## _wp08158 - 1. Parameter Values ....................................................................................................

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---

### Background and motivation
- Observed facts:
  - Life expectancy in middle-income countries increased about 24 years (about 46 to 70) over the 1960-2004 period; in advanced economies it increased about 10 years (69 to 79); low-income countries experienced about 16 years’ (43 to 59) increase (1960-2004).
- Puzzle addressed:
  - Per capita GDPs of developing countries have not converged much toward advanced economies, yet life expectancy has converged substantially.
- Literature gap:
  - Existing labor-economics-based estimates of the value of life use a one-generation, partial-equilibrium concept and ignore general equilibrium and dynastic intergenerational substitution.

### Dynastic general equilibrium (GE) value of life: concept and modeling approach
- New concept:
  - Dynastic GE value of life: welfare gains from increased life expectancy evaluated within a dynastic general equilibrium growth model with intergenerational altruism.
- Key modeling assumptions and structure:
  - Barro-Becker perfect altruism: γ = β (Assumption 1).
  - Fertility given and population stationary; replacement ratio ρt = 1/T under deterministic or stochastically stationary interpretation.
  - Production: Cobb-Douglas y_t = A k_t^α h_t^{1−α}, α ∈ (0,1).
  - Physical capital law of motion: k_{t+1} = (1−δ_k) k_t + i_{k,t}, δ_k ∈ (0,1).
  - Human capital law of motion with generation-dependent depreciation:
    - h_{t+1} = (1−δ_w) h_t + i_{h,t} for t ≠ nT
    - h_{t+1} = (1−δ_o) h_t + i_{h,t} for t = nT
    - δ_o ∈ (0,1), δ_w ∈ (0,1).
  - Perfect life insurance market with actuarially fair premium π_t = ρ_t/(1−ρ_t), benefits b_t chosen to offset extra intergenerational depreciation.
- Definition of dynastic GE value of life:
  - Increase in dynastic utility (discounted sum over dynasty) when life expectancy increases from T = ξ to T = ξ̃; measured equivalently as a wealth transfer τ such that V_ξ(K(1+τ)) = V_{ξ̃}(K).

### Main theoretical results
- Neutrality under neoclassical assumptions:
  - Assumption 2 (Smooth human capital transmission): δ_o = δ_w.
  - Theorem 1 (Neutrality of Longevity): Under Assumptions 1 and 2, the dynastic GE value of life is zero.
- Positive value with costly intergenerational human capital transfer:
  - Assumption 3 (Costly human capital transfer): δ_o > δ_w.
  - Proposition 1 (Non-Neutrality of Longevity): Under Assumptions 1 and 3, the dynastic GE value of life is strictly positive.
  - Economic mechanism: Longer longevity reduces aggregate human-capital depreciation δ_h and raises steady-state human and physical capital accumulation, economizing transmission costs across generations.
- Robustness to imperfect altruism:
  - Assumption 4 (Imperfect altruism): 0 < γ < β.
  - Proposition 2 (Imperfect altruism properties):
    - (i) Consumption growth rate within a generation is g, same as perfect altruism case.
    - (ii) Consumption growth rate over a generation is φ g, where φ ≡ (γ/β)^{1/σ} (φ < 1).
    - (iii) All households keep the same optimal human-to-physical-capital ratio x as in perfect altruism case.
    - (iv) Human and physical capital grow at the same rate g as consumption.
  - Proposition 3: Increase in the dynastic GE value of life is independent of the degree of altruism; wealth transfer τ that compensates for a change in life expectancy does not vary with γ.

### Quantitative assessment: computable form and calibration
- Composite human-capital depreciation:
  - δ_h ≡ (1−ρ_t) δ_w + ρ_t δ_o.
- Representative-agent reduction and optimal H/K ratio:
  - Optimal human-to-physical-capital ratio x solves (1−α)A(H/K)^{−α} − δ_h = αA(H/K)^{1−α} − δ_k (equivalently equation (19)).
  - x is decreasing in δ_h (∂x/∂δ_h < 0) and increasing with longevity (∂x/∂T > 0).
- Euler equation (CRRA utility c^{1−σ}/(1−σ), σ > 0):
  - (c_{t+1}/c_t)^σ = β G(x, δ_h), and assume β G(x, δ_h) > 1 with β(β G(x, δ_h))^{1/σ} < 1 to ensure positive perpetual growth and finite value function.
- Benchmark parameter values (Table 1 summary):
  - β = 0.96
  - σ = 1.2
  - α = 1/3
  - δ_k = 0.05
  - δ_w = 0.02
  - δ_o = 0.7
  - A = 0.25
- Life insurance calibration target:
  - Life insurance coverage per annual income in U.S. data ≈ 6 (Hong and Ríos-Rull (2006)). Under benchmark parameters, optimal x ≈ 12 and equilibrium life insurance benefit-to-income ratio ≈ 6.

### Key quantitative findings (benchmark calibration; Table 2)
- Implied increase in dynastic GE value of life τ from baseline life expectancy 40:
  - Life expectancy 50: N/A
  - Life expectancy 60: 12.20 percent
  - Life expectancy 70: 15.91 percent
  - Life expectancy 80: 18.75 percent
  - Life expectancy ∞: 40.33 percent
  - Example: increase in life expectancy from 40 to 60 is equivalent to a 12.2 percent increase in permanent consumption (τ = 12.201).
- Associated equilibrium statistics:
  - Life insurance benefits (ratio to annual income): 5.966 (40), 6.00 (50), 6.04 (60), 6.06 (70), 6.07 (80).
  - Consumption growth (%): 1.80 (40), 2.03 (50), 2.19 (60), 2.30 (70), 2.38 (80), 2.97 (∞).
  - Optimal H/K ratio: 11.35 (40), 11.44 (50), 11.51 (60), 11.55 (70), 11.58 (80), 11.82 (∞).

### Sensitivity analysis highlights (Table 3)
- Higher β = 0.99 (other parameters same):
  - Raises implied increase in dynastic GE value of life (e.g., from 40 to 60: 26.26 percent).
  - Consumption growth becomes high (≈ 4.44–5.64 percent).
- Higher σ = 2:
  - Reduces implied value (e.g., from 40 to 60: 8.95 percent).
  - Consumption growth low (≈ 1.07–1.77 percent).
- Higher δ_w = 0.04:
  - Dramatically raises implied value (e.g., from 40 to 60: 40.31 percent).
  - Consumption growth becomes very low (≈ 0.47–1.59 percent).
- Higher δ_o = 0.9:
  - Raises implied value (e.g., from 40 to 60: 16.32 percent).
  - Increases equilibrium life-insurance benefits to income ratio to about 7.68 (not consistent with empirical estimates).
- Conclusion: parameter choices are constrained near benchmark values to match realistic consumption growth (~2 percent) and life-insurance-to-income ratio (~6).

### Effects on measured income convergence (empirical exercise, sample 96 countries)
- Full income (GDP corrected for dynastic GE value of life) shows better convergence than GDP per capita, but gains are small.
- Improvements in convergence from accounting for dynastic GE value of life are less than half the improvements reported by Becker, Philipson, and Soares using partial-equilibrium correction.
- Representative metrics reported:
  - Relative mean deviation (GDP per capita): 1960 = 0.48; 1990 = 0.48; 2000 = 0.44. Improvements in full income shown as Improvements (%) around 3.2 and 4.6 in Table 4.
  - Coefficient of variation (GDP per capita): 1960 = 1.24; 1990 = 1.31; 2000 = 1.23. Improvements in full income around 3.4–4.5 percent in Table 4.
- Overall conclusion: increase in life expectancy does not substantially change the view of inequality based on GDP per capita; GDP-based measure is a good approximation for world income convergence even when adjusting for life expectancy.

### Imperfect altruism and private risk premia
- Main invariance:
  - Even with imperfect altruism (γ < β), aggregate optimal x and aggregate growth g remain as in perfect altruism once adjusted, and τ is invariant to γ.
- Life insurance benefits with imperfect altruism:
  - b_t / Y_t = (1−ρ_t)[(δ_o − δ_w) − (1−φ)(s + g/x + g)] x^{α} / A  (equation (40)), where φ = (γ/β)^{1/σ}.
- Wage premium for risky jobs:
  - Under perfect altruism, no wage premium attributable to early death.
  - Under imperfect altruism, a wage premium exists; its sign and magnitude depend on γ relative to β and model scaling.

### Conclusions, caveats, and policy interpretation
- Summary:
  - Introduced dynastic GE value of life consistent with growth theory; shown theoretically positive under costly intergenerational human-capital transmission and quantitatively sizable under benchmark calibration.
  - Quantitative welfare gains from increases in life expectancy can be large in percent terms of permanent consumption (e.g., 12.2 percent for 40→60), yet corrections to measured cross-country income convergence are small (less than half of partial-equilibrium-based estimates).
- Policy interpretation:
  - Dynastic GE value of life is appropriate for valuing exogenous, economy-wide mortality-reducing interventions (e.g., new medicine, sanitation improvements).
  - Partial-equilibrium, one-generation value of life remains appropriate for valuing individual choices associated with risk (e.g., risky jobs, personal health investments).
- Caveats and directions for future work:
  - Model simplifications: steady-state focus, no life-cycle explicitness, fertility exogenous, simplified stochastic death interpretation, omission of within-country income heterogeneity.
  - Important omitted dynamics: life-cycle productivity effects, transitional population dynamics, endogenous fertility and longevity choices could change transitional and cross-sectional implications.
  - Empirical mismatches: model predicts slightly higher growth and H/K with longer life expectancy contrary to some empirical findings; effects are small and may be offset by omitted mechanisms.

*Source: INTRODUCTION section of working paper _wp08158 (IMF PDF).*

### 1. Parameter Values ....................................................................................................

### _wp08158 - 1. Parameter Values ....................................................................................................

### Main sections (with page references)
- 1. Parameter Values — page 25
- 2. Benchmark Quantitative Assessment — page 25
- 3. Sensitivity Analysis — page 26
- 4. Convergence of Income and Full Incom — page 27

### Figures
- Figure 1. Evolution of Life Expectancy — page 24

### Appendices and subcomponents
- I. Solutions — page 28
  - A. Optimal H/K Ratio — page 28
  - B. Euler Equation — page 29
- II. Imperfect Altruism Case — page 29
  - A. Proof of Proposition 2 — page 29
  - B. Proof of Proposition 3 — page 33

### Reference and pagination
- Reference — page 22
- Document pagination indicator: 3

*Content unit: _wp08158 - 1. Parameter Values ....................................................................................................*

### INTRODUCTION

### INTRODUCTION

### Background and motivation
- Observed facts:
  - Life expectancy in middle-income countries increased about 24 years (about 46 to 70) over the 1960-2004 period; in advanced economies it increased about 10 years (69 to 79); low-income countries experienced about 16 years’ (43 to 59) increase (1960-2004).
- Puzzle addressed:
  - Per capita GDPs of developing countries have not converged much toward advanced economies, yet life expectancy has converged substantially. This challenges the lack-of-convergence puzzle in growth literature.
- Literature gap:
  - Existing labor-economics-based estimates of the value of life use a one-generation, partial-equilibrium concept (willingness to accept higher wage for higher fatality rate). These ignore general equilibrium effects from economy-wide longevity changes and dynastic substitution across generations.

### Dynastic general equilibrium (GE) value of life: concept and modeling approach
- New concept introduced:
  - Dynastic general equilibrium value of life: welfare gains from increased life expectancy evaluated within a dynastic general equilibrium growth model with intergenerational altruism.
- Key modeling assumptions:
  - Barro-Becker perfect altruism: γ = β (Assumption 1).
  - Fertility given and population stationary; replacement ratio ρt = 1/T under deterministic or stochastically stationary interpretation.
  - Production: Cobb-Douglas y_t = A k_t^α h_t^{1−α}, α ∈ (0,1).
  - Physical capital law of motion: k_{t+1} = (1−δ_k) k_t + i_{k,t}, δ_k ∈ (0,1).
  - Human capital law of motion allows higher depreciation over generations: h_{t+1} = (1−δ_w) h_t + i_{h,t} for t ≠ nT and = (1−δ_o) h_t + i_{h,t} for t = nT, with δ_o ∈ (0,1), δ_w ∈ (0,1).
  - Perfect life insurance market with actuarially fair premium π_t = ρ_t/(1−ρ_t), benefits b_t chosen to offset extra intergenerational depreciation.
- Definition of dynastic GE value of life:
  - Increase in dynastic utility (discounted sum over dynasty) when life expectancy increases from T = ξ to T = ξ̃; measured equivalently as a wealth transfer τ such that V_ξ(K(1+τ)) = V_{ξ̃}(K).

### Main theoretical results
- Neutrality under neoclassical assumptions:
  - Assumption 2 (Smooth human capital transmission): δ_o = δ_w.
  - Theorem 1 (Neutrality of Longevity): Under Assumptions 1 and 2, the dynastic GE value of life is zero.
- Positive value with costly intergenerational human capital transfer:
  - Assumption 3 (Costly human capital transfer): δ_o > δ_w.
  - Proposition 1 (Non-Neutrality of Longevity): Under Assumptions 1 and 3, the dynastic GE value of life is strictly positive.
  - Economic mechanism: Value of life arises from economizing depreciation cost when generations change less frequently (i.e., longer longevity reduces aggregate human-capital depreciation δ_h and raises steady-state human and physical capital accumulation), rather than from direct extra utility from additional years for the current generation.
- Robustness to imperfect altruism:
  - Assumption 4 (Imperfect altruism): 0 < γ < β.
  - Proposition 2 (Imperfect altruism properties):
    - (i) Consumption growth rate within a generation is g, same as perfect altruism case.
    - (ii) Consumption growth rate over a generation is φ g, where φ ≡ (γ/β)^{1/σ}, i.e., φ < 1.
    - (iii) All households keep the same optimal human-to-physical-capital ratio x as in perfect altruism case.
    - (iv) Human and physical capital grow at the same rate g as consumption.
  - Proposition 3: Increase in the dynastic GE value of life is independent of the degree of altruism; wealth transfer τ that compensates for a change in life expectancy does not vary with γ.

### Quantitative assessment: computable form and calibration
- Composite human-capital depreciation:
  - δ_h ≡ (1−ρ_t) δ_w + ρ_t δ_o.
- Representative-agent reduction and optimal H/K ratio:
  - Optimal human-to-physical-capital ratio x solves (1−α)A(H/K)^{−α} − δ_h = αA(H/K)^{1−α} − δ_k (equivalently equation (19) in text).
  - x is decreasing in δ_h (∂x/∂δ_h < 0) and increasing with longevity (∂x/∂T > 0).
- Euler equation (CRRA utility c^{1−σ}/(1−σ), σ > 0):
  - (c_{t+1}/c_t)^σ = β G(x, δ_h), and assume β G(x, δ_h) > 1 with β(β G(x, δ_h))^{1/σ} < 1 to ensure positive perpetual growth and finite value function.
- Benchmark parameter values (Table 1 summary):
  - β = 0.96
  - σ = 1.2
  - α = 1/3
  - δ_k = 0.05
  - δ_w = 0.02
  - δ_o = 0.7
  - A = 0.25
- Life insurance calibration target:
  - Life insurance coverage per annual income in U.S. data ≈ 6 (Hong and Ríos-Rull (2006)). Under benchmark parameters, optimal x ≈ 12 and equilibrium life insurance benefit-to-income ratio ≈ 6 (see Table 2).

### Key quantitative findings
- Wealth-transfer measure τ (dynastic GE value of life) under benchmark calibration (Table 2):
  - Implied increase in dynastic GE value of life from baseline 40:
    - Life expectancy 50: N/A; 60: 12.20 percent; 70: 15.91 percent; 80: 18.75 percent; ∞: 40.33 percent.
  - Example: increase in life expectancy from 40 to 60 is equivalent to a 12.2 percent increase in permanent consumption (τ = 12.201).
- Associated equilibrium statistics (Table 2):
  - Life insurance benefits (ratio to annual income): 5.966 (Life expectancy 40), 6.00 (50), 6.04 (60), 6.06 (70), 6.07 (80).
  - Consumption growth (%): 1.80 (40), 2.03 (50), 2.19 (60), 2.30 (70), 2.38 (80), 2.97 (∞).
  - Optimal H/K ratio: 11.35 (40), 11.44 (50), 11.51 (60), 11.55 (70), 11.58 (80), 11.82 (∞).
- Sensitivity analysis highlights (Table 3):
  - Higher β = 0.99 (other parameters same) raises implied increase in dynastic GE value of life (e.g., from 40 to 60: 26.26 percent) and consumption growth becomes high (≈ 4.44–5.64 percent).
  - Higher σ = 2 reduces implied value (e.g., from 40 to 60: 8.95 percent) and consumption growth low (≈ 1.07–1.77 percent).
  - Higher δ_w = 0.04 dramatically raises implied value (e.g., from 40 to 60: 40.31 percent) but consumption growth becomes very low (≈ 0.47–1.59 percent).
  - Higher δ_o = 0.9 raises implied value (e.g., from 40 to 60: 16.32 percent) and increases equilibrium life-insurance benefits to income ratio to about 7.68 (not consistent with empirical estimates).
  - Conclusion: parameter choices are constrained near benchmark values to match realistic consumption growth (~2 percent) and life-insurance-to-income ratio (~6).
- Effects on measured income convergence (empirical exercise, sample 96 countries):
  - Full income (GDP corrected for dynastic GE value of life) shows better convergence than GDP per capita, but gains are small.
  - Improvements in convergence from accounting for dynastic GE value of life are less than half the improvements reported by Becker, Philipson, and Soares (BPS) using partial-equilibrium correction.
  - Representative reported metrics (selected):
    - Relative mean deviation (GDP per capita): 1960 = 0.48; 1990 = 0.48; 2000 = 0.44. Full income relative mean deviation improvements reported under Improvements (%) around 3.2 and 4.6 in Table 4.
    - Coefficient of variation (GDP per capita): 1960 = 1.24; 1990 = 1.31; 2000 = 1.23. Improvements in full income in Table 4 around 3.4–4.5 percent.
  - Overall: increase in life expectancy does not substantially change the view of inequality based on GDP per capita; GDP-based measure is a good approximation for world income convergence even when adjusting for life expectancy.

### Implications of imperfect altruism and private risk premia
- Even with imperfect altruism (γ < β), main mechanisms and quantitative increase in dynastic GE value of life are unchanged: aggregate optimal x and aggregate growth g remain as in perfect altruism once appropriately adjusted, and τ is invariant to γ.
- Life insurance benefits formula with imperfect altruism adds an extra term that can alter equilibrium benefit-to-income ratio:
  - b_t / Y_t = (1−ρ_t)[(δ_o − δ_w) − (1−φ)(s + g/x + g)] x^{α} / A  (equation (40) in text), where φ = (γ/β)^{1/σ}.
- Wage premium for risky jobs:
  - Under perfect altruism, no wage premium attributable to early death (descendants perfectly substitute current generation).
  - Under imperfect altruism, a wage premium exists; its sign and magnitude depend on γ relative to β and the model scaling.

### Conclusions, caveats, and policy interpretation
- Summary:
  - Introduced dynastic GE value of life consistent with growth theory; shown theoretically positive under costly intergenerational human-capital transmission and quantitatively sizable under benchmark calibration.
  - Quantitative welfare gains from increases in life expectancy can be large in percent terms of permanent consumption (e.g., 12.2 percent for 40→60), yet corrections to measured cross-country income convergence are small (less than half of partial-equilibrium-based estimates).
- Policy interpretation:
  - Dynastic GE value of life is appropriate for valuing exogenous, economy-wide mortality-reducing interventions (e.g., new medicine, sanitation improvements).
  - Partial-equilibrium, one-generation value of life remains appropriate for valuing individual choices associated with risk (e.g., risky jobs, personal health investments).
- Caveats and directions for future work:
  - Model simplifications: steady-state focus, no life-cycle explicitness, fertility exogenous, simplified stochastic death interpretation, omission of within-country income heterogeneity.
  - Important omitted dynamics: life-cycle productivity effects, transitional population dynamics, endogenous fertility and longevity choices could change transitional and cross-sectional implications (e.g., Acemoglu and Johnson (2006) mechanisms).
  - Empirical mismatches: model predicts slightly higher growth and H/K with longer life expectancy contrary to some empirical findings; effects are small and may be offset by omitted mechanisms.

*Source: INTRODUCTION section of working paper _wp08158 (IMF PDF).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08158.pdf_