## 3. Progressive Capacity and Market Gini Coefficients

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### Introduction
- Tax policy roles:
  - Ensure poor individuals pay little or no tax.
  - Mimic spending via refundable tax credits (e.g., EITC) or negative income tax.
  - Raise revenues for inequality-reducing spending.
- Redistribution at the top mainly relies on tax policy.
- Main mechanism for reducing income inequality: progressive income taxation (average tax rate rises with income).
- Two components of progressivity:
  - Structure of tax system (personal allowance + flat rate vs multiple rates).
  - Taxation of income sources (capital income often taxed at lower rates than labor income).
- Measurement challenges:
  - Progressivity varies across incomes; measures sensitive to income range chosen.
  - Many measures depend on pre-tax income distribution; very progressive systems may appear to redistribute little if pre-tax distribution is even.
  - Paper proposes a measure of progressive capacity independent of pre-tax income distribution.

### Methodology and Data

- Scope and definitions:
  - Progressive tax system defined as one with rising average tax rate.
  - Focus on direct taxes on individuals (income taxes including regional/local where available) and social security contributions (SSC). Indirect taxes excluded.
- Treatment of SSCs:
  - SSCs often mandatory and income-determined; pension contributions can resemble forced savings.
  - Progressivity measures calculated both including and excluding SSCs when data permit.
- Choice of income category: employed labor used where taxation differs by category.
- Existing measures described:
  - Pigou (1928) style: ratio of change in average tax rate to change in income.
  - Sabirianova Peter, Buttrick, and Duncan (2010): slope of average tax rate on income across 100 points.
  - Kakwani (1977) index: twice the area between income and tax-payment Lorenz curves (depends on pre-tax distribution).
  - Suits (1977): share of taxes paid vs share of income.
- Progressive capacity (new measure):
  - Kakwani index calculated over a fixed range of incomes, each given equal weight (income treated as uniformly distributed).
  - Under uniform income assumption, income Lorenz curve simplifies to s^2; area below equals 1/3.
  - Practical range used: 0–500 percent of per capita GDP for cross-country comparability.
  - Tax payable computed at 500 equally spaced income points and area below tax Lorenz curve estimated by summing 500 rectangles (or integrating for simple schedules).
- Data sources and limitations:
  - Pigou-type measures: OECD Taxing Wages at 67, 100, 167 percent of median income; upper income range limited.
  - Sabirianova et al. data: broad coverage but only through 2005 and exclude SSC.
  - Progressive capacity requires detailed tax schedule data; OECD provides federal personal income tax data from 1981 onward; from 2000 onward sub-central taxes and SSCs available with supplementation.

### Progressivity Trends
- Personal income tax progressivity:
  - Declined steeply in the 1980s and 1990s and has remained broadly stable since then.
  - Progressive capacity and average rate progression both show strong downward trend, especially the 1980s; progressive capacity shows stabilization in the 2000s.
  - Average rate progression higher among OECD members compared to overall sample.
  - Decline in progressivity consistent with decline in top income tax rates globally (with a slight recent reversal for advanced economies).
- Inclusion of SSCs:
  - For 2000–2017 subsample, progressive capacity including SSCs is flat over the shorter period and much lower in level than tax-only progressive capacity.
  - Reason: SSCs often have fairly flat structure, small or no personal allowance, and often an upper earnings limit (or reduced rate above a threshold).
- Self-employed:
  - Progressive capacity for self-employed shows no strong trend over the later period, and a slightly higher level than for employees.
- Pigou-style OECD Taxing Wages comparisons (increase from 100 to 167 percent of average earnings):
  - Progressivity higher for single individuals than for a second earner in a two-earner family with two children.
  - Since 2000, progressivity for singles has fallen slightly; for second earners entering workforce, no change.
  - Including SSC (tax wedge) reduces differences and shows no clear trend.
- Practical caveat: actual progressivity may be lower due to tax reliefs, deductions, tax planning, and tax evasion concentrated at upper incomes.

### Growth Regressions

- Specification (annual OECD data, 1981–2017; 5-year growth horizon):
  - ȳ_{i,t} = a + β1 p_{i,t−5} + β2 y_{i,t−5} + γ′X_{i,t−5} + f_i + g_t + ε_{i,t}
  - ȳ_{i,t} = growth over 5 years; p_{i,t−5} = initial progressivity; X includes: population growth rate, per capita GDP, Gini coefficient, capital account openness; f_i country FE; g_t year FE.
- Main empirical findings:
  - Regressions generally do not reveal a significant impact of progressivity on growth.
  - Selected results from fixed effects regressions (Table 1):
    - Avg. rate progression, 0-400% pc GDP: coefficient = 189.3* (97.05).
    - Δ Avg. tax rate / 100-167% avg wage: 182.6 (136.6).
    - Δ Avg. tax rate / 67-100% avg wage: -12.87 (29.48).
    - Δ Avg. wedge measures: 77.77 (168.5) and -17.12 (21.69).
    - Progressive capacity t-5: 17.52 (29.01).
    - Progressive capacity including SSC t-5: -50.46 (38.80).
    - Top statutory rate and Top rate^2 insignificant.
  - Control highlights:
    - ln(GDP pc) t-5 frequently negative and significant (e.g., -7.965*** (1.960) in (1)).
    - Capital account openness often positive and significant (e.g., 16.04** (6.225) in (1)).
  - Observations vary (e.g., 2,019 in (1); 2,665 in (4)).
- GMM (System GMM) results (Table 2):
  - Progressive capacity t-5: 29.05 (33.87) — insignificant.
  - Progressive capacity including SSC t-5: 89.25 (73.58) — positive but not significant.
  - Example Pigou-style GMM result: Δ Avg. tax rate / 67-100% avg wage t-5 = -92.57** (45.20) in one specification.
  - GMM diagnostics: AR1 p small (expected), AR2 p not rejecting absence of second-order serial correlation; Hansen p = 1 (may indicate many instruments).
  - Gini t-5 becomes consistently negative and significant in GMM (e.g., -1.946*** (0.355) in (1)).
- Robustness checks:
  - Annual regressions with one-year lags: insignificant coefficients for all progressivity measures.
  - Rolling 10-year subsamples: no consistent significant sub-period results.
  - Quantile regressions and interaction with high-progressivity dummy (95th percentile): no significant negative result.
  - Additional controls (top personal rate, personal income tax revenue/GDP, government spending/GDP): results unchanged.
  - Progressive capacity computed up to 1000 and 2000 percent per capita GDP: insignificant.
  - Excluding Gini: progressivity measures remain insignificant.
- Interpretation:
  - No clear detrimental effect on growth from progressivity levels observed since 1981 in OECD countries.
  - Lack of significance does not prove absence of effect, especially for extreme progressivity levels.
  - Theoretical channels exist for potentially positive effects of progressivity (e.g., addressing inefficient labor markets, reducing rent-seeking).

### Progressivity and Pre-Tax Inequality

- Advantage of progressive capacity: constructed to be exogenous to actual income distribution, enabling analysis of the relationship between policy design and market inequality.
- Empirical specification for market Gini:
  - G_ijt = a + β1 p_i,t-1 + γ′X_i,t-1 + f_i + g_i + ε_ijt.
- Political endogeneity noted as a concern; authors argue it would bias coefficients upwards (since more unequal societies could introduce more progressive systems), so observed negative coefficients are not driven by this channel.

### 2017 Methods and Empirical Findings on Market Inequality

- Progressivity measures used in this exercise: (1) top statutory rate and its square; and (2) the proposed measure of progressive capacity.
- Estimation approach: regular and bootstrapped standard error quantile regressions for quantiles 75, 80, and 90.
- Main empirical findings (selected coefficients from Table 3):
  - Progressive capacity t-1, 0-500% of income:
    - Coefficient: -25.25***; standard error: (7.807).
    - Alternate specification coefficient: -26.48***; standard error: (8.840).
  - Progressive capacity t-1, 0-1000% of income:
    - Coefficient: -39.08**; standard error: (18.07).
  - Progressive capacity t-1, 0-2000% of income:
    - Coefficient: -13.64*; standard error: (7.477).
- Control variable examples (standard errors in parentheses):
  - Pop. growth t-1: -0.925 (0.781); -0.964 (0.841); -1.058 (0.800).
  - ln(GDP pc) t-1: 1.754 (1.508); 1.768 (1.461); 1.804 (1.574).
  - Capital account openness t-1: -0.862 (1.511); -0.921 (1.551); -0.218 (1.553).
- Sample sizes and panel dimensions:
  - Observations: 707; 678; 678; 678 (by column).
  - Number of countries: 35; 34; 34; 34 (by column).
- Robustness:
  - Regressions on concurrent values of progressive capacity produced the same results.
- Interpretation emphasized by authors:
  - Progressive tax systems, as measured by progressive capacity, are associated with reductions in market (pre-tax) inequality across income ranges used.
  - Political endogeneity would bias coefficients upwards, reinforcing confidence in negative association.
- Policy implication highlighted:
  - “Making a tax system more progressive does have a real impact on market economy outcomes by reducing inequality, but nevertheless no detrimental impact on growth.”
  - “Progressivity-enhancing measures could be taken without major risks to growth. This would be especially relevant in countries that are marked by great inequality.”

### Broader context and unresolved questions
- Capital income dynamics:
  - Capital income is distributed more unequally than labor income, has risen as a share of total income, and is often taxed at a lower rate than labor income.
  - Corporate income tax affects progressivity directly (borne partly by owners) and indirectly by supporting enforcement of labor income taxation and mitigating arbitrage.
  - International tax competition and capital mobility have produced a downward trend in corporate income tax rates, reducing overall tax progressivity.
- Optimal top income tax rate with income shifting (formula cited):
  - t* = (1 + s·τ·aea) / (1 + aea)  (notation and parameter definitions as given in the source).
- Areas for further research:
  - Need for progressivity measures that take the entire tax and benefit system, and public spending, into account while remaining independent of pre-tax and spending distributions.
  - Remaining doubts about the impact of progressivity on growth warrant further study.

*Source: wp18246 — IMF Working Paper chapter "3. Progressive Capacity and Market Gini Coefficients".*

### REFERENCES ___________________________________________________________________________________ 23

### REFERENCES

### Figures
- 1. Average Tax Rate across Incomes ______________________________________________________________ 7
- 2. Lorenz Curves __________________________________________________________________________________ 8
- 3. Tax Progressivity  ______________________________________________________________________________ 11
- 4. Average Top (Combined) Personal Income Tax Rates _________________________________________ 11
- 5. Average Progressive Capacity, OECD countries ________________________________________________ 13
- 6. Rate Progression, Increase from 100 to 167 Percent of Average Earnings _____________________ 14
- 7. Average (Combined) Corporate Income Tax Rates ____________________________________________ 22

### Tables
- 1. Regressions of 5-Year Growth on Progressivity Measures _____________________________________ 17
- 2. Regressions of 5-Year Growth on Progressivity Measures, GMM results _______________________ 18

*Source: wp18246 - REFERENCES; https://www.imf.org/-/media/files/publications/wp/2018/wp18246.pdf*

### 3. Progressive Capacity and Market Gini Coefficients ____________________________________________ 20

### 3. Progressive Capacity and Market Gini Coefficients

### Introduction
- Tax policy is a primary tool to address income inequality; at lower incomes spending policies can be more important, but tax policy can: (i) ensure poor individuals pay little or no tax; (ii) mimic spending via refundable tax credits (e.g., EITC) or negative income tax; (iii) raise revenues for inequality-reducing spending.
- At the top of the distribution, redistribution mainly relies on tax policy because spending cannot further target the highest earners once they pass qualification thresholds.
- Main mechanism for reducing income inequality: progressive income taxation (average tax rate rises with income). Two components:
  - Structure of tax system (personal allowance + flat rate vs multiple rates).
  - Taxation of income sources (capital income often taxed at lower rates than labor income; capital typically more unequally distributed than wages).
- Measurement challenges:
  - Progressivity varies widely across incomes; measures are sensitive to the income range chosen.
  - Many measures depend on the pre-tax income distribution; very progressive tax systems may appear to redistribute little if pre-tax distribution is even.
  - The paper proposes a measure of progressive capacity that is independent of the pre-tax income distribution.

### Methodology and Data

#### A. Methodology
- Definition: progressive tax system = rising average tax rate.
- Desired properties of a cross-country measure: (i) summarize progressivity over the entire rate schedule in a single figure; (ii) focus on redistributive power independent of pre-tax income distribution. No existing measure fully meets both.
- Scope: focus on direct taxes on individuals (income taxes including regional/local where available) and social security contributions (SSC). Indirect taxes excluded.
- Treatment of social security contributions:
  - Many SSCs are mandatory and income determined; pension contributions can resemble forced savings.
  - Progressivity measures are calculated both including and excluding SSCs when data permit.
- Choice of income category: employed labor used where taxation differs by category.

#### Measures of Progressivity (existing)
- Pigou (1928) style: ratio of change in average tax rate to change in income (equals difference between marginal and average tax rate divided by income). Sensitive to income point chosen.
- Sabirianova Peter, Buttrick, and Duncan (2010): estimate average tax rate progression over 100 data points (e.g., 4 to 400 percent of per capita GDP) as slope of regression of average tax rate on income.
- Kakwani (1977) index: twice the area between income and tax-payment Lorenz curves; incorporates entire rate schedule but depends on pre-tax income distribution.
- Suits (1977) is similar (plots share of taxes paid against share of income).

#### A Measure of Progressive Capacity (new)
- Goal: abstract from underlying income distribution and measure redistributive potential of tax system.
- Defined as the Kakwani index calculated over a fixed range of incomes, each given equal weight (income treated as if uniformly distributed).
- Under uniform income assumption, the income Lorenz curve simplifies to s^2; area below this curve equals 1/3.
- Practical implementation: use a range of 0–500 percent of per capita GDP for cross-country comparability (choice of range is the only link to actual income distribution). Tax payable computed at 500 equally spaced income points and area below tax Lorenz curve estimated by summing 500 rectangles (or integrating for simple schedules).

#### B. Data
- Pigou-type measures: OECD Taxing Wages data provide average tax rates at 67, 100, 167 percent of median income; also family configurations and tax wedge including employer SSC. Limitation: upper income range limited (mostly up to 167 percent of median).
- Sabirianova Peter et al. (2010) data: broad country coverage including emerging and developing economies, but only through 2005 and do not include SSC.
- Progressive capacity requires detailed tax schedule data (allowances, brackets, rates, surtaxes). OECD provides federal personal income tax data from 1981 onward; from 2000 onward data on sub-central income taxes and SSCs are available (though less complete and supplemented with IBFD, Ernst & Young, national sources).
- When including SSCs: both employer and employee contributions included; labor cost defined as gross income plus employer-paid contributions. Separate self-employed SSC rates used where available.

### Progressivity Trends
- Personal income tax progressivity declined steeply in the 1980s and 1990s and has remained broadly stable since then.
- Evidence from multiple measures:
  - Progressive capacity measure (new) and average rate progression (Sabirianova Peter et al., 2010) both show strong downward trend, especially the 1980s; progressive capacity shows stabilization in the 2000s.
  - Average rate progression higher among OECD members compared to overall sample.
  - Decline in progressivity consistent with decline in top income tax rates globally (with a slight recent reversal for advanced economies).
- Inclusion of social security contributions:
  - For 2000–2017 subsample, progressive capacity including SSCs is flat over the shorter period and is much lower in level than tax-only progressive capacity.
  - Reason: SSCs often have fairly flat structure, small or no personal allowance, and often an upper earnings limit (or reduced rate above a threshold).
- Self-employed: progressive capacity for self-employed shows no strong trend over the later period, and a slightly higher level than for employees; cross-country variation in SSC design for self-employed explains some differences.
- Pigou-style OECD Taxing Wages comparisons (increase from 100 to 167 percent of average earnings):
  - On average, progressivity is higher for single individuals than for a second earner in a two-earner family with two children.
  - Two reasons: (i) second earner benefits from additional personal allowances and lower rates; (ii) dependents reduce taxable base in many systems.
  - Since 2000, progressivity for singles has fallen slightly; for second earners entering workforce, no change in progressivity.
  - Including SSC (tax wedge) reduces differences and shows no clear trend.
- Practical caveat: actual progressivity may be lower due to tax reliefs, deductions (e.g., mortgage interest), tax planning, and tax evasion concentrated at upper incomes (Alstadsaeter, Johannesen, and Zucman (2017) evidence).

### Growth Regressions

#### A. Previous studies
- Mixed evidence on relationship between progressivity and growth; no strong empirical consensus on a pronounced negative relationship.
- Some studies on fiscal redistribution find no or even positive effects for non-extreme redistribution (Ostry, Berg, and Tsangarides, 2014).
- Rhee (2013) finds negative relationship across U.S. states (greater labor mobility may drive different results than cross-country analysis).
- Other work uses average and marginal tax rates as crude proxies (e.g., Padovano and Galli (2005) find negative relation for marginal tax rate).

#### B. New results (this paper)
- Regression specification (annual OECD data, 1981–2017; 5-year growth horizon):
  - ȳ_{i,t} = a + β1 p_{i,t−5} + β2 y_{i,t−5} + γ′X_{i,t−5} + f_i + g_t + ε_{i,t}
  - ȳ_{i,t} = growth over 5 years; p_{i,t−5} = initial progressivity; y_{i,t−5} = initial real per capita GDP; X includes controls: population growth rate, per capita GDP, Gini coefficient, capital account openness; f_i country FE; g_t year FE.
- Main finding: regressions generally do not reveal a significant impact of progressivity on growth.
  - Table 1 summary (fixed effects regressions):
    - Regression (1) Avg. rate progression, 0-400% pc GDP: coefficient = 189.3* (standard error 97.05) — the one significant positive result.
    - Regression (2)-(6): most progressivity measures insignificant, including:
      - Δ Avg. tax rate / 100-167% avg wage: 182.6 (136.6)
      - Δ Avg. tax rate / 67-100% avg wage: -12.87 (29.48)
      - Δ Avg. wedge measures: 77.77 (168.5) and -17.12 (21.69)
      - Progressive capacity t-5: 17.52 (29.01)
      - Progressive capacity including SSC t-5: -50.46 (38.80)
      - Top statutory rate and Top rate^2 insignificant.
    - Control variable highlights:
      - ln(GDP pc) t-5 coefficients strongly negative and significant in most specs (e.g., -7.965*** (1.960) in (1)).
      - Gini t-5 mixed; capital account openness often positive and significant (e.g., 16.04** (6.225) in (1)).
    - Observations vary across specifications (e.g., 2,019 in (1); 2,665 in (4); smaller samples for OECD-specific measures).
  - GMM robustness (System GMM) produces broadly similar stories (Table 2):
    - Progressive capacity t-5: 29.05 (33.87) — insignificant.
    - Progressive capacity including SSC t-5: 89.25 (73.58) — positive but not significant at conventional levels.
    - Some Pigou-style measures show varying signs; Δ Avg. tax rate / 67-100% avg wage t-5 = -92.57** (45.20) in one GMM specification.
    - GMM diagnostics: AR1 p small (expected), AR2 p not rejecting absence of second-order serial correlation; Hansen p = 1 (may indicate many instruments).
    - Gini t-5 becomes consistently negative and significant in GMM (e.g., -1.946*** (0.355) in (1)).
- Robustness checks performed:
  - Annual regressions with one-year lags: insignificant coefficients for all progressivity measures.
  - Regressions on rolling 10-year subsamples: no changes or significant sub-period results.
  - Quantile regressions and interaction with high-progressivity dummy (95th percentile): no significant negative result.
  - Additional controls added (top personal rate, personal income tax revenue/GDP, government spending/GDP): results unchanged.
  - Progressive capacity computed up to 1000 and 2000 percent per capita GDP (to capture very top incomes): insignificant.
  - Fixed-effect regressions excluding Gini: progressivity measures remain insignificant.
- Interpretation:
  - Failure to find significant negative impact does not prove absence of effect, especially for extreme levels of progressivity (e.g., near-100 percent rates historically).
  - Evidence suggests no clear detrimental effect on growth from progressivity levels observed since 1981 in OECD countries.
  - Theoretically, there are arguments for potentially positive effects of progressivity (e.g., addressing inefficient labor markets, reducing rent-seeking within firms).

#### C. Progressivity and Pre-Tax Inequality
- Progressive capacity measure's independence from actual income distribution is an advantage: unlike measures defined as differences between market and net Gini coefficients, it is exogenous to income behavior.
- This allows analysis of the relationship between policy design (progressive capacity) and market inequality without confounding by actual redistribution outcomes driven by pre-tax income distribution.

*Source: IMF Working Paper "3. Progressive Capacity and Market Gini Coefficients" (chapter/section content).*

### 2017. Because of data availability limitations, this exercise is carried out only for the following progressivity

### wp18246 - 2017. Because of data availability limitations, this exercise is carried out only for the following progressivity

### Methods and measures
- Progressivity measures used: (1) the top statutory rate and its square; and (2) the proposed measure of progressive capacity.
- Estimation approach: regular and bootstrapped standard error quantile regressions for quantiles 75, 80, and 90.
- Empirical test of channel: regressions of market (gross income) Gini coefficients (G) on tax progressivity using the specification:
  - G_ijt = a + β1 p_i,t-1 + γ′X_i,t-1 + f_i + g_i + ε_ijt  (equation (2) as presented).

### Empirical findings on market inequality
- The progressive capacity variable is constructed to be exogenous with respect to the actual distribution (i.e., not mechanically affected by current distributional outcomes), though political endogeneity remains a concern.
- Authors note political endogeneity would bias coefficients upwards (because governments in more unequal societies could introduce more progressive systems), implying observed negative coefficients are not driven by this channel.
- Main result summary: “we find quite strong evidence of such an effect, across various specifications, and with the progressive capacity calculated over different income ranges.”

### Table 3 — Progressive Capacity and Market Gini Coefficients (selected coefficients and statistics)
- Progressive capacity t-1, 0-500% of income:
  - Coefficient: -25.25***
  - Coefficient (alternate specification): -26.48***
  - Standard errors (corresponding): (7.807), (8.840)
- Progressive capacity t-1, 0-1000% of income:
  - Coefficient: -39.08**
  - Standard error: (18.07)
- Progressive capacity t-1, 0-2000% of income:
  - Coefficient: -13.64*
  - Standard error: (7.477)
- Control variable coefficients (selected; standard errors in parentheses):
  - Pop. growth t-1: -0.925 (0.781); -0.964 (0.841); -1.058 (0.800) across specifications
  - ln(GDP pc) t-1: 1.754 (1.508); 1.768 (1.461); 1.804 (1.574)
  - Capital account openness t-1: -0.862 (1.511); -0.921 (1.551); -0.218 (1.553)
- Sample sizes and panel dimensions:
  - Observations: 707; 678; 678; 678 (by column)
  - Number of countries: 35; 34; 34; 34 (by column)
- Source reported: Authors' Estimates.

### Interpretation and robustness
- Robustness note: regressions on concurrent values of the progressive capacity measure produced the same results.
- Interpretation emphasized by authors:
  - Progressive tax systems, as measured by progressive capacity, are associated with reductions in market (pre-tax) inequality.
  - This effect is detected across different income ranges used to calculate progressive capacity.
  - Political endogeneity, if present, would work against the observed sign, reinforcing confidence in the negative association.

### Comparison with growth results and policy implication
- Authors contrast the inequality impact with the absence of an impact on economic growth (as found elsewhere in the paper).
- Implication highlighted: “making a tax system more progressive does have a real impact on market economy outcomes by reducing inequality, but nevertheless no detrimental impact on growth.”
- Policy-relevant conclusion: “progressivity-enhancing measures could be taken without major risks to growth. This would be especially relevant in countries that are marked by great inequality.”

### Broader context and related mechanisms
- Developments in capital income taxation likely contributed to reducing overall progressivity:
  - Capital income is distributed more unequally than labor income, has risen as a share of total income, and is often taxed at a lower rate than labor income (IMF, 2017b referenced in text).
  - Corporate income tax affects progressivity directly (borne partly by owners) and indirectly by supporting enforcement of labor income taxation and mitigating arbitrage.
- International tax competition and capital mobility have produced a downward trend in corporate income tax rates, which reduces overall tax progressivity and may put downward pressure on personal income tax rates.
- The paper cites the optimal top income tax rate formula allowing for income shifting:
  - t* = (1 + s·τ·aea) / (1 + aea)  (notation and parameter definitions as given in the source, with reference to Saez, Slemrod, and Giertz 2012).

### Unresolved questions and areas for further research
- Need for progressivity measures that take the entire tax and benefit system, and public spending, into account to better capture overall progressivity.
- Challenges: finding a comprehensive measure that remains independent of pre-tax and spending distributions is described as “enormous.”
- Authors emphasize remaining doubts about the impact of progressivity on growth and call for further research.

*Source: Authors' Estimates and text from wp18246 chapter content provided.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18246.pdf_