c(t)−n(ρ+η)˜k(t) (1.22) plus an initial k(0)˜ >0 and a steady state k˜∗ normalized capital. There is a unique saddle-path stable steady state equilibrium (˜k∗,c˜∗), which solves d˜k(t)dt = d˜c(t)dt = 0.
Aggregate variables determine prices, and cohort-specic consumption solves c(t|τ˙ )
c(t|τ) =r(t)−ρ (1.23)
c(τ|τ) = (ρ+η)W(τ)`h (1.24)
for W(τ)≡ Z ∞
τ
e−
Rs
τ(r(υ)+η)dυw(s)ds
Proof. See Appendix 1.A.
1.3 Human capital and productivity-equivalent welfare
1.3.1 Productivity and welfare
Now we turn to welfare and value of life considerations. As households' choices change with cohorts, there is no representative agent, and we must choose a welfare criterion. We follow the common practice in the literature based on the value of life approach in Rosen (1988) and associate welfare in timetto the lifetime expected utilityV(t)of an agent born int, as inBecker et al. (2005) andJones and Klenow (2016).
Jones and Klenow(2016) associate this welfare criterion to the veil of ignorance concept from Rawls(1971). By this principle, normative judgments are promoted from the viewpoint of an in-dividual that ranks alternative societies without any prior knowledge of which preferences, talent or social position she will draw from the available positions in that society. Jones and Klenow (2016) suggest a version of this principle in which individuals behind the veil of ignorance are aware of their preferences, and they compare societies that dier in life expectancy, leisure time,
and the mean and variance of consumption streams. Here we assume the same methodological principle, but considering that the individual behind the veil of ignorance compares societies that are heterogeneous regarding life expectancy and years of schooling.
Basu and Fernald(2002) argue that, in a representative agent economy and under very general conditions, productivity growth measured by Solow's residual is the correct measure of welfare improvement. They claim that to be true even under imperfect competition or non-constant returns to scale. In these cases, aggregate productivity mismeasures technological change, yet growth in a modied Solow's residual still represents welfare change.
Basu et al.(2016) show that, to a rst-order approximation around the steady-state, the in-tertemporal utility for a representative consumer reects the present discounted value of produc-tivity residuals plus the initial stock of assets, under broad assumptions for production function and market competitiveness. They conclude that productivity growth and initial assets together comprise a sucient statistic for households' welfare.
The next proposition shows that a similar result holds for our OLG economy.
Propositon 2. Dene welfare at timeτ as the lifetime expected utility of a household born inτ. Then at the steady-state equilibrium, for ˜k∗ andg dened as in Proposition1, welfare is equal to
V∗(τ) = 1 (ρ+η)
¯
u+ logc∗(τ|τ) + r∗−ρ (ρ+η)
(1.25)
where
c∗(τ|τ) = (ρ+η)w∗(τ)`h
r∗+η−g (1.26)
r∗ =α˜k∗α−1−δ (1.27)
w∗(τ) = (1−α)˜k∗αA(τ)1−α1 (1.28) Furthermore, at the steady-state equilibrium and conditioned on the death rate and education endowments, productivity growth rate γ determines welfare improvement:
dV∗(τ)
dτ = γ
(1−α)(ρ+η) (1.29)
Proof. See Appendix 1.A.
1.3.2 Welfare TFP-equivalent
We have stated that there is an immediate relationship between productivity growth and welfare for a newborn household in the steady-state of our standard OLG economy, assuming health and education as xed. Now we propose a new productivity-equivalent welfare measure for changes in human capital endowments, which builds on the consumption-equivalence tradition of Lucas (1987,2003),Becker et al.(2005), and Jones and Klenow (2016).
These authors use models of expected intertemporal utility under exogenous consumption ows to perform welfare evaluations by calculating consumption-equivalent changes in utility. Lu-cas(1987,2003) computes the compensation in the consumption level of a risk-averse household that is equivalent to eliminating consumption variance. Becker et al.(2005) nd the consumption-equivalent in intertemporal utility for the rise in life expectancy in many countries between 1960 and 2000. Jones and Klenow(2016) calculate consumption compensation to compare welfare in dierent countries taking into account leisure, life expectancy, and consumption inequality.
We evaluate the welfare eects of human capital improvements by computing the productivity-equivalent change in expected lifetime utility of a newborn household. Using our OLG model, we calculate the productivity compensation that is equivalent in welfare terms to the gains in life expectancy and years of schooling observed for each country.
Our approach diers from the literature in that we compute productivity-equivalent welfare in an overlapping generation's general equilibrium model instead of consumption-equivalent for a single household that just consumes its endowments. By proceeding this way, we are able to compute welfare equivalents for changes in human capital, whose eects on utility are indi-rect. Consumption is endogenous and total factor productivity is the exogenous variable that is compensated for by changes in human capital and mortality.
Denition. Let V (A(τ), k(τ), h, η, n, `) denote the lifetime expected utility of a newborn indi-vidual in time τ, as a function of initial productivity A(τ), initial physical capital k(τ), human capital h, death rate η, population birth rate n, and the labor supply scaling factor `.
Now consider another newborn individual that lives in a reference economy with the same parameters, birth rate, and initial physical capital, but dierent death rate and human capital endowments (h0, η0).
Then the welfare TFP-equivalent is dened as the adjusted productivity λA(τ) that makes the newborn from the reference economy indierent to possessing the health and education
endowments of the rst economy:
V (λA(τ), k(τ), h0, η0, n, `) =V (A(τ), k(τ), h, η, n, `) (1.30)
In our empirical exercise, we nd the welfare-equivalent productivity change that accounts for the gains in longevity and schooling endowments, from 1960 to 2014 and from 1987 to 2014, for each country in our dataset. We take the US economy in 2014 as our reference to dene (h0, η0), such that λU S2014 = 1, and for each country we computeλ for the years 1960, 1987, and 2014.