Convergence of Life Expectancy and Living Standards in the World Kenichi Ueda* *The University of Tokyo PRI-ADBI Joint Workshop January 13, 2017 The views are those of the author and should not be attributed to any institutions that the author has been affiliated with. Ueda (UTokyo) DGE Life PRI-ADBI 1 / 27
Motivation Slow (or lack of) income convergence in the world is one of the biggest puzzle from the viewpoint of the neoclassical growth theory. Yet, life expectancy has converged a lot. 80 75 70 65 Life Expectancy 60 55 50 45 40 High Income Country Average Upper Middle Income Country Average Middle Income Country Average Lower Middle Income Country Average Low Income Country Average 35 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year Ueda (UTokyo) DGE Life PRI-ADBI 2 / 27
Motivation Once the increase in life expectancy is converted to monetary values and added to GDP, then world income seems to be quite converged (Becker, Phillipson, and Soares, 2005). However, they compute increase in the value of life based on one generation, partial equilibrium concept, using estimates of wage premium for a job with a higher fatality rate (e.g., Viscusi and Aldy, 2003). This concept is theoretically inconsistent with a growth theory in that it fails to account for General equilibrium effects through possible economy-wide acceleration in accumulation of human and physical capital; and Dynastic consideration substitutability between parents and kids. Ueda (UTokyo) DGE Life PRI-ADBI 3 / 27
Objective Introduce a new concept, the dynastic general equilibrium value of life. Compute it to measure welfare gains from an increase in life expectancy based on a neoclassical growth model with altruism; and actual data long-run U.S. growth rate and life insurance coverage. Ueda (UTokyo) DGE Life PRI-ADBI 4 / 27
Results DGE value of life is zero under the standard neoclassical assumptions. It is positive with slight modification larger human capital depreciation over generations than within. Calibration confirms sizable welfare gains from increase in life expectancy. However, these gains hardly mitigate the large inequality among countries GDP based measure is a good approximation for (slow) convergence in world living standards. Ueda (UTokyo) DGE Life PRI-ADBI 5 / 27
Model, Demography and Production Demography: A continuum of dynasty with measure one; In any period, a dynasty consists of an individual (no overlapping); People live T period Death is modeled as deterministic for now, but stochastic death can be analyzed in a same way. Production function: LOM of physical capital: LOM of human capital: y t = Ak α t h 1 α t. (1) k t+1 = (1 δ k )k t +i kt. (2) h t+1 = (1 δ w )h t +i ht, for t nt and; = (1 δ o )h t +i o ht, for t = nt. (3) Ueda (UTokyo) DGE Life PRI-ADBI 6 / 27
Model, Consumption Life Insurance: an individual pays a premium π t b t for b t benefits for her child in case she dies. Budget Constraint: c t +i kt +i ht = r t k t +w t h t π t b t, for t nt and; c t +i kt +i o ht = r tk t +w t h t +b t, for t = nt. (4) Dynastic Utility Maximization: max {c t} t= t=1 u(c 1 )+βu(c 2 )+ +β T u(c T ) +γ T+1 u(c T+1 )+γ T+1 βu(c T+2 )+. (5) Ueda (UTokyo) DGE Life PRI-ADBI 7 / 27
Perfect Altruism Assumption (1 Barro-Becker, Perfect Altruism) γ = β. The dynastic utility maximization becomes max {c t} t= t=1 β t 1 u(c t ). (6) t=1 Ueda (UTokyo) DGE Life PRI-ADBI 8 / 27
Consumption Smoothing using Life Insurance The only possible drop of income within a dynasty occurs when a parent dies. To smooth consumption, it is optimal to set the benefits to compensate the extra depreciation, b t = (δ o δ w )h t π t b t. (7) Life insurance is assumed to be actuarially fair, π t = ρ t 1 ρ t, (8) where ρ t [0,1] is the fraction of dynasties that change generations (replacement ratio). Ueda (UTokyo) DGE Life PRI-ADBI 9 / 27
Representative Agent Economy Combining two conditions, benefits and premium can be solved: b t = (1 ρ t )(δ o δ w )h t ; (9) π t b t = ρ t (δ o δ w )h t. (10) There is no dip in consumption, human capital evolution, and physical capital evolution over time. The economy can be regarded as a representative agent economy. Ueda (UTokyo) DGE Life PRI-ADBI 10 / 27
Stationary Environment and Stochastic Death For the sake of simplicity, I compare stationary economy under a given life expectancy with the one with a different life expectancy. Assume additional assumptions for stationarity: The longevity is the same for any age cohort; The initial physical and human capital are equal across all households; The initial population within each age cohort is identical. The population replacement ratio becomes equal to household level replacement ratio, ρ t = 1/T. The model now can be interpreted as a model with stochastic death with ρ t being the same death probability for everyone. Ueda (UTokyo) DGE Life PRI-ADBI 11 / 27
Neutrality of Longevity Assumption (2 Smooth Human Capital Transmission) Theorem (Neutrality of Longevity) δ o = δ w. Under the standard neoclassical growth assumptions 1 and 2, the dynastic general equilibrium value of life is zero. Sketch of proof: Compare one dynasty replacing generations every 30 years to the other dynasty replacing generations every 60 years. Ueda (UTokyo) DGE Life PRI-ADBI 12 / 27
Assumption (3 Costly Human Capital Transfer) δ o > δ w. Proposition (1 Non-Neutrality of Longevity) Under Assumptions 1 and 3, the dynastic general equilibrium value of life is strictly positive. Sketch of proof: Infrequent replacement of generations save depreciation costs over time. Ueda (UTokyo) DGE Life PRI-ADBI 13 / 27
Quantitative Assessment Computable Representative Agent Model W(K,H) = subject to resource constraint and law of motions, and where max,h + ), (11) I k,i h,c,k +,H +u(c)+βw(k+ c +I k +I h = AK α H 1 α, (12) K t+1 = (1 δ k )K t +I kt (13) H t+1 = (1 δ h )H t +I ht, (14) δ h (1 ρ t )δ w +ρ t δ o. (15) Obtain optimal x = H/K, which is constant, from first order conditions. Ueda (UTokyo) DGE Life PRI-ADBI 14 / 27
One Capital Representation subject to V(K) = max c,k +u(c)+βv(k+ ), (16) c +(1+x)I k +x(δ h δ k )K = Ax 1 α K. (17) The Euler equation with c 1 σ /(1 σ), (u ) ( c ) σ u = = β (A x1 α c 1+x x ) 1+x (δ h δ k )+(1 δ k ) = βg(x,δ h ). (18) Note: I restrict attention to perpetual growth βg(x,δ h ) 1 but assume no explosion β(βg(x,δ h )) 1/σ < 1. Ueda (UTokyo) DGE Life PRI-ADBI 15 / 27
α 1/3 δ k 0.05 δ w 0.02 0.04 δ o 0.7 0.9 A 0.25 0.3 β 0.96 0.99 0.99 Ueda (UTokyo) DGE Life PRI-ADBI 16 / 27 Benchmark Parameters Standard parameter values. Growth rate is approximately equal to 2 percent, consistent with long-run U.S. data. Life insurance coverage is about 6 times high as the annual income, consistent with the actual U.S. data (Hong and Ríos-Rull, 2006). Table 1: Parameter Values Benchmark Higher δ w Higher δ o Higher A Higher β Higher β High (recalibrated)
Dynastic General Equilibrium Value of Life The wealth transfer that compensates a possible welfare increase by a change in longevity at the steady state. Let V ξ (K) associated with the value of the value function with the state variable K under the specific life expectancy T = ξ. Suppose some policies can increase the longevity from T = ξ to T = ξ. Then, the percentage increase in the dynastic general equilibrium value of life τ is defined as V ξ (K(1+τ)) = V ξ (K). (19) Ueda (UTokyo) DGE Life PRI-ADBI 17 / 27
Calculation of Value of Life If then V(K) = ΨK 1 σ = ((1 s)ax1 α ) 1 σ (1 βg 1 σ )(1 σ) K1 σ. (20) Ψ ξk 1 σ = (Ψ ξ (1+ ))K 1 σ, 1+τ = (1+ ) 1 1 σ. (21) Note: Ψ, and thus τ, can be obtained numerically only x needs to be obtained numerically; then, analytical formula gives us s, g, and Ψ. Since the model exhibits constant growth, the wealth transfer τ is equivalent to the increase in annual output and thus permanent consumption, typically used as the measure of welfare gains in the business cycle literature. Ueda (UTokyo) DGE Life PRI-ADBI 18 / 27
Benchmark Quantitative Assessment Life Expectancy 40 50 60 70 80 Implied Increase in Dynastic G.E. Value of Life from 40 N/A 7.11 12.08 15.74 18.56 39.88 (% increase in annual income) Life Insurance Benefits 5.98 6.00 6.01 6.02 6.02 N/A (ratio to annual income) Consumption Growth (%) 1.61 1.84 2.00 2.10 2.19 2.76 Optimal H/K Ratio 11.28 11.37 11.43 11.47 11.51 11.74 The longer the life expectancy is, the higher is the welfare gain. However, the marginal gain is smaller as the life expectancy rises. This creates income convergence. Sensitivity analysis shows that growth rate and life insurance coverage is sensitive to alteration of parameter values so that I focus on the benchmark case. Ueda (UTokyo) DGE Life PRI-ADBI 19 / 27
World Income Convergence Compute convergence in world income, corrected for the increase in value of life. Compare the dynastic general equilibrium value of life with the one generation, partial equilibrium value (BPS). 96 countries; GDP per capita from Penn World Table 6.1; Life expectancy at birth from World Development Indicators. Stata s standard inequality measures relative mean deviation, coefficient of variation, standard deviation in log values, and the Gini coefficient plus the regression to the mean from 1960 values. For each measures, improvements are calculated as measure using GDP per capita - measure using full income per cpaita. measure using GDP per capita (22) Ueda (UTokyo) DGE Life PRI-ADBI 20 / 27
Convergence of Income and Full Income GDP per capita Full Income Improvements (%) 1960 1990 2000 1990 2000 1990 2000 Rel mean dev 0.48 0.48 0.44 0.47 0.42 3.2 4.6 Coef of var 1.24 1.31 1.23 1.26 1.18 3.4 4.5 Std dev log 1.03 1.01 0.97 0.98 0.95 3.2 1.8 Gini 0.57 0.57 0.54 0.56 0.52 2.7 3.4 Reg to mean -0.01-0.13-0.05-0.18 82.1 24.5 (0.82) (0.03) (0.22) (0.01) BPS Rel mean dev 0.48 0.47 0.42 0.44 0.38 7.1 10.8 Coef of var 1.23 1.25 1.17 1.17 1.05 6.9 10.3 Std dev log 1.02 1.03 0.96 0.98 0.95 5.3 1.5 Gini 0.51 0.52 0.49 0.49 0.46 4.9 6.4 Reg to mean -0.01-0.13-0.10-0.26 93.1 49.3 (0.86) (0.01) (0.02) (0.00) Ueda (UTokyo) DGE Life PRI-ADBI 21 / 27
Imperfect Altruism All the results hold under imperfect altruism. Assumption (4 Imperfect Altruism) 0 < γ < β. Proposition (2 Equilibrium with Imperfect Altruism) Under Assumption 4: (i) The consumption growth rate within a generation is the same as in the case with perfect altruism; (ii) The consumption growth rate over a generation is φ (γ/β) 1/σ times lower than within a generation; (iii) All households keep the same optimal ratio of human to physical capital x as in the case with perfect altruism. Ueda (UTokyo) DGE Life PRI-ADBI 22 / 27
Irrelevance of Altruism Proposition (3 Irrelevance of Altruism) Increase in the dynastic general equilibrium value of life is independent of the degree of altruism. Sketch of proof: Given constant growth rates and the optimal human-to-physical-capital ratio, the economy can be expressed again as a representative agent economy with aggregate physical capital as the only state variable. The value function with imperfect altruism is shown as a multiplicative form of the value function with perfect altruism Ṽ(K) = κv(k). By construction, the wealth compensation is not affected by κ. Ueda (UTokyo) DGE Life PRI-ADBI 23 / 27
Life Insurance Benefits Even if the increase in dynastic general equilibrium value of life is the same, life insurance coverage may well be different if parents do not care about children much. This is qualitatively true, but as shown in sensitivity analysis, there is not much room to change parameter values. To match with the data, only near perfect altruism (e.g., γ 0.9β) is plausible and the life insurance benefit is still around 6 times as annual income. b t Y t = (1 ρ t ) ( ( s +g (δ o δ w ) (1 φ) x )) x α +g t A. (23) Ueda (UTokyo) DGE Life PRI-ADBI 24 / 27
Wage Premium for a Risky Job Now consider the wage premium that a person (measure zero) demands for an early death, given the exogenously given life expectancy for everyone else, including her descendants. This wage premium is positive only under imperfect altruism. The wage premium depends on the level of value function Ṽ(K). It can be matched with the data easily by adjusting the exchange rate between the model unit of K and U.S. dollar. Similar adjustment has been commonly used since Rosen (1988) In the partial equilibrium setting, it is the intercept term in the (period) utility function to be set to freely to obtain the observed wage premium. Ueda (UTokyo) DGE Life PRI-ADBI 25 / 27
Conclusion Summary Introduced a new concept, the dynastic general equilibrium value of life, which is consistent with the standard growth theory. The calibration study shows that the welfare gains from increase in life expectancy is sizable. However, improvements in world income inequality are small, implying that a GDP-based measure is a good approximation for (non)convergence of the world living standards. 3 percent consumption tax for 10 year longer life. Ueda (UTokyo) DGE Life PRI-ADBI 26 / 27
Conclusion Further Work Several caveats due to focus on simple, stationary environment: life cycle effects, transitional effects (population dynamics), endogenous choice on fertility and longevity, and so forth. These may explain inability of the model to replicate more rich information in the cross-country and time-series data. A philosophical dilemma: Life is precious, but too much emphasis on value of life overlook the big difference in living standards in the world. Only way to progress is through scientific evaluation of value of life, based on a rigorous theory with support of actual data. Ueda (UTokyo) DGE Life PRI-ADBI 27 / 27