## _wp0717

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---

### I. Introduction and purpose
- Evaluates Robert Lucas’ (2004) assertion that economic growth outweighs redistributive policies using Lucas (1987) welfare-evaluation framework extended in three directions:
  - welfare gains from growth for intertemporal elasticity of substitution (IES) values other than one;
  - introduction of consumption inequality across individuals;
  - assessment of optimal inequality and optimal growth by introducing a production possibility frontier (technological restrictions) for inequality, consumption levels, and growth.
- Counterfactual experiments:
  - how much growth society would give up for perfect equality;
  - what level of inequality compensates for lack of growth;
  - welfare consequences of eliminating all growth and inequality simultaneously.

### Methodology and model
- Welfare evaluated in present-value terms using an isoelastic utility function and log-normal distribution of consumption (Lucas (1987) framework extended).
- Social welfare function: classical utilitarian (Benthamite) weighting everyone equally; identical individual preferences assumed.
- Inequality measured by the standard deviation of the log of consumption.
- World model: continuum of countries; each country populated by a continuum of individuals; world population normalized to 1.
- Individual log consumption follows an autoregressive process with persistence ρ (ρ ∈ [0,1]); shocks: εt ~ N(0,1) (individual), ηt ~ N(0,1) (country).
- Unconditional cross-sectional distribution of log consumption: ln ct ~ N(abt + λ, σx^2 + σy^2), where σx^2 = cross-country inequality and σy^2 = within-country inequality.
- Individual expected lifetime utility: U = E0[Σ β^t u(ct)] with u(c) = (c^(1−γ) − 1)/(1 − γ) and 1/γ = IES.
- Social welfare W = ∫ U(c) dF(c); comparative statics: ∂W/∂(σ^2) < 0 and ∂W/∂μ > 0.
- Social planner treats aversion to inequality equal to aversion to risk: coefficient of aversion to inequality equals γ (inverse of IES).

### Welfare measures and closed-form expressions
- Compensation-rate measures λ:
  - μλ: proportional increase in consumption required to leave planner indifferent between baseline path and a path with no growth.
  - xλ, yλ, 0λ: compensations for cross-country, within-country, and total inequality respectively.
- Alternative compensation rates relative to baseline path: λ̂μ, λ̂x, λ̂y, λ̂0.
- Growth-equivalent measures μx, μy, μ0: percentage-point changes in growth the planner would trade for eliminating respective inequalities.
- Inequality-change measures xθ, yθ, 0θ: percentage reductions in inequality required to compensate for no growth.
- Marginal rate of substitution (shadow price) between inequality and growth: MRS_i (around baseline) expressed in closed form (see model equations referenced in text).

### Calibration choices (preserved exact values and statements)
- Growth of per-capita consumption (Penn World Tables 6.1, 108 economies, 1960–2000):
  - Unweighted average growth rate: 2.3 percent (standard deviation 1.17 percent).
  - Weighted average growth rate: 2.1 percent (standard deviation 0.99 percent).
  - Calibration choice: 0 2.1%μ=.
- Cross-country dispersion calibration stated as: 0 1 x σ = .
- Within-country dispersion calibration: 0 0.5 y σ = .
- Discount factor: 0.95 β = .
- Risk aversion γ (inverse of IES) evaluated over parameter set presented in the source (examples reported for γ = 1, 2, 5, 10, 20).

### Main quantitative findings — welfare gains from growth
- Welfare gains from one additional percentage point of annual economic growth range from 2 percent to 21 percent across IES values considered.
- At per-capita consumption growth rate of 2.1 percent (1960–2000 average for the 108 economies analyzed), welfare gains from total economic growth range from 7.6 percent to 51 percent.
- These gains are typically lower than the 20 percent reported by Lucas for the log-utility (IES = 1) case.
- Reproduction of Lucas (1987) result: μλ and 1%λ substantial when γ ≈ 1 and decline as γ increases; example: if γ = 5, gains are 8.5 percent (reported in text).

### Main quantitative findings — welfare costs of inequality
- Time-series standard deviation of log aggregate consumption: around 1 to 2 percent (business-cycle costs small).
- Cross-sectional standard deviation of log consumption:
  - within countries: around 50 percent;
  - across countries: around 100 percent.
- Welfare costs of inequality (permanent compensation rates) range from:
  - within-country: 12 percent to 91 percent of current welfare;
  - cross-country: 40 percent to almost 100 percent of current welfare.
- Shadow price of inequality in terms of growth (μ = ∂growth/∂σinequality):
  - within-country range: 0.026 to 6.55;
  - cross-country range: 0.0521 to 13.11.

### Counterfactual numerical examples (IES = 1/2)
- Planner would give up 1.62 percentage points of economic growth (out of 2.1 points) to eliminate all within-country inequality.
- Planner would give up 4.49 percentage points of economic growth to eliminate all cross-country inequality.
- A reduction of 34 percent in cross-country inequality would compensate for the lack of growth (no growth).
- A reduction of 136 percent in within-country inequality would compensate for the lack of growth (no growth).
- Eliminating all growth and cross-country inequality improves welfare by 48 percent.
- Eliminating all growth and within-country inequality reduces welfare by 9 percent.

### Welfare costs: business cycles versus inequality
- Business-cycle cost estimates: "the cost of the business cycle can reach 3–4 percent of baseline consumption for households with no wealth" and "as large as 12 percent" under less restrictive preferences.
- Comparative statement preserved: "Even these estimates of the costs of business cycles seem small, however, relative to the costs of inequality."
- For the 2γ= interpretation quoted in text:
  - "Eliminating economic growth is equivalent to introducing a permanent 28 percent tax on consumption;"
  - "eliminating within-country inequality is equivalent to introducing a permanent 28 percent subsidy on consumption;"
  - "eliminating cross-country inequality is equivalent to introducing a permanent 249 percent subsidy on consumption!"

### Equality–growth tradeoffs and planner willingness to substitute (selected quantified examples)
- Example: "3.1 y μ = − for 5γ = means that the planner would accept a reduction of 3.1 points in the growth rate of consumption in exchange for eliminating all within-country inequality." (Baseline growth = 2.1 percent → new growth = -1 percent.)
- Example: "36.9% y θ = for 5γ = means that a 36.9 percent reduction in within-country inequality would be sufficient to compensate the planner for a total lack of growth."
- Example: "33% y λ = − for 5γ = means that a large welfare gain ensues from eliminating inequality while simultaneously stopping growth."
- Shadow price example: for 5γ =, "1 0.377 y y MRS μ σ ∂ == ∂ means that the shadow price of one point of inequality, measured by σ, is around 1 3 of a point of economic growth."
- Marginal willingness to give up consumption for inequality reduction example: for 5γ =, "1 2.5 y y MRS λ σ ∂ == ∂ means that the planner would be willing to permanently give up 2.5 percent of consumption for a permanent reduction of one point of inequality."

### Technological-constraint (optimality) results — analytical solutions
- Reduced-form technological frontier specified as tradeoffs between inequality, growth, and consumption (equation references preserved in text).
- Optimal inequality: *2 2 1 σ εγ = (optimal inequality depends positively on IES 1 γ and negatively on elasticity of inequality w.r.t. consumption).
- Optimal growth: 1 1 1 * 2 11 γ ε μ β ε − ⎡⎤ ⎛⎞ +=  + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ (condition for positive growth: 1 2 11 ε β ε ⎛⎞ + > ⎜⎟ ⎝⎠).
- Optimal consumption level: () 2 1 1 * * * 1 1 A ε ε σ λ μ ⎡⎤ ⎢⎥ += ⎢⎥ + ⎣⎦.
- Optimal social welfare: W**** = W(λ*, μ*, σ*). Welfare cost of baseline choices c λ defined by solving (22) in text.

### Calibration of technological elasticities (US vs Scandinavia; East vs West Germany)
- US vs Scandinavia (Aaberge and others (2002)):
  - Per capita income difference used: "use 25% λ = or () 1ln1.25 λ ∆ + =."
  - Gini (1990 averages): US = 0.346; Scandinavia = 0.2173.
  - Transformed standard deviations: US σ = 0.634; Scandinavia σ = 0.39; ratio = 1.63 → ln ln 1.63 σ ∆ =.
  - Calculated elasticity: () 2 ln ln 1.63 2.19 ln  1ln 1.25 σ ε λ ∂ = ≅ = ∂ + (i.e., ε2 ≅ 2.19).
- East vs West Germany (Biewen (2000) and related data):
  - 1990 log-variance ratio: ywest / yeast = 1.41 → ln ln 1.41 σ ∆ =.
  - Mean income ratios and alternative growth-accounting assumptions discussed; second estimate of 1 ε constructed from these comparisons.

### Calibration results and optimality (exact reported values from Table 5)
- Optimal inequality *σ:
  - γ = 1: 0.6724
  - γ = 2: 0.4778
  - γ = 5: 0.3022
  - γ = 10: 0.2137
  - γ = 20: 0.1511
- Calibrated values of 1 a ε:
  - γ = 1: 41.44
  - γ = 2: 29.30
  - γ = 5: 15.22
  - γ = 10: 8.14
  - γ = 20: 3.89
- Case a (using 1 a ε):
  - *a μ (optimal growth) = 0.0210 for all γ reported.
  - *a λ (optimal consumption level):
    - γ = 1: 0.1448
    - γ = 2: -0.0205
    - γ = 5: -0.2054
    - γ = 10: -0.3217
    - γ = 20: -0.4210
  - c a λ (welfare cost of baseline, assuming *0yσσ=):
    - γ = 1: -0.0327
    - γ = 2: -0.0010
    - γ = 5: -0.1536
    - γ = 10: -0.4693
    - γ = 20: -0.8919
- Case b (using 1 b ε = 30.00):
  - 1 b ε = 30.00 for all γ reported.
  - *b μ (optimal growth):
    - γ = 1: ∞
    - γ = 2: 0.0194
    - γ = 5: 0.0048
    - γ = 10: 0.0021
    - γ = 20: 0.0010
  - *b λ (optimal consumption level):
    - γ = 1: -1.0000
    - γ = 2: 0.0014
    - γ = 5: -0.0108
    - γ = 10: -0.1242
    - γ = 20: -0.2408
  - c b λ (welfare cost of baseline, using German data):
    - γ = 1: -0.9885
    - γ = 2: -0.0012
    - γ = 5: -0.2072
    - γ = 10: -0.5336
    - γ = 20: -0.8517
- Illustrative example preserved:
  - If γ = 5, reducing inequality to an optimal level of *0.3σ would cost a reduction in consumption (*a λ) of 20 percent but would yield a welfare gain (c a λ) of 15 percent.
  - For γ = 5, the b-calibration implies reducing inequality would imply a fall in the growth rate from 2.1 percent (*a μ) to 0.48 percent (*b μ).

### Interpretation, implications, and policy conclusions
- Three factors explain why inequality costs can appear larger than growth gains:
  - Most gains from growth accrue in the future while inequality costs are borne every period.
  - Consumption dispersion (cross-sectional) is large relative to mean consumption.
  - Commonly used values of the IES imply substantial social aversion to consumption dispersion and inequality.
- Cross-country inequality is identified as the major determinant of worldwide welfare; for individual countries, within-country inequality is as important a determinant of welfare as growth.
- Policy implications (examples from paper):
  - Assess tradeoffs when evaluating progressivity of tax systems, trade liberalization, labor market reforms, law enforcement to curb evasion/informal markets, and migration policies.
  - Any evaluation of institutions must weigh welfare gains from efficiency and growth against welfare costs of increased inequality.
- Newborn-child thought experiment: contrary to Lucas (2004), a newborn may be willing to give up all growth to avoid birthplace risk; large fraction of growth could be given up to avoid family risk.
- Ranking of macro issues by potential social welfare impact (suggested by paper):
  - Cross-country inequality
  - Within-country inequality
  - Economic growth
  - Business cycles

### Caveats and directions for future research
- Technological restrictions linking inequality, growth, and consumption are modeled in reduced form; micro-foundations not fully specified.
- Empirical literature on inequality–growth relationship lacks consensus.
- Authors suspect main tradeoff may be between inequality and consumption levels rather than inequality and growth given similar long-run growth rates across countries.
- Future research: improve measurement of technological constraints using panel data for inequality and consumption levels.

*Source: _wp0717 - References*

### References..............................................................................................................

### References

### I. Introduction and purpose
- Evaluates Robert Lucas’ (2004) assertion that economic growth outweighs redistributive policies, using Lucas’ (1987) welfare-evaluation framework extended in three directions:
  - welfare gains from growth for intertemporal elasticity of substitution (IES) values other than one;
  - introduction of consumption inequality across individuals;
  - assessment of optimal inequality and optimal growth by introducing a production possibility frontier (technological restrictions) for inequality, consumption levels, and growth.

### Key methodological features
- Welfare evaluated in present-value terms using an isoelastic utility function and log-normal distribution of consumption (Lucas (1987) framework extended).
- Social welfare function: classical utilitarian (Benthamite) weighting everyone equally; assumes identical individual preferences so inequality emerges from unequal opportunities rather than tastes.
- Inequality measured by the standard deviation of cross-sectional consumption (log of consumption).
- Cost of inequality defined as the permanent compensation rate on consumption required to leave the social planner indifferent between observed situation and an ideal situation with no cross-sectional dispersion.
- Counterfactual experiments performed:
  - how much growth society would give up for perfect equality;
  - what level of inequality compensates for lack of growth;
  - welfare consequences of eliminating all growth and inequality simultaneously.
- Technological frontier (reduced-form) specified in the space of inequality, growth, and consumption levels; two calibrations using US and Scandinavian data (Aaberge and others (2002)) and data from West and East Germany.

### Main quantitative findings — welfare gains from growth
- Welfare gains from one additional percentage point of annual economic growth range from 2 percent to 21 percent, across the IES values considered by Lucas (1987).
- At a per-capita consumption growth rate of 2.1 percent (the 1960–2000 average for the 108 economies analyzed), welfare gains from total economic growth range from 7.6 percent to 51 percent.
- These gains are typically lower than the 20 percent reported by Lucas for the log-utility (IES = 1) case.

### Main quantitative findings — welfare costs of inequality
- Standard deviation of the log of aggregate consumption (time series) is around 1 to 2 percent (explains small welfare costs of business cycles in Lucas).
- Standard deviation of the cross-sectional distribution of the log of consumption:
  - around 50 percent within countries;
  - around 100 percent across countries.
- Welfare costs of inequality (permanent compensation rates) range, for various IES values, from:
  - 12 percent to 91 percent of current welfare for within-country inequality;
  - 40 percent to almost 100 percent of current welfare for cross-country inequality.
- Shadow price of inequality in terms of growth (marginal willingness to substitute inequality for growth, μ = ∂growth/∂σinequality):
  - ranges from 0.026 to 6.55 for within-country inequality;
  - ranges from 0.0521 to 13.11 for cross-country inequality.

### Counterfactual numerical examples (IES = 1/2)
- Planner would give up 1.62 percentage points of economic growth (out of 2.1 points) to eliminate all within-country inequality.
- Planner would give up 4.49 percentage points of economic growth to eliminate all cross-country inequality.
- A reduction of 34 percent in cross-country inequality would compensate for the lack of growth (no growth).
- A reduction of 136 percent in within-country inequality would compensate for the lack of growth (no growth).
- Eliminating all growth and cross-country inequality improves welfare by 48 percent.
- Eliminating all growth and within-country inequality reduces welfare by 9 percent.

### Technological-constraint (optimality) results
- For within-country analysis and two calibrations (US/Scandinavia; West/East Germany):
  - Current US values of inequality, growth, and consumption are close to their optimal values if IES ≈ 1/2.
  - If IES is smaller (e.g., IES = 1/5), optimal level of inequality in the US is close to that observed in Scandinavian countries.
  - If IES = 1/5, overall welfare cost of maintaining current suboptimal choices in the US is around 15 percent.

### Interpretation and implications
- Three factors explain why inequality costs appear larger than growth gains:
  - Most gains from growth accrue in the future while inequality costs are borne every period.
  - Consumption dispersion (cross-sectional) is large relative to mean consumption.
  - Commonly used values of the IES imply substantial social aversion to consumption dispersion and inequality.
- Cross-country inequality is identified as the major determinant of worldwide welfare.
- For individual countries, within-country inequality is as important a determinant of welfare as growth.
- The gross and net costs of inequality are likely large; estimated welfare measures provide upper bounds because they ignore redistributive policy costs and potential efficiency–inequality tradeoffs (e.g., eliminating inequality may reduce incentives and growth).

### Concluding view
- Findings support Okun’s view of a pervasive and important tradeoff between inequality and efficiency, suggesting that maximizing social welfare is not equivalent to maximizing economic growth.

*Source: _wp0717 - References*

### Chapter III extends the model of Chapter II by introducing technological restrictions, derives

### _wp0717 - Chapter III extends the model of Chapter II by introducing technological restrictions, derives

### Model and distribution of consumption
- World: continuum of countries; each country populated by a continuum of individuals. World population normalized to 1.
- Individual log consumption follows an autoregressive process with persistence ρ (ρ ∈ [0,1]):
  - Individual-specific shock: εt ~ N(0,1).
  - Country-specific shock: ηt ~ N(0,1), common to individuals in same country.
- Unconditional cross-sectional distribution of log consumption: ln ct ~ N(abt + λ, σx^2 + σy^2).
  - σx^2 measures cross-country inequality (country-specific factors).
  - σy^2 measures within-country inequality (individual factors).
- No aggregate uncertainty at the worldwide level (law of large numbers).
- Baseline initial distribution is the unconditional distribution evaluated at t = 0.
- Parameters a and b chosen so that aggregate worldwide consumption E(c_t) = λ + μ t + (1)(1) (specification in text); baseline λ set to 0 in baseline case.

### Individual and social welfare framework
- Individual expected lifetime utility (starting from c0): U = E0[Σ β^t u(ct)] with momentary utility u(c) = (c^(1−γ) − 1)/(1 − γ) and 1/γ > 0 is the intertemporal elasticity of substitution.
- Social welfare W defined as the population average of individual welfare: W = ∫ U(c) dF(c), where F is the cross-sectional cdf at time 0.
- Social welfare depends on:
  - Initial consumption level λ,
  - Growth rate μ,
  - Total dispersion σx^2 + σy^2,
  - Preference parameters γ and β.
- Key comparative statics:
  - ∂W/∂(σ^2) < 0 (inequality reduces social welfare).
  - ∂W/∂μ > 0 (growth increases social welfare).
- Persistence ρ affects welfare only via its effect on dispersion σx^2 + σy^2 (social mobility matters only to the extent it affects inequality).
- Social planner treats aversion to inequality equal to aversion to risk: coefficient of aversion to inequality equals γ (inverse of IES), following Atkinson (1970).

### Welfare measures (definitions and closed-form expressions)
- Compensation-rate measures λ:
  - μλ: proportional increase in consumption (uniform across all periods, countries, and individuals) required to leave planner indifferent between baseline path and a path with no growth.
  - xλ: compensation for cross-country inequality.
  - yλ: compensation for within-country inequality.
  - 0λ: compensation for total inequality.
- Closed-form solutions:
  - μλ = [1 − β^(1/γ)] / [β^(1/γ) (1 − β)] × (−βμ?) (expression presented in text; see equation (9) in source).
  - For i ∈ {x,y,0}: λ_i = − (1/2) e^{(1−γ) σ_i^2/2} (formula represented in text; see equation (10) in source).
- Alternative compensation rates on the baseline consumption path (hatted λ̂):
  - λ̂μ, λ̂x, λ̂y, λ̂0 defined as percentage changes relative to the baseline path.
  - Relationships: λ̂μ = 1 − 1/(1 + μλ) (equation (12)); λ̂_i = 1 − 1/(1 + λ_i) for i ∈ {x,y,0} (equation (13)).
- Welfare measures in terms of growth rates μ_i:
  - μx, μy, μ0 are percentage-point changes in growth the planner would trade for eliminating respective inequalities; closed-form expressions given in equation (14).
- Inequality-change measures θ:
  - xθ, yθ, 0θ: percentage reductions in inequality a planner would forgo in exchange for zero economic growth; closed-form expressions in equation (15).
- Net welfare consequences of eliminating both growth and inequality:
  - λ (for x, y, 0 cases) represent net welfare gains (or costs if negative) of eliminating both growth and the corresponding inequality; closed-form expression given in text.
- Marginal rate of substitution (shadow price) between inequality and growth:
  - MRS_i (around baseline) = (1 + β μ^1/γ) / (γ μ) × σ_i (expression summarized from text; see relevant equation).
  - Shadow price between inequality and consumption level given similarly in terms of λ.

### Calibration (parameters chosen)
- Growth of per-capita consumption (based on Penn World Tables 6.1, 108 economies, 1960–2000):
  - Unweighted average growth rate: 2.3 percent (standard deviation 1.17 percent).
  - Weighted average growth rate: 2.1 percent (standard deviation 0.99 percent).
  - Calibration choice: 0 2.1%μ=.
- Cross-country dispersion:
  - Evidence from Figure 2 suggests stability; calibration choice: 0 1 x σ = .
  - (Text presents this as "0 1 x σ = ." — preserved as in source.)
- Within-country dispersion:
  - Based on Krueger and Perri (2002) for U.S. consumption (std dev ~0.48); calibration choice: 0 0.5 y σ = .
- Risk aversion γ (inverse of IES):
  - Literature disagreement noted (IES close to zero → γ ≅ ∞; or IES close to 1 → γ ≅ 1).
  - Calibration computes welfare measures for different values of the IES 1 γ or for "[] 1, 2, 5, 1 0, 2 0γ∈" (parameter set as presented in source).
- Discount factor: 0.95 β = .
- Baseline parameter collection summarized in Table 1 (referenced in text).

### Results (key findings from calibrated experiments)
- Reproduction of Lucas (1987) result:
  - Welfare gains from economic growth (μλ and 1%λ) are substantial when γ is close to 1 and decline as γ increases.
  - Example: Lucas reports a 20 percent gain from one additional point of economic growth; Table 2 shows that if γ = 5, the gains are 8.5 percent.
  - Total gains from economic growth as measured by μ are between 11 percent and 48 percent.
- Large welfare costs associated with consumption inequality:
  - Within-country inequality costs range from around 12 percent to 92 percent.
  - Cross-country inequality costs range from around 40 percent to almost 100 percent.
- Comparative thresholds:
  - Gains from economic growth are smaller than the cost of total inequality if 1.11γ≥.
  - Gains from economic growth are smaller than the cost of cross-country inequality if 1.28γ≥.
  - Gains from economic growth are smaller than the cost of within-country inequality if 3.2γ≥.
- Overall interpretation:
  - Inequality has a very large impact on aggregate welfare under the calibrated parameter values.
  - These results challenge Lucas’ (2004) assertion that benefits of growth dwarf costs of inequality.

### Implications and interpretation
- Social mobility per se matters only through its effect on inequality (ρ enters welfare via σx^2 + σy^2).
- Preference parameters crucially shape welfare trade-offs:
  - Higher γ (greater risk aversion / lower IES) → smaller welfare gains from growth and exponentially larger welfare costs of inequality.
- Policymaking interpretation:
  - Under the model and calibrations used here, policies that reduce cross-country and within-country consumption dispersion can yield very large welfare gains, potentially exceeding gains from growth depending on γ.
  - The shadow price (MRS) between inequality and growth increases with the degree of inequality and exponentially with γ, implying that a risk-averse planner places a much higher value on inequality reduction.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0717.pdf*

### references.

### _wp0717 - references

### Welfare costs: business cycles versus inequality
- Estimates of business-cycle costs: "the cost of the business cycle can reach 3–4 percent of baseline consumption for households with no wealth" and "as large as 12 percent" under less restrictive preferences.
- Comparative statement: "Even these estimates of the costs of business cycles seem small, however, relative to the costs of inequality."
- Welfare measures introduced in Section A:
  - ˆμλ is the welfare cost of no growth.
  - ˆxλ is the welfare gain of eliminating cross-country inequality.
  - ˆyλ is the welfare gain of eliminating within-country inequality.
- Quantified interpretations (for 2γ=):
  - "Eliminating economic growth is equivalent to introducing a permanent 28 percent tax on consumption;"
  - "eliminating within-country inequality is equivalent to introducing a permanent 28 percent subsidy on consumption;"
  - "eliminating cross-country inequality is equivalent to introducing a permanent 249 percent subsidy on consumption!"

### Equality–growth tradeoffs and planner willingness to substitute
- First measure: reduction in growth that offsets gains from eliminating inequality (reported in Table 4 as i μ, {},,0ixy= for different γ).
  - Example: "3.1 y μ = − for 5γ = means that the planner would accept a reduction of 3.1 points in the growth rate of consumption in exchange for eliminating all within-country inequality." Baseline growth = 2.1 percent → new growth = -1 percent.
  - "All the μ-welfare measures are negative, in most cases implying negative net growth rates."
- Second measure: reduction in inequality that compensates for elimination of growth (reported in Table 4 as i θ, {},,0ixy=).
  - Example: "36.9% y θ = for 5γ = means that a 36.9 percent reduction in within-country inequality would be sufficient to compensate the planner for a total lack of growth."
  - "The fact that most θ’s are below 37 percent suggests that relatively small reductions in inequality would compensate for the lack of growth."
- Third measure: welfare consequences of eliminating growth and inequality simultaneously (reported as i λ, {},,0ixy=).
  - Example: "33% y λ = − for 5γ = means that a large welfare gain ensues from eliminating inequality while simultaneously stopping growth."
  - "The fact that most i λ’s are negative suggests that there is too much inequality relative to growth."
- Social marginal rate of substitution between inequality and growth (reported as 1 i MRS for {},,0ixy=):
  - Indifference curve defined by () 2 0 0,   ,WWμσ=.
  - Example: for 5γ =, "1 0.377 y y MRS μ σ ∂ == ∂ means that the shadow price of one point of inequality, measured by σ, is around 1 3 of a point of economic growth."
  - Even for 1γ =, "the shadow price of inequality differs significantly from zero."
- Marginal rate of substitution between inequality and consumption level (2 i MRS):
  - Example: "1 2.5 y y MRS λ σ ∂ == ∂ for 5γ = means that the planner would be willing to permanently give up 2.5 percent of consumption for a permanent reduction of one point of inequality."
- Interpretation: as risk aversion rises, increases in inequality must be compensated with much higher growth. Observed social choices lie on both social indifference curves and the production possibility frontier if choices are optimal.

### Planner’s problem and optimal inequality (Chapter III summary)
- Social welfare function (recast; equation (16)):
  - () () 22 (1)() / 2 1 22 1 (1) ,,; , (1)  1(1) xy xy e WW γγσσ γ γ λ λμσ  σ γβ γβμ −−  + − − + =+≡ −−+ 
- Technological frontier (equation (17)):
  - () () 12 12 11,0,0,0AA εε σμλ εε =+  +   > ≥ ≥
  - Interpretation: reduced-form technological tradeoffs between inequality, growth, and consumption levels.
  - Focus of analysis: within-country inequality due to limited knowledge about cross-country enforcement and redistribution feasibility.
- Planner’s maximization (first-order conditions and solutions):
  - Optimality conditions lead to (18): () () 2 1 2 2 2 2 111A ε εγ μ λε γ σ = ++ =
  - Optimal inequality (19): *2 2 1 σ εγ =
    - "optimal inequality depends positively on the intertemporal elasticity of substitution, 1 γ, and negatively on the elasticity of inequality with respect to the level of consumption."
    - "The discount factor and, in particular, 1 ε, play no role in determining optimal inequality."
  - Optimal growth (20): 1 1 1 * 2 11 γ ε μ β ε − ⎡⎤ ⎛⎞ +=  + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦
    - If 1γ >, then optimal growth depends positively on β and 2 ε and negatively on 1 ε.
    - Condition for positive growth: 1 2 11 ε β ε ⎛⎞ + > ⎜⎟ ⎝⎠.
  - Optimal consumption level (21): () 2 1 1 * * * 1 1 A ε ε σ λ μ ⎡⎤ ⎢⎥ += ⎢⎥ + ⎣⎦
  - Optimal social welfare: W**** = W(λ*, μ*, σ*).
  - Welfare cost of baseline choices c λ defined by solving (22).

### Calibration: estimating elasticities and parameters
- Required parameter for optimal inequality: 2 ε (elasticity of inequality w.r.t. consumption).
- Procedure: compute ε2 as percentage difference in inequality relative to percentage difference in consumption per capita for country pairs with similar growth rates (US vs Scandinavia example).
- US vs Scandinavia data (Penn World Table 6.1; Aaberge and others (2002)):
  - Per capita income: US ≈ 24 percent higher than Scandinavian countries in 1990 and ≈ 25 percent higher in 2000 → use 25% λ = or () 1ln1.25 λ ∆ + =.
  - Gini coefficients (1990 averages): US = 0.346; Scandinavia = 0.2173.
  - Transforming Gini to standard deviation (assuming log-normal):
    - σ = Φ−1( (1 + Gini)/2 ) × sqrt(2)? (formula given as 21 2 Gini σ ⎛⎞ =Φ − ⎜⎟ ⎝⎠ — solved to obtain standard deviations.)
    - Standard deviation of disposable income: US = 0.634; Scandinavia = 0.39; ratio = 1.63 → ln ln 1.63 σ ∆ =.
  - Calculated elasticity: () 2 ln ln 1.63 2.19 ln  1ln 1.25 σ ε λ ∂ = ≅ = ∂ +
- Additional parameters needed: A (constant in (17)) and 1 ε (elasticity of inequality w.r.t. consumption level) to compute λ* and μ* and welfare cost c λ.
- Two estimates of 1 ε:
  - First estimate (assume baseline growth rate is optimal): solve from (20) for 1 ε using observed μ0; denote resulting optimal choices as *a λ and c a λ with μ* = μ0.
  - Second estimate (use natural experiment of 1945 East vs West Germany to estimate 1 ε):
    - Biewen (2000) data (1990): log-variance of income per capita (2 y σ) = 0.23 in West Germany and 0.1150 in East Germany → ratio 1.41 (ywest / yeast = 1.41) → ln ln 1.41 σ ∆ =.
    - Mean income in 1990: West ≈ 2.15 times East (Biewen 2000); Burda and Hunt (2001) report 2.32. Accounting for post-reunification drops suggests alternative ratios; example calculation produces 1.37 or other intermediate values; Penn World Table Mark 5.6: ratio 1.95 in 1970, 1.28 in 1988.
    - Assume per-capita consumption ratio 1 in 1945 and 1.65 in 1989 → ratio of annual growth rates: (1 + μwest) / (1 + μeast) = 1.65^(1/44).
    - Second estimate of 1 ε: () 1 ln 30 ln  1 b σ ε μ ∆ == ∆ +.
- Empirical literature caveat: relationship between inequality and growth lacks consensus (references to Benabou (1996), Alesina and Rodrik (1994), Persson and Tabellini (1994), Partridge (1997), Deninger and Squire (1998), Forbes (2000)).

### Calibration results (preview)
- Table 5 (first row) reports optimal σ for different γ:
  - σ* ranges from 0.67 for 1γ = to 0.15 for 20γ =.
  - Interpretation: "They suggest that the current degree of inequality in the US is optimal if the coefficient of risk aversion is around" (text truncated at that point in provided content).

*Source: _wp0717 - references*

### 2. However, if the coefficient is larger than 2, as the equity-premium puzzle suggests, then

### _wp0717 - 2. However, if the coefficient is larger than 2, as the equity-premium puzzle suggests, then

### Key findings on inequality, growth, and welfare
- If γ = 5 then the optimal inequality level is more like that of Scandinavian countries and current US inequality is excessive.
- For the US at the actual choice, the production possibility frontier and the social indifference curve do not touch, indicating that US consumers on average would benefit from lower inequality.
- If γ is around 2 then current US choices are approximately optimal and the welfare costs are close to zero.
- If γ ≠ 2 the current US choices have significant welfare costs.
- These findings contradict Lucas’s suggestion that the gains from economic growth always dwarf the costs of inequality; the burden of inequality can be large enough to merit some sacrifice in economic growth.

### Calibrated parameters and Table 5 highlights (exact reported values)
- Optimal inequality *
σ:
  - γ = 1: 0.6724
  - γ = 2: 0.4778
  - γ = 5: 0.3022
  - γ = 10: 0.2137
  - γ = 20: 0.1511
- Calibrated values of 1 a ε (row 2 in Table 5):
  - γ = 1: 41.44
  - γ = 2: 29.30
  - γ = 5: 15.22
  - γ = 10: 8.14
  - γ = 20: 3.89
- Case a (associated with 1 a ε):
  - *
a
μ (optimal growth) = 0.0210 for all γ reported
  - *
a
λ (optimal consumption level):
    - γ = 1: 0.1448
    - γ = 2: -0.0205
    - γ = 5: -0.2054
    - γ = 10: -0.3217
    - γ = 20: -0.4210
  - c a λ (welfare cost of baseline, assuming *
0y
σσ=):
    - γ = 1: -0.0327
    - γ = 2: -0.0010
    - γ = 5: -0.1536
    - γ = 10: -0.4693
    - γ = 20: -0.8919
- Case b (associated with b ε = 0.30):
  - 1 b ε = 30.00 for all γ reported
  - *
b
μ (optimal growth):
    - γ = 1: ∞
    - γ = 2: 0.0194
    - γ = 5: 0.0048
    - γ = 10: 0.0021
    - γ = 20: 0.0010
  - *
b
λ (optimal consumption level):
    - γ = 1: -1.0000
    - γ = 2: 0.0014
    - γ = 5: -0.0108
    - γ = 10: -0.1242
    - γ = 20: -0.2408
  - c b λ (welfare cost of baseline, using German data):
    - γ = 1: -0.9885
    - γ = 2: -0.0012
    - γ = 5: -0.2072
    - γ = 10: -0.5336
    - γ = 20: -0.8517
- Specific illustrative example (text):
  - If γ = 5, reducing inequality to an optimal level of *
0.3σ
would cost a reduction in consumption (*
a
λ) of 20 percent but would yield a welfare gain (c a λ) of 15 percent.
  - For γ = 5, the optimal policy under the b-calibration would require consumption levels to stay around their baseline (*
0.0108
b
λ=− is close to 0), but reducing inequality would imply a fall in the growth rate from 2.1 percent (*
a
μ) to 0.48 percent (*
b
μ).

### Mechanisms and interpretation
- Critical elements driving results:
  - Time discounting and risk aversion downplay the role of growth for welfare.
  - Risk aversion amplifies the benefits of more equal outcomes.
  - The size of the risk involved at birth (birthplace and family risk) is enormous and central to results.
- Under some parameterizations (e.g., as γ → 1 in the b-case), it becomes optimal to sacrifice all consumption (*
1
b
λ→−) to obtain an infinite growth rate of output (*
b
μ = ∞).
- For γ = 2 there is approximate equivalence between the a- and b-calibrations (noted: 11 ab εε≈ for 2γ=), reinforcing that current US choices for inequality, growth, and consumption are close to optimal when γ = 2.

### Policy implications and tradeoffs
- There can be a "big tradeoff" between inequality and efficiency (growth), consistent with Okun (1975).
- Societies may not always find it best to adopt growth-enhancing institutions if those institutions foster further inequality.
- Examples of policy choices that reflect the equality-efficiency tradeoff:
  - Degree of progressivity of the tax system.
  - Trade liberalization and labor market reforms.
  - Law enforcement efforts (e.g., cracking down on tax evasion or informal markets).
  - Migration policies.
- Any evaluation of institutions must weigh welfare gains from efficiency and growth against welfare costs of more inequality; inequality concerns should be explicitly considered in aggregate institutional evaluations.

### Welfare interpretation framed by the newborn-child thought experiment
- Contrary to Lucas (2004), a newborn child may be willing to give up all growth to avoid birthplace risk, and a large fraction (if not all) of growth to avoid family risk.
- The paper quantifies social cost of inequality under standard macroeconomic assumptions and suggests societies could greatly benefit from reducing inequality.

### Caveats and directions for future research
- The micro-foundations of the technological restrictions linking inequality, growth, and consumption have not been fully specified; the paper uses a reduced-form technology and calibration from natural experiments rather than cross-country regressions.
- Empirical literature on the inequality-growth tradeoff lacks consensus.
- The authors suspect the main tradeoff is between inequality and consumption levels rather than between inequality and growth, given similar long-run growth rates across countries.
- Future research aims to improve measurement of technological constraints using panel data for inequality and consumption levels.

### Suggested ranking of macro issues by potential social welfare impact (from paper)
- Cross-country inequality
- Within-country inequality
- Economic growth
- Business cycles

*Source: _wp0717 - 2. However, if the coefficient is larger than 2, as the equity-premium puzzle suggests, then (IMF working paper content provided)*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0717.pdf_