## Annex I. Proofs for Two Propositions

## Source details

**Canonical URL:** [Annex I. Proofs for Two Propositions](https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf.json)

---

### Overview
- Location: page 22.
- Content: Proofs for two propositions regarding the impact of social versus individual work preferences on the optimal marginal income tax schedule and on progressivity. (Details summarized from supplied excerpts.)

### Model and key assumptions
- Framework extends Diamond (1998) by allowing government and individual preferences over labor/leisure to differ:
  - Homogeneous preference difference: 훿 (government utility u_g(c,l) = c + (1 + 훿) v(1 − l)).
  - Heterogeneous preference differences: 훿(n) = 훿 + 휑(n) with 휑(·) monotone in skill n.
- Focus case: 훿 < 0 (government values work relatively more than individuals), and cases with 훿(·) < 0.
- Government chooses non-linear tax T(z) as function of earnings z; informational asymmetry: earnings observed, labor supply not fully observable.
- Government budget constraint: aggregate consumption ≤ aggregate earnings − government expenditures E.
- Social welfare: W = ∫ G[U_n(c(n), l(n))] f(n) dn with G increasing and strictly concave.

### Proposition 1 — Impact on the optimal marginal income tax schedule
- Statement:
  - For a given negative 훿, the optimal marginal income tax schedule T′(z) shifts downward for all income levels relative to the benchmark model with 훿 = 0.
  - The magnitude of the downward shift is inverted U-shaped across z (largest decreases at middle incomes).
- Proof highlights and expressions (verbatim expressions summarized from source):
  - Definitions used: x(z) ≡ ∫[...] (expression in source) and ∆x(z) ≡ ...; Taylor expansion applied to relate ∆T′ to 훿.
  - Key monotonicities noted in source:
    - x(z) is roughly monotonically decreasing in z so that [...] is monotonically increasing in z.
    - Another factor [...] is monotonically decreasing in z.
  - Conclusion: For negative 훿 (example 훿 = −0.2), ∆T′(z) < 0 for all z and |∆T′(z)| first increases and then decreases in z. Q.E.D.

### Proposition 2 — Impact on the progressivity of the optimal tax-transfer system
- Statement:
  - If 훿 ≠ 0 (specifically 훿 < 0 in the focus case), the optimal income tax schedule tends to be less progressive compared to the benchmark model with 훿 = 0.
- Proof highlights:
  - When 훿 decreases from 0 to a negative value, both marginal tax rates T′ and the optimal demogrant −T(0) decrease.
  - For low-income individuals: the decrease in −T(0) tends to dominate the decrease in T′ → average tax rate for low-income individuals tends to increase.
  - For high-income individuals: the decrease in T′ tends to dominate the decrease in −T(0) → average tax rate for high-income individuals tends to decrease.
  - Therefore, overall progressivity tends to decline when 훿 < 0, though local patterns reflect Proposition 1's inverted U-shape in ∆T′. Q.E.D.

### Numerical simulations — setup and calibration (as reported)
- Pre-tax income distribution:
  - Log-normal with µ = ln(30000) = 10.31 and σ = sqrt{2 * (ln(35000) − ln(30000))} = 0.56.
  - Top tail replaced by Pareto with parameter α = 3 starting at €75,649 (95th percentile of log-normal).
  - Median income €30,000; mean ~ €35,000.
- Behavioral and welfare parameters:
  - Elasticity of taxable income benchmark ε(z) ≡ 0.25.
  - Marginal social welfare curvature g(z) = 1 / z^β with benchmark β = 1.
  - Marginal cost of public funds λ equals average of g(z) under quasi-linear preferences.
  - Government must collect 10 percent of total labor income to finance E.

### Simulation results — benchmark (훿 = 0)
- Optimal marginal income tax rates:
  - Non-negative everywhere (Mirrlees result).
  - U-shaped pattern across income (Diamond (1998) result).
- Optimal demogrant −T(0):
  - Ratio to median income (€30,000) = 66 percent.
- Average tax rates:
  - Negative for low-income individuals because of lump-sum grant.
  - Average tax rates increase in z → schedule is progressive.

### Simulation results — homogeneous preference differences (훿 = −0.2, −0.6, −1)
- General patterns for 훿 < 0:
  - Optimal marginal tax rates shift down for all incomes relative to 훿 = 0; the downward shift is inverted U-shaped across income.
  - Optimal schedule becomes less progressive as 훿 decreases.
- Demogrant as share of median (€30,000):
  - 66 percent when 훿 = 0.
  - 62 percent when 훿 = −0.2.
  - 52 percent when 훿 = −0.6.
  - 31 percent when 훿 = −1.
- Sensitivity to redistributive taste β (with 훿 = −1):
  - Increasing β from 1 to 5 raises marginal tax rates at all incomes and raises demogrant/median from 31 percent (β = 1) to 87 percent (β = 5).
- Sensitivity to elasticity ε (with 훿 = −1, β = 1):
  - Increasing ε from 0.25 to 0.275 decreases marginal tax rates at all incomes and makes them negative at low incomes.
  - Demogrant/median falls from 31 percent (ε = 0.25) to 21 percent (ε = 0.275).

### Simulation results — heterogeneous preference differences (specification and outcomes)
- Specification used: 훿(z) = −1 − γ * log(H(z)), H(z) = CDF of z; benchmark γ values:
  - γ = 0 → homogeneous case (훿 = −1).
  - γ = 0.1 → heterogeneous case with 휑(z) = −γ * log(H(z)) negative and increasing in z.
- For 훿 = −1 and γ = 0.1 (β = 1, ε(z) ≡ 0.25):
  - Optimal marginal tax rates shift down relative to γ = 0 for all incomes.
  - Marginal tax rates are negative at low incomes → heterogeneous differences reinforce the potential role for wage subsidies.
  - Demogrant/median decreases from 31 percent (γ = 0) to 21 percent (γ = 0.1).
  - Optimal schedule becomes less progressive as γ increases.

### Key theoretical and policy findings (concise)
- Allowing government preferences to weight work more (훿 < 0) lowers optimal marginal income tax rates across the earnings distribution relative to the benchmark; decreases are largest at middle incomes (inverted U-shape).
- Lower marginal income taxes imply lower revenues and a lower uniform demogrant, which tends to reduce overall progressivity.
- Higher elasticity of taxable income increases the case for wage subsidies at the bottom of the income distribution under 훿 < 0.
- Heterogeneous social preferences that weight work more for lower earners amplify the case for wage subsidies at low incomes and further reduce progressivity.
- The framework rationalizes policy variation: groups with high labor supply elasticity and social preference for work → low benefit generosity and low MITs or wage subsidies; groups constrained by care responsibilities → higher benefit generosity with medium-to-high MITs, potentially phased down with conditional subsidies or ALMP participation.

### Concluding remarks (as presented)
- Introducing social versus individual preference differences over labor/leisure provides a mechanism by which negative marginal income taxes (wage subsidies) can be optimal under certain parameters.
- Principal equilibrium effects of a social preference for work (훿 < 0):
  - Reduction in optimal MITs across earnings distribution.
  - Reduction in uniform demogrant and overall revenues.
  - Lower progressivity of the tax-transfer system.
- Heterogeneous social preferences that weight work more for lower earners magnify these effects and support targeted wage subsidies or in-work conditional transfers.

*Source: wpiea2022062-print-pdf - Annex I. Proofs for Two Propositions; canonical URL: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf*

### Annex I. Proofs for Two Propositions ...................................................................................

### Annex I. Proofs for Two Propositions

### Proofs for Two Propositions
- Location: page 22.
- Content: Proofs for two propositions (details not provided in the supplied excerpt).

### References
- Location: page 23.
- Content: References list (details not provided in the supplied excerpt).

### Figures
- Figure 1. Income Distribution .................................................................................................................................. 15
- Figure 2. Homogeneous Preference Differences and Optimal Income Tax Schedule ............................................ 17
- Figure 3. Inequality Aversion and Optimal Income Tax Schedule ........................................................................... 18
- Figure 4. Elasticity of Taxable Income and Optimal Income Tax Schedule ............................................................ 19

*Source: wpiea2022062-print-pdf - Annex I. Proofs for Two Propositions; canonical URL: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf*

### 5. Heterogeneous Preference Differences and Optimal Income Tax Schedule ........................................... 20

### 5. Heterogeneous Preference Differences and Optimal Income Tax Schedule

### Introduction and motivation
- Wage subsidy programs have expanded in many advanced economies and were widely used in response to the COVID-19 pandemic.
- Rationale: wage subsidies are seen as a policy to strengthen work incentives while supporting incomes of the working poor.
- Standard Mirrlees (1971) model (intensive-margin focus) implies no role for wage subsidies (optimal MIT non-negative everywhere).
- This chapter extends Diamond (1998) by allowing differences between social (government) and individual preferences over labor/leisure, both:
  - homogeneous across individuals (single parameter 훿), and
  - heterogeneous across individuals (훿(푛) = 훿 + 휑(푛)).
- Key conceptual focus: government may place a higher social value on work than individuals (훿 < 0), which can justify lower MITs and possibly wage subsidies at low earnings.

### Literature overview (extensions that can justify wage subsidies)
- Intensive vs. extensive labor supply margins (Diamond, 1980; Saez, 2002): subsidies can be optimal when participation elasticities are large relative to intensive-margin elasticities.
- Two-dimensional informational asymmetry (skill and taste for leisure) (Choné and Laroque, 2010): negative MITs can be optimal if low-skill individuals have stronger tastes for leisure.
- Non-welfarist social welfare functions (Kanbur et al., 1994): if government cares only about consumption of the poor, negative MITs may be optimal for the poorest.
- Individual information failures (Gerritsen, 2016; Farhi and Gabaix, 2020; Lockwood, 2020): corrective tax adjustments may imply negative MITs at low incomes.

### Benchmark model (Diamond 1998 special case; no preference difference)
- Individual utility quasi-linear: u(c,l) = c + v(1 - l) = n l - T(n l) + v(1 - l). No income effects on labor supply.
- Government chooses non-linear tax T(z) as function of earnings z = n l; informational asymmetry: government observes earnings only.
- Government budget constraint: aggregate consumption ≤ aggregate earnings − government expenditures E (equation (1)).
- Social welfare: W = ∫ G[U_n(c(n), l(n))] f(n) dn, with G increasing and strictly concave (equation (6)).
- Diamond (1998) first-order condition yields optimal T′ in range [0,1]; key implication: T′ ≥ 0 everywhere (no wage subsidies) (equation (8)).

### Extended model with homogeneous preference differences (single 훿)
- Preference difference modeled as (equation (9)):
  - government utility u_g(c,l) = c + (1 + 훿) v(1 - l) = u(c,l) + 훿 v(1 - l);
  - 훿 measures social vs. individual valuation of leisure (훿 ≠ 0 implies preference difference).
- Focus on case 훿 < 0 (government values leisure less / values work more than individuals).
- Optimal tax FOC becomes additive-shift version of benchmark (equation (12)); implications:
  - The formula for optimal T′ includes an additive term depending on 훿 and the ratio of social marginal utility of income to marginal cost of public funds.
  - T′ is increasing in 훿: larger 훿 (government values leisure more) → higher T′; smaller 훿 (government values work more) → lower T′.
  - If 훿 < 0, T′ shifts downward across income distribution; decreases are inverted U-shaped across income (largest decreases at middle incomes).
  - Possibility that T′ < 0 (wage subsidies) at bottom of income distribution under some parameter constellations.
  - Optimal demogrant −T(0) is increasing in 훿 (for fixed E): lower 훿 → lower demogrant.
  - Progressivity: as 훿 increases, progressivity tends to increase; therefore, 훿 < 0 implies less progressive optimal schedule relative to benchmark.
  - Kanbur et al. (1994) is nested: 훿 = −1 implies government cares only about consumption (poverty alleviation), allowing negative MITs for the very poor.

### Extended model with heterogeneous preference differences (훿(푛) = 훿 + 휑(푛))
- Allow preference difference to vary with skill/income: 휑(푛) is monotonically increasing in n; authors focus on case 훿 < 0 and 훿(푛) < 0.
- Optimal tax FOC remains additive with 훿 + 휑(푛) replacing 훿 (equation (15)); implications:
  - T′ is increasing in 훿 + 휑(푛).
  - If 훿 < 0, heterogeneous preference differences (휑 negative and increasing in n) further decrease T′ for low incomes relative to homogeneous case — enhancing role for wage subsidies.
  - Optimal demogrant −T(0) is increasing in 훿 + 휑(푛); however, if 훿 < 0 and heterogeneous differences lower T′, the optimal demogrant must decrease (to finance fixed E).
  - Heterogeneous preference differences with 훿 < 0 tend to further decrease progressivity: lower average tax rates for high-income individuals and higher average tax rates for low-income individuals (because decrease in demogrant tends to dominate decrease in T′ for low incomes).

### Numerical simulations: setup and calibration
- Approach follows Saez (2001) and Hummel and Jacobs (2016); formulas translated to observable income distribution (equations (16)–(18)).
- Calibration details:
  - Pre-tax income z assumed log-normal with mean parameters:
    - mean input used: ln(30000) = 10.31 (휇 = ln(30000) = 10.31),
    - standard deviation 휎 = sqrt{2 * (ln(35000) − ln(30000))} = 0.56.
  - Top tail replaced by Pareto with Pareto parameter 훼 = 3 and Pareto start at €75,649 (95th percentile of log-normal).
  - Elasticity of taxable income benchmark: ε(z) ≡ 0.25 (mid-range from literature).
  - Marginal social welfare curvature: g(z) = 1 / z^β (parametric form); benchmark β = 1.
  - Marginal cost of public funds λ equals average of g(z) under quasi-linear preferences.
  - Government expenditures financed: government must collect 10 percent of total labor income to finance E.
- Income distribution summary: median income €30,000; mean ~ €35,000.

### Simulation results — benchmark (훿 = 0)
- Optimal marginal income tax rates:
  - Non-negative everywhere, confirming Mirrlees result.
  - U-shaped pattern across income (Diamond (1998) result).
- Optimal demogrant −T(0):
  - Positive lump-sum grant; ratio of optimal demogrant to median income (€30,000) = 66 percent.
- Average tax rates:
  - Negative for low-income individuals because of lump-sum grant.
  - Average tax rates increase in z ⇒ schedule is progressive.

### Simulation results — homogeneous preference differences (훿 = -0.2, -0.6, -1)
- For negative 훿 (government values work more):
  - Optimal marginal tax rates shift down for all income levels relative to 훿 = 0; downward shift magnitude is inverted U-shaped across income.
  - Optimal demogrant to median income decreases:
    - 66 percent when 훿 = 0
    - 62 percent when 훿 = −0.2
    - 52 percent when 훿 = −0.6
    - 31 percent when 훿 = −1
  - Optimal schedule becomes less progressive as 훿 decreases.
- Redistributive taste sensitivity (훿 fixed at −1; increase β from 1 to 5):
  - Stronger redistributive taste (β = 5) → higher marginal income tax rates at all incomes and higher demogrant.
  - Demogrant/median increases from 31 percent (β = 1) to 87 percent (β = 5).
  - Stronger redistributive taste → more progressive tax schedule.
- Elasticity sensitivity (훿 = −1, β = 1):
  - Increase ε(z) from 0.25 to 0.275:
    - Optimal marginal tax rates decrease for all incomes; become negative at low incomes (wage subsidies possible).
    - Demogrant/median decreases from 31 percent (ε = 0.25) to 21 percent (ε = 0.275).
    - Higher elasticity → less progressive optimal schedule.

### Simulation results — heterogeneous preference differences (specification and outcomes)
- Specification used: 훿(z) = −1 − γ * log(H(z)), where H(z) is the CDF of z and γ ≥ 0; benchmark γ values:
  - γ = 0 → homogeneous case with 훿 = −1.
  - γ = 0.1 → heterogeneous preference differences with 휑(z) = −γ * log(H(z)) negative and increasing in z.
- For 훿 = −1 and γ = 0.1 (with β = 1 and ε(z) ≡ 0.25):
  - Optimal marginal tax rates shift down relative to γ = 0 for all incomes.
  - Marginal tax rates are negative at low incomes → heterogeneous differences reinforce potential role for wage subsidies.
  - Demogrant/median decreases from 31 percent (γ = 0) to 21 percent (γ = 0.1).
  - Optimal schedule becomes less progressive as γ increases.

### Key theoretical and policy findings
- Allowing government preferences to place a higher social value on work (훿 < 0) lowers optimal marginal income tax rates across the earnings distribution relative to the benchmark; decreases are largest at middle incomes (inverted U-shape).
- Lower MITs imply lower revenues and a lower uniform demogrant, which tends to reduce the progressivity of the overall tax and transfer system.
- Under homogeneous preference differences, higher elasticity of taxable income increases the case for wage subsidies at the bottom of the income distribution.
- Heterogeneous preference differences, where social preference for work is relatively stronger at low earnings, further enhance the case for wage subsidies at low incomes and further reduce progressivity.
- The framework can rationalize observed policy variation:
  - High labor supply elasticity groups (e.g., single adults) with social preference for work → low benefit generosity and low MITs or wage subsidies.
  - Groups constrained by family care (social preference for leisure) and low labor supply elasticity → high benefit generosity with medium to high MITs, possibly phased down as children age and combined with conditional subsidies or ALMP participation.

### Concluding remarks
- Extending optimal income taxation to allow social vs. individual preference differences over labor/leisure introduces a plausible mechanism by which wage subsidies (negative MITs) can be optimal under certain parameter constellations.
- Main equilibrium effects of social preference for work (훿 < 0):
  - Reduction in optimal MITs across earnings distribution.
  - Reduction in uniform demogrant and overall revenues.
  - Lower progressivity of the tax-transfer system.
- Heterogeneous social preferences that weight work more for lower earners amplify these effects and reinforce the potential role for targeted wage subsidies or in-work conditional transfers.

*Source: IMF Working Paper — chapter “5. Heterogeneous Preference Differences and Optimal Income Tax Schedule.”*

### Annex I. Proofs for Two Propositions

### Annex I. Proofs for Two Propositions

### Impact on the optimal marginal income tax schedule
- Proposition 1. For a given negative 훿
, the optimal marginal income tax schedule will shift downward for all income levels relative to the benchmark model with 훿
ൌ0, with the magnitude of the downward shift being inverted U-shaped (i.e., the decreases in 푇′ are higher for middle than bottom and top).
- Proof highlights and expressions (verbatim from source):
  - If we let 푥
ሺ
푧
ሻ
≡
׬
൫
ఒି௚
ሺ
௦
ሻ
൯
௛
ሺ
௦
ሻ
ௗ௦
೥
೘ೌೣ
೥
ఌ
ሺ
௭
ሻ
ఒ௭௛
ሺ
௭
ሻ
, then (16) suggests that 푇
ᇱ
ൌ
௫
ሺ
௭
ሻ
ଵା௫
ሺ
௭
ሻ
 so that 
 ௗ்
ᇲ
ௗ௫
ሺ
௭
ሻ
ൌ
ଵ
൫
ଵା௫
ሺ
௭
ሻ
൯
మ
.
  - If we further let ∆푥
ሺ
푧
ሻ
≡
ఋ௚
ሺ
௭
ሻ
ఒ
, then by applying the Taylor expansion, (17) suggests that ∆푇
ᇱ
ൎ
∆௫
ሺ
௭
ሻ
൫
ଵା௫
ሺ
௭
ሻ
൯
మ
ൌ훿∗
ଵ
൫
ଵା௫
ሺ
௭
ሻ
൯
మ
∗
௚
ሺ
௭
ሻ
ఒ
.
  - Figure 2(a) for the case of 훿
ൌ0 shows that 푥
ሺ
푧
ሻ
 is, roughly speaking, monotonically decreasing in 푧 so that 
 ଵ
൫
ଵା௫
ሺ
௭
ሻ
൯
మ
 is monotonically increasing in 푧. Also,  
  ௚
ሺ
௭
ሻ
ఒ
 is monotonically decreasing in 푧.
  - Therefore, for a given negative 훿
 (e.g., 훿ൌെ0.2), the optimal marginal income tax schedule will shift downward for all income levels relative to the benchmark model with 훿
ൌ0, with the magnitude of ∆푇
ᇱ
 (i.e., 
 |
 ∆푇
ᇱ
 |
 ) first increasing and then decreasing in 푧. Q.E.D.

### Impact on the progressivity of the optimal tax-transfer system
- Proposition 2. If 훿
൏0, the optimal income tax schedule tends to be less progressive compared to the benchmark model with 훿
ൌ0.
- Proof highlights:
  - When 훿
 decreases from 0 to a negative value (e.g., 훿ൌെ0.2), the optimal marginal income tax rates 푇′ and the optimal demogrant െ푇
ሺ
0
ሻ
 both decrease.
  - As the decrease in െ푇
ሺ
0
ሻ
 tends to dominate the decrease in 푇′ for low-income individuals, the average tax rate for low-income individuals tends to increase.
  - In contrast, as the decrease in 푇′ tends to dominate the decrease in െ푇
ሺ
0
ሻ
 for high-income individuals, the average tax rate for high-income individuals tends to decrease.
  - Therefore, if 훿
൏0, the optimal income tax schedule tends to be less progressive compared to the benchmark model with 훿
ൌ0.
  - Note: Although the overall profile tends to be less progressive when 훿
൏0, Proposition 1 implies this may not hold locally: when 훿
 decreases from 0 to a negative value (e.g., 훿ൌെ0.2), the decreases in the optimal marginal income tax rates 푇′ are higher for middle than bottom and top. Q.E.D.

*IMF WORKING PAPERS Social versus Individual Work Preferences: Implications for Optimal Income Taxation*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022062-print-pdf.pdf_