policy and empirical tax studies.
- This new approach is summarized in Diamond-Saez JEP'11 and Piketty-Saez Handbook'13.
Social Welfare Function
- The goal of the government is to maximize social welfare.
- The social welfare function, W, is a concave function that transforms individual utility into social welfare, with a focus on redistributive tastes.
- The function is defined as W = G(ui)f(wi)dw, where G(·) is an increasing and concave function.
Optimal Linear Tax
- A linear tax system consists of a lump-sum grant, R, and a marginal tax rate, Ï„, on taxable income, z.
- The tax function is T(z) = Ï„z - R, funded by taxes Ï„Z, where Z is aggregate earnings.
- The revenue-maximizing linear tax rate, Ï„max, is given by Ï„max = 1/(1+e), where e is the elasticity of aggregate earnings with respect to the net-of-tax rate (1 - Ï„).
Optimal Top Income Tax
- The optimal top income tax rate, Ï„top, is derived using a perturbation argument.
- It depends on the weighted average of private utility effects, dB, and the mechanical effect, dM.
- The optimal tax rate is given by Ï„top = 1 - gtop/(1 - gtop + etop a), where gtop is the average social welfare weight of the rich, etop is the top income elasticity, and a is a constant.
A Tale of Three Elasticities
- Piketty, Saez, and Stantcheva (2014) analyze the channels through which the top 1% income share has increased.
- The channels are: standard supply side (e1), tax avoidance and income shifting (e2), and compensation bargaining and rent seeking (e3).
- In the simplest case, e = e1 + e2 + e3, where e is the total taxable income elasticity.
- The optimal top tax rate depends on the three elasticities and the redistributive tastes of society.### Optimal Taxation
- The Laffer curve suggests that there is an optimal tax rate (Ï„*) where tax revenue is maximized, beyond which increasing tax rates lead to decreased revenue.
Optimal Linear Tax
- The optimal linear tax rate (τ) is a function of elasticity (e), income (Z), and the social welfare weight (ḡ)
- The formula captures the equity-efficiency trade-off: τ* ↓ ḡ, τ* ↓ e
- Without efficiency costs (e = 0), τ* = 1; if society does not value redistribution (ḡ = 1), τ* = 0
- If social preferences are Rawlsian (ḡ = 0), τ* is the revenue-maximizing tax rate
Social Welfare Function
- Government's revenue constraint: ∫[Z · T(z)]f(w)dw = 0
- Social marginal welfare weight (gi) measures the dollar value to society of increasing consumption of individual i by $1
- gi = G'(ui)/GÌ„', where G'(ui) is the marginal utility of individual i and GÌ„' is the mean of G'(ui)
Optimal Top Income Tax
- The optimal top income tax rate (Ï„top*) is a function of elasticity (etop), income (z), and the social welfare weight (g.top)
- Ï„top* decreases with g.top (redistributive tastes), etop (efficiency), and a (thickness of top tail)
- Empirical estimates suggest a ≈ 1.5 in US tax return data, resulting in an optimal top tax rate of 0.73
Empirical Evidence
- Top income shares in Switzerland have remained relatively stable since 1980
- a ≈ 1.9 in Switzerland, implying a different optimal top tax rate than in the US
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