Health externalities to labor productivity and optimal policies withendogenous fertility, labor and longevity
Siew Ling Yew a and Jie Zhang b,ca

Department of Economics, Monash University, Melbourne, Australiasiew.ling.yew@monash.edubSchool of Economics and Business Administration, Chongqing University, Chongqing,
China 400030c

Department of Economics, National University of Singapore, Singapore 117570
zhangjiecqu@cqu.edu.cn; ecszj@nus.edu.sg
February 21, 2024
Abstract

We explore how health externalities to labor productivity affect efficiency and optimal policies
in a lifecycle-dynastic model with endogenous fertility, labor and longevity. The healthexternalities decrease health spending, labor productivity, longevity, savings and labor but
increase fertility from the socially optimal levels. Public health subsidies through universal
public health insurance or private health subsidies through employer-based health insurance
increase the marginal benefit of health spending and the marginal cost of childrearing. Taxes
on income and consumption reduce the benefit of health spending and the cost of childrearing.
Appropriate taxes and subsidies, such as age-specific health subsidies and age-specific laborincome taxes, can fully internalize the health externalities to achieve the social optimum.
Without such taxes, private health subsidies alone cannot achieve the social optimum.
Calibration results suggest a larger welfare gain of optimal policies in the US with private
health subsidies for workers than in Australia with universal public health subsidies.
Keywords: Health externality, Longevity, Labor productivity, Fertility, Savings
JEL Classification Codes: H21; I13, I15; J13; O41
Declarations of Interest: noneWe would like to thank seminar participants at the World Bank EAPCE Research Center KL Hub, CDES
Sustainable development conference 2023, Asia-Pacific Academy of Economics and Management at Universityof Macau, Australian Public Choice Conference 2023, Workshop of the Australasian Macroeconomics Society
2022 and Productivity Commission Victoria Australia for their helpful comments.

1. Introduction
Health is an important form of human capital for our capabilities and wellbeing in life. For
workers, in particular, health is a key determinant of labor productivity. Thus, achieving good
health for all is an important on-going call by the United Nations to achieve the Sustainable
Development Goals. However, a major challenge to achieve good health for all is the market
failure arising from health externalities (Bloom and Canning, 2003). The external costs of poor
health or the external benefits of health improvements through preventive and curative healthspending are massive. Empirical evidence indicates that infectious diseases lead to adverse
macroeconomic effects, workplace productivity losses and mortality unrecognized byindividuals (e.g., Bloom et al., 2022b; de Courville at al., 2022), while influenza vaccination
effectively reduces influenza-related mortality and illness-related work absences (White, 2021;
Acton et al., 2022). Empirical evidence also indicates that workers’ utilization of preventive
healthcare services depends positively on the share of peers at work utilizing such services (e.g.,
Pruckner at al., 2020). Yet, it is unclear how health externalities to labor productivity affecthealth spending, savings, longevity and fertility and whether social policies such as public or
employer-based health insurance and taxes can internalize the health externalities.
This paper investigates the consequences of health externalities to labor productivity andthe roles of public or private health subsidies and taxes in internalizing the health externalities.
As shown in empirical studies, health investment or health spending improves health, laborproductivity and longevity.1 The core idea of health externalities to labor productivity in thispaper is that individual workers’ health is private information not perfectly observed by firms(Sauermann, 2016). With asymmetric information on health, health externalities to laborproductivity occur when healthier and more productive workers improve the average
productivity of labor at workplaces but firms lack the information to reward the workers.
Specifically, market failure occurs when firms benefit from having healthy workers but do
not pay them the wage rate that reflects their contributions to average labor productivity at
workplaces. Consequently, individual workers overlook the contributions of their healthinvestment to average labor productivity and under-invest in their health because the perceived
private return to health spending is below the social return. Under-investment in health results

1

Empirical evidence suggests a positive effect of health spending on health outcomes (Wolfe and Gabay, 1987;
Hitiris and Posnett, 1992; Cremieux et al., 1999; Or, 2000; Martin et al., 2008; and Kim and Lane, 2013) andpositive effects of health spending and better health on labor productivity (Rivera and Currais, 2004; Suhrcke and
Urban, 2010; and Prettner et al., 2013). See Tompa (2002) for a thorough survey on the links among health, healthexpenditure and labor productivity.

1

in low levels of average labor productivity, wage rates (time cost of childrearing), interest rates,
longevity, savings and output but high fertility relative to the socially optimal levels.
It is essential to measure the efficiency consequences of the health externalities for policy
implications. In many developed countries such as Australia, workers are covered by universal
public health insurance financed by taxes on income and consumption, while most workers in
the US are covered by employer-based health insurance (EHI).2 It is important to ask whether
the public or private health subsidies can internalize the health externalities and achieve thesocial optimum.
To gain insight into these questions, we develop and calibrate a lifecycle-dynastic modelwith two-sided intergenerational transfers 3 and endogenous fertility, labor and longevity to
explore the macroeconomic impacts of health externalities to labor productivity and optimal
health and tax policies, both theoretically and quantitatively. Departing from lifecycle models
where health spending only extends longevity and retirement lives, health spending here has
external benefits for the average health and labor productivity and private benefits for longevity
and working life at old age. By incorporating two-sided intergenerational transfers along withendogenous fertility and labor, the costs of raising a child in our model consist of the time costof childrearing (forgone earnings for time-intensive childrearing) as well as bequests and intervivos transfers between young and old parents.4 These features provide several novel resultsthat have important implications for optimal health and tax policies as summarized below.
First, we show that appropriate taxes and universal public health can internalize the healthexternalities to achieve the social optimum. Intuitively, young-age and old-age public healthsubsidies increase young-age and old-age health spending, lifetime labor supply and longevity
by lowering the private cost of health spending. Old-age public health subsidies also increase
parental transfers to children (the transfer cost of childrearing). The transfer cost of childrearing
rises further if there are public transfers to the elderly. On the contrary, taxes on young-age andold-age labor income reduce the time cost and the transfer cost of childrearing, respectively,
while consumption taxes reduce the time cost, bequest cost and transfer cost of childrearing.
Capital income taxes reduce the relative price of current consumption, while savings subsidies
increase the relative price of current consumption and decrease the bequest cost of childrearing.
Thus, appropriate public health subsidies and taxes can fully internalize the health externalities.

2

The EHI is popular in the US where public health only covers 35% of the population (OECD, 2016).
Existing evidence finds intergenerational transfers within families (Laitner and Juster, 1996; Altonji et al., 1997).
4
For instance, altruistic parents are willing to ease the tax burdens of public transfers to the elderly on children
by giving bequests as in Zhang (1995).
32

Second, the EHI chosen by firms to maximize profit can substitute for public healthsubsidies to fully internalize the health externalities when it is accompanied by appropriate
taxes, with or without public health subsidies. However, without taxes, the EHI alone cannot
achieve the social optimum because taxes on income and consumption and the EHI exert
opposite effects on the cost of childrearing, and thus it is necessary to combine them to attainsocially optimal fertility. The EHI can also ease the financial burden of universal publichealthcare. Third, health subsidies and labor income taxes should have different age-specificrates to achieve the social optimum due to the age-dependent impacts on the marginalcontributions of health spending and on the costs of childrearing.5
For quantitative assessment, we calibrate the model to Australian and US data as twointeresting case studies since both countries share similar levels of economic development but
have different health systems: universal public health in Australia but the EHI for most workers
aged below 65 in the US. We calibrate key parameters to match certain data moments in eachcountry, such as longevity, the investment to output ratio, fertility, health-spending shares of
output, taxes, subsidies and public transfers to the elderly. We assume the same degree of healthexternalities to labor productivity in both countries based on the estimated range in Bloom etal. (2022a) for 133 countries.
Our model generates some interesting quantitative results as follows. The benchmark tax
and health systems in both countries have lower welfare than their laissez faire and social
optimum as the welfare loss of distortionary taxes outweighs the welfare gain of healthsubsidies, mainly due to their high income taxes. The net welfare loss is larger in the US thanin Australia as US taxes on capital and labor income are relatively high.
Since the most important source of funding public health subsidies in Australia and the US
is income taxes, we also compute optimal public health subsidies financed by taxes on laborand capital income by shutting down consumption taxes and public transfers. For Australia
with universal public health, the optimal public health subsidy is higher for young workers thanfor old workers, hence young workers should pay higher labor income taxes than old workers
to achieve the social optimum. On the contrary, for the US with public health subsidies to oldworkers (65 or above), old workers should pay higher labor income taxes than young workers
to achieve the social optimum. This means that different health systems have important
implications for optimal tax policies. The optimal capital income tax should be lower than the5

Weinzierl (2011) argues that age-dependent taxes are useful for Pareto improvements in a model with private
information for earnings ability in the absence of investment in capital and health.

3benchmark rate to raise savings, especially in the US, given the relatively low investment
subsidy rates. If using the benchmark consumption tax, optimal health subsidies and labor
income taxes would be lower in both countries (especially the optimal old-age labor incometax) as consumption taxes reduce the childrearing costs of time, bequests and transfers.
We further compute the sensitivity of the optimal allocation and optimal policies todifferent degrees of the health externalities when public health subsidies are financed byincome taxes. Internalizing a stronger health externality increases labor productivity or the time
cost of childrearing, thus reducing fertility and consumption over output per young worker atyoung and old ages and raising health spending at young and old ages, longevity and savings.
Hence, stronger health externalities increase optimal health subsidies and income taxes. Whenpublic health is fully replaced by the universal EHI in a counterfactual experiment,internalizing a stronger health externality increases the optimal young-age and old-age EHI
coverage, optimal old-age labor income tax and optimal capital income tax but reduces the
optimal young-age labor income tax. That is, the intergenerational gap of income-tax rates
increases with the degree of health externalities. The age-specific labor income taxes in favorof young workers under such health reform are relevant features for policy design.
The rest of this study proceeds as follows. Section 2 reviews related studies. Section 3
introduces the model. Section 4 establishes the socially optimal allocation. Section 5
characterizes equilibrium allocations and derives the socially optimal EHI, public subsidies
and taxes. Section 6 characterizes equilibrium allocations and derives the socially optimal EHI,
public subsidies and taxes in steady state with specific functions. Section 7 explores
quantitative implications in steady state with specific functions. Section 8 concludes the study.
The Appendix contains all proofs.
2. Related literatureThis paper is related to several strands of the literature. The first strand is theoretical studies on
health improvements and macroeconomic outcomes. Public health enhances longevity and isbeneficial for economic development (Chakraborty, 2004; Aisa and Pueyo, 2006; Bhattacharya
and Qiao, 2007; and Tang and Zhang, 2007). A decline in fertility can increase healthinvestment and effective labor supply (Prettner et al., 2013).
The second strand theoretically analyzes the effects of health externalities, the welfare
effect of health policies and optimal health policies. In Davies and Kuhn (1992) and Philipson
and Becker (1998), longevity externalities in pension annuity returns result in overspending on
health for longer life and over-savings. In Yew and Zhang (2018) with two-sided altruism and4

endogenous fertility, longevity externalities in annuity returns also yield overspending on
health but under-savings and excessive fertility. Fang and Gavazza (2011) show that the EHI
may lead to under-investments in workers’ health due to labor turnover. In Zhang et al. (2006),
public health subsidies financed by labor income taxes enhance longevity and welfare but
reduce savings and future output. Expanding public healthcare beyond the growth-maximizing
level can be Pareto superior in an R&D-based growth model (Kuhn and Prettner, 2016). Usinglabor income taxes to fund public pension and health subsidies, optimal health subsidies depend
negatively on public pensions (Pestieau et al., 2008).
The third strand focuses on the quantitative implication of public or employer-based healthinsurance. In Kelly (2020), both public and private health insurance systems improve thewelfare of young households at the expense of old households. In Frankovic and Kuhn (2023),
the welfare gain of public health insurance in enhancing life expectancy exceeds the welfare
loss in increasing health spending. Studies on health insurance against individual health shocks
have explored its effects on wage, labor, welfare and income distribution (e.g., Jeske and Kitao,
2009; Feng and Zhao, 2018; Feng and Villamil, 2022; Jung and Tran, 2022; Chen et al., 2022).
For example, in Jung and Tran (2022), a consumption-tax financed expansion of public healthinsurance leads to smaller distortions compared to an income-tax financed expansion; and amix of public and private health insurance results in larger welfare gains. Chen et al. (2022)
find that universal public health reduces the gaps in health and life expectancy among workers
with different skills and is beneficial to low-skilled workers.
However, these studies abstract from health externalities to labor productivity and often
ignore two-sided intergenerational transfers, endogenous fertility or endogenous old-age labor.
This paper explores the effects of health externalities to labor productivity and theirimplications for optimal policies, both analytically and quantitatively, in a model with twosided intergenerational transfers, endogenous fertility and endogenous old-age labor.

3. The model
The model has an infinite number of periods. Each period has three overlapping generations:children, young parents and old parents. Children make no economic decision. The length ofyoung parenthood equals one and that of old parenthood, 𝑇(ℎ𝑡−1 , 𝑚𝑡 ): ℝ2+ → (0,1), increases
with health spending at young age ℎ𝑡−1 and old age 𝑚𝑡 at diminishing rates.
3.1. Households

5The wellbeing of a dynastic family increases with the consumption of the young parent 𝑐𝑡 , theconsumption of the old parent adjusted for old-age longevity 𝑑𝑡 𝑇(ℎ𝑡−1 , 𝑚𝑡 ) and the young
parent’s number of children 𝑛𝑡 at diminishing rates:𝑡(1) ∑∞
𝑡=0 𝛼 {𝛽𝑈(𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑑𝑡 ) + 𝛼[𝑈(𝑐𝑡 ) + 𝜌𝐺(𝑛𝑡 )]},

where 𝛼 ∈ (0,1) is the subjective discount factor, 𝛽 ∈ (0,1) is the taste for old-age
consumption and 𝜌 > 0 is the taste for the number of children. Raising a child needs 𝑣 ∈ (0,1)
fixed units of a young parent’s time that set an upper bound on fertility, 𝑛𝑡 ≤ 1/𝑣. Given thewage rate 𝑤𝑡 , each young parent allocates one unit of time endowment to rearing children 𝑣𝑛𝑡
and working 1 − 𝑣𝑛𝑡 , receives a bequest from the old parent 𝑏𝑡 > 0, or gives a gift to the oldparent 𝑏𝑡 < 0. The young parent allocates resources to young-age consumption 𝑐𝑡 , healthspending ℎ𝑡 and savings 𝑠𝑡 as follows:(2) 𝑐𝑡 = 𝑏𝑡 + (1 − 𝑣𝑛𝑡 )𝑤𝑡 − 𝑠𝑡 − ℎ𝑡 .
Given the gross interest rate (rental price of capital) 𝑅𝑡 , an old parent spends the wage
income 𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑤𝑡 and capital income 𝑅𝑡 𝑠𝑡−1 on old-age consumption and health,adjusted for old-age longevity, (𝑑𝑡 + 𝑚𝑡 )𝑇(ℎ𝑡−1 , 𝑚𝑡 ), respectively, and leaves a bequest to or
receives a gift from each child. Thus, the budget constraint of an old parent is(3) 𝑑𝑡 𝑇(ℎ𝑡−1 , 𝑚𝑡 ) = 𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑤𝑡 + 𝑅𝑡 𝑠𝑡−1 − 𝑚𝑡 𝑇(ℎ𝑡−1 , 𝑚𝑡 ) − 𝑏𝑡 𝑛𝑡−1.
This model abstracts from retirement at old age for simplicity as scaling down old-age laborfrom longevity for retirement will not change the essence of the results.
3.2. Production
The production for a final good per young worker increases with capital per young worker 𝑘𝑡 ,̅ (ℎ̅𝑡 , 𝑚effective labor per young worker Ω̅ 𝑡 )𝑙𝑡 , where 𝑙𝑡 is the total labor time per young̅ (ℎ̅𝑡 , 𝑚worker6 and Ω̅ 𝑡 ) is the average health of the labor force as a labor-augmenting factor: 7̅ (ℎ̅𝑡 , 𝑚(4) 𝑦𝑡 = 𝑓(𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ),̅ (ℎ̅𝑡 , 𝑚with constant returns to scale in (𝑘𝑡 , 𝑙𝑡 ) but increasing returns to scale in (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ). In
line with the aforementioned empirical evidence, the average health status of the labor force
has positive external effects on average labor productivity and depends positively on the
average health spending by young and old workers, ℎ̅𝑡 and 𝑚
̅ 𝑡 , at diminishing rates.

6

Total labor time per young worker is the total labor of young and old workers over the number of young workers.
Results remain the same if average health of labor is modeled as a separate production factor, 𝑦𝑡 =̅ (ℎ̅𝑡 , 𝑚𝑓(𝑘𝑡 , Ω̅ 𝑡 ), 𝑙𝑡 ). Equation (4) agrees with the notion that health is a form of general human capital (e.g.,
Becker, 1964; Fang and Gavazza, 2011).
7

6

Since the health status of an individual worker is private information, firms cannot directly
link the wage rate to the worker’s health status or health investment, despite the contributionof the worker’s health to the average health status of the labor force. This leads to healthexternalities to labor productivity. Consequently, workers overlook the contributions of theirhealth investment to the average health of the labor force, and thus their perceived private
returns to health spending are lower than the social return.
Competitive firms compensate production factors by their marginal products as follows:̅ (ℎ̅𝑡 , 𝑚̅ (ℎ̅𝑡 , 𝑚̅ (ℎ̅𝑡 , 𝑚(5) 𝑤𝑡 = 𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) = 𝑓2 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 )Ω
̅ 𝑡 ),̅ (ℎ̅𝑡 , 𝑚(6) 𝑅𝑡 = 𝑓𝑘 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ).
Hence, the marginal product of capital increases with the average health status of workers. By
contrast, the average health status has opposing forces on the marginal product of labor. For a
Cobb-Douglas production function, the net effect of the average health status on the marginal
product of labor is positive.
As one period here corresponds to 30 years, we assume that physical capital depreciates
fully within one period. The size of the young generation in the economy 𝑁𝑡 evolves via
𝑁𝑡+1 = 𝑁𝑡 𝑛𝑡 . Hence, markets clear when𝑘𝑡+1 = 𝑠𝑡 /𝑛𝑡 ,𝑙𝑡 = 1 − 𝑣𝑛𝑡 + 𝑇(ℎ𝑡−1 , 𝑚𝑡 )/𝑛𝑡−1 .
The labor supply of all young workers (1 − 𝑣𝑛𝑡 )𝑛𝑡−1 in a dynastic family at time 𝑡 decreases
with fertility 𝑛𝑡 but increases with the number of young workers 𝑛𝑡−1. The labor supply of theold worker 𝑇(ℎ𝑡−1 , 𝑚𝑡 ) increases with lifetime health investment. Feasibility for the allocationof output is(𝑑 +𝑚 )𝑇(ℎ,𝑚 )
̅ (ℎ̅𝑡 , 𝑚(7) 𝑐𝑡 = 𝑓(𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) − 𝑘𝑡+1 𝑛𝑡 − ℎ𝑡 − 𝑡 𝑡𝑛 𝑡−1 𝑡 .
𝑡−1

To gauge the efficiency loss of the health externalities and to explore optimal policies to attainthe social optimum, the next section describes the socially optimal allocation.

4. The socially optimal allocation
Given an initial state (ℎ−1 , 𝑘0 , 𝑛−1 ), the social planner chooses {𝑑𝑡 , 𝑚𝑡 , 𝑐𝑡 , 𝑛𝑡 , 𝑘𝑡+1 , ℎ𝑡 }∞𝑡=0 tomaximize utility in (1) subject to feasibility in (7) by internalizing health externalities ℎ𝑡 = ℎ̅𝑡and 𝑚𝑡 = 𝑚
̅ 𝑡 as follows:
𝑉(𝑘𝑡 , ℎ𝑡−1 , 𝑛𝑡−1 ) =max{𝛽𝑈(𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑑𝑡 ) +{𝑑𝑡 ,𝑚𝑡 ,𝑛𝑡 ,𝑘𝑡+1 ,ℎ𝑡 }̅ (ℎ̅𝑡 , 𝑚𝛼𝑈(𝑓(𝑘𝑡 , Ω̅ 𝑡 )(1 − 𝑣𝑛𝑡 + 𝑇(ℎ𝑡−1 , 𝑚𝑡 )⁄𝑛𝑡−1 )) − 𝑘𝑡+1 𝑛𝑡 − ℎ𝑡 −7

(𝑑𝑡 + 𝑚𝑡 )𝑇(ℎ𝑡−1 , 𝑚𝑡 )⁄𝑛𝑡−1 ) + 𝛼𝜌𝐺(𝑛𝑡 ) + 𝛼 𝑉(𝑘𝑡+1 , ℎ𝑡 , 𝑛𝑡 )}.
The planner chooses inter-vivos transfers to equate the marginal rate of substitution
between the old and young agents’ consumption with old-age dependency as follows:
𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )1=.
′
𝛼𝑈 (𝑐𝑡 )𝑛𝑡−1(8)Higher health spending in the previous period ℎ𝑡−1 or in the current period 𝑚𝑡 contributes tohigher old-age labor and longevity, thus motivating transfers from old to young agents that
decrease old agents’ consumption but increase young agents’ consumption. Higher fertility inthe previous period results in lower old-age dependency, thus generating more transfers fromyoung to old agents.
The planner chooses capital investment or bequests to children to equate the marginal rateof substitution between young-age consumption across generations with the marginal productof capital per young worker in the next period:̅ (ℎ̅𝑡+1 ,𝑚
̅ 𝑡+1 )𝑙𝑡+1 )𝑈 ′ (𝑐𝑡 )𝑓𝑘 (𝑘𝑡+1 ,Ω
=.
′𝛼𝑈 (𝑐𝑡+1 )𝑛𝑡(9)

The average health of young and old agents in the next period contributes to the marginal
product of capital, thus creating a positive substitution effect on savings or investment in capital.
By equalizing the marginal benefit and cost of young-age health spending, the plannerinternalizes young-age health externalities to labor productivity as follows:(10)𝑇ℎ (ℎ𝑡 ,𝑚𝑡+1 )[𝛽𝑈 ′ (𝑇(ℎ𝑡 ,𝑚𝑡+1 )𝑑𝑡+1 )𝑑𝑡+1 +̅ 𝑡+1 ,𝑚̅ (ℎ̅̅̅𝑡+1 )𝑙𝑡+1 )−𝑑𝑡+1 −𝑚𝑡+1 )
𝛼𝑈′ (𝑐𝑡+1 )(𝑓𝑙 (𝑘𝑡+1 ,Ω
]
𝑛𝑡

̅ (ℎ
̅𝑡 ,𝑚
̅ 𝑡 )𝑙𝑡 )]𝑈 ′ (𝑐𝑡 )[1−𝑓ℎ̅ (𝑘𝑡 ,Ω= 1.

The marginal cost of young-age health spending is the forgone marginal utility of young-age
consumption. The marginal benefit is twofold in the next period: the marginal utility of oldage consumption from extended life and the marginal utility of children’s young-age
consumption from the increase in bequests to children owing to the extended working life at̅ (ℎ̅𝑡 , 𝑚old age. The marginal product of average young-age health spending 𝑓ℎ̅ (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 )contributes to the internalized marginal benefit of young-age health spending.
By equalizing the marginal gain and loss of old-age health spending, the plannerinternalizes old-age health externalities to labor productivity as follows:(11)

𝑇𝑚 (ℎ𝑡−1 ,𝑚𝑡 )[𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )𝑑𝑡 +̅ 𝑡 ,𝑚̅ (ℎ̅̅̅𝑡 )𝑙𝑡 )−𝑑𝑡−𝑚𝑡 )𝛼𝑈′ (𝑐𝑡 )(𝑓𝑙 (𝑘𝑡 ,Ω
]
𝑛𝑡−1

̅ (ℎ
̅𝑡 ,𝑚
̅ 𝑡 )𝑙𝑡 )]𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )[𝑇(ℎ𝑡−1 ,𝑚𝑡 )−𝑛𝑡−1 𝑓𝑚̅̅̅ (𝑘𝑡 ,Ω

= 1.

The marginal cost of old-age health spending is the forgone marginal utility of old-age
consumption. The marginal benefit of old-age health spending includes the marginal utility of
old-age consumption from extended life and the marginal utility of children’s young-age

8

consumption from the increase in inter-vivos transfers from parents to children owing to theextended working life at old age. The marginal product of average old-age health spending̅ (ℎ̅𝑡 , 𝑚
𝑓𝑚̅ (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) contributes to the internalized marginal benefit of old-age health spending.
The planner also equalizes the marginal rate of substitution between fertility and youngage consumption and their relative costs as follows:(12)

𝜌𝐺 ′ (𝑛𝑡 )̅ (ℎ̅𝑡 , 𝑚= 𝑣𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) + 𝑘𝑡+1 +𝑈 ′ (𝑐𝑡 )̅ (ℎ̅𝑡+1 ,𝑚
̅ 𝑡+1 )𝑙𝑡+1 )−𝑑𝑡+1 −𝑚𝑡+1 )𝛼𝑈 ′ (𝑐𝑡+1 )𝑇(ℎ𝑡 ,𝑚𝑡+1 )(𝑓𝑙 (𝑘𝑡+1 ,Ω
.
𝑈 ′ (𝑐𝑡 )𝑛𝑡2

The first term of the relative cost of fertility is the time cost of childrearing; the second term iscapital per child from parental investment (bequests); and the last term is the discounted
marginal cost of inter-vivos transfers from parents to children in the next period. Particularly,
old-age labor relates positively with lifetime health investment and interacts with capital
accumulation, inter-vivos transfers and fertility in contrast to standard overlapping-generations
models that assume retirement at old age without intergenerational transfers and old-age labor.
The transversality condition islim 𝛼 𝑡 𝑈′(𝑐𝑡 )𝑛𝑡 𝑘𝑡+1 = 0.

𝑡→∞̅ (ℎ̅𝑡 , 𝑚
Denoting 𝑓𝑘 (𝑡) ≡ 𝑓𝑘 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) and combining this transversality condition withsuccessive substitutions on 𝛼𝑢′(𝑐𝑡 ) = 𝑢′(𝑐𝑡−1 )𝑛𝑡−1 ⁄𝑓𝑘 (𝑡) from condition (9), we have
lim 𝑘𝑡+1 ∏𝑡𝑗=0

𝑡→∞

𝑛𝑗

𝑓𝑘 (𝑗)= 0.

From this condition, the marginal product of capital should exceed the growth rate of aggregate
capital for dynamic efficiency when time approaches infinity. This condition ensures a boundedvalue function of the state 𝑉(𝑘𝑡 , ℎ𝑡−1 , 𝑛𝑡−1 ).
The socially optimal allocation from an initial state ( ℎ−1 , 𝑛−1 , 𝑘0 ) is a sequence{𝑐𝑡 , 𝑑𝑡 , ℎ𝑡 , 𝑚𝑡 , 𝑛𝑡 , 𝑘𝑡+1 }∞𝑡=0 that satisfies technology (4), feasibility (7), first-order conditions (8)to (12) and the transversality condition. We now turn to the competitive equilibrium allocations.

5. Competitive equilibrium allocations
This section first determines the competitive equilibrium allocation without private healthsubsidies and government interventions for comparisons with the socially optimal allocation toreveal the effects of the health externalities. It then determines the competitive equilibrium
allocation with private and public health subsidies, investment subsidies as well as taxes on
income and consumption to explore their roles in internalizing the health externalities.
5.1. Laissez faire9

From budget constraints (2) and (3), the dynasty faces a constraint in each period as follows:(13)

𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑅𝑠

𝑐𝑡 = 𝑛𝑡 𝑡−1 + (
𝑡−1

𝑛𝑡−1

+ 1 − 𝑣𝑛𝑡 ) 𝑤𝑡 − 𝑠𝑡 − ℎ𝑡 −(𝑑𝑡 +𝑚𝑡 )𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑛𝑡−1.

The dynasty maximizes utility in (1) subject to (13), taking prices as given. The respective
intergenerational and intertemporal substitution conditions are given as follows:(14)

𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )1=𝑛 ,𝛼𝑈 ′ (𝑐𝑡 )
𝑡−1

(15)

𝑈 ′ (𝑐𝑡 )𝑅= 𝑛𝑡+1.
𝛼𝑈 ′ (𝑐𝑡+1 )𝑡̅ (ℎ̅𝑡+1 , 𝑚
From 𝑅𝑡+1 = 𝑓𝑘 (𝑘𝑡+1 , Ω̅ 𝑡+1 )𝑙𝑡+1 ) in (6), conditions (14) and (15) are analogous to (8)and (9) chosen by the social planner. Yet, the health externalities affect the intertemporal̅ (ℎ̅𝑡+1 , 𝑚
substitution via the investment return 𝑅𝑡+1 = 𝑓𝑘 (𝑘𝑡+1 , Ω̅ 𝑡+1 )𝑙𝑡+1 ) as agents disregard

the contribution of their health to average health and labor productivity in the economy.
The first-order conditions with respect to health spending ℎ𝑡 and 𝑚𝑡 are
(16)
(17)

𝑇ℎ (ℎ𝑡 ,𝑚𝑡+1 )[𝛽𝑈 ′ (𝑇(ℎ𝑡 ,𝑚𝑡+1 )𝑑𝑡+1 )𝑑𝑡+1 +

𝛼𝑈′ (𝑐𝑡+1 )(𝑤𝑡+1 −𝑑𝑡+1 −𝑚𝑡+1 )]𝑛𝑡

𝑈 ′ (𝑐𝑡 )𝑇𝑚 (ℎ𝑡−1 ,𝑚𝑡 )[𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )𝑑𝑡 +𝛼𝑈′ (𝑐𝑡 )𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑛𝑡−1

𝛼𝑈′ (𝑐𝑡 )(𝑤𝑡 −𝑑𝑡 −𝑚𝑡 )
]
𝑛𝑡−1

= 1,

= 1.

Since the perceived private returns to health spending are below the social returns, private
health spending in (16) and (17) is below the optimal levels in (10) and (11).
The first-order condition with respect to the number of children is(18)

𝜌𝐺′ (𝑛𝑡 )𝑈 ′ (𝑐𝑡 )𝑠= 𝑣𝑤𝑡 + 𝑛𝑡 +𝑡

𝛼𝑈 ′ (𝑐𝑡+1 )𝑇(ℎ𝑡 ,𝑚𝑡+1 )(𝑤𝑡+1 −𝑑𝑡+1 −𝑚𝑡+1 ),𝑈 ′ (𝑐𝑡 )𝑛𝑡2

̅ (ℎ̅𝑡 , 𝑚where 𝑤𝑡 = 𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) in (5). The health externalities result in lower health spending,lower labor productivity, and hence a lower wage rate or time cost of childrearing 𝑣𝑤𝑡 in (18)than their socially optimal levels. Lower labor productivity implies lower returns to capital,
thus leading to lower savings in (15). The decline in savings in turn reduces the bequest costof childrearing 𝑠𝑡 ⁄𝑛𝑡 in (18). Lower health spending results in lower old-age earnings andparental transfers to children 𝑇(ℎ𝑡 , 𝑚𝑡+1 )(𝑤𝑡+1 − 𝑑𝑡+1 − 𝑚𝑡+1 ) which reduce the transfer costof childrearing in (18). Thus, fertility is higher and young-age labor is lower than the socially
optimal level in (12). Thus, output per young worker is lower than the socially optimal level.
Since 𝑠𝑡 = 𝑛𝑡 𝑘𝑡+1 , the transversality condition associated with assets is the same as thatfor the social planner’s allocation. Combining the transversality condition with successive
substitutions on 𝛼𝑢′(𝑐𝑡 ) = 𝑢′(𝑐𝑡−1 )𝑛𝑡−1 ⁄𝑅𝑡 from condition (15) yields binding solvency𝑛lim 𝑘𝑡+1 ∏𝑡𝑗=0 𝑗 = 0.

𝑡→∞

𝑅𝑗

10

From the binding solvency, the gross return to savings should exceed the growth rate ofaggregate capital for dynamic efficiency, thus ensuring a bounded value function of the state
when time approaches infinity.8
Definition 1. A competitive equilibrium from an initial state (ℎ−1 , 𝑘0 , 𝑛−1 ) is a sequence of∞allocations {𝑏𝑡 , 𝑐𝑡 , 𝑑𝑡 , ℎ𝑡 , 𝑚𝑡 , 𝑛𝑡 , 𝑠𝑡 , 𝑙𝑡 , 𝑘𝑡+1 , 𝑦𝑡 }∞𝑡=0 and prices {𝑅𝑡 , 𝑤𝑡 }𝑡=0 such that: (i) given

average health and prices, firms and families optimize, satisfying budget constraints (2) and(3), technology (4), conditions (5), (6) and (14) to (18), the transversality condition, andbinding solvency; (ii) all markets clear; (iii) consistency holds: ℎ𝑡 = ℎ̅𝑡 and 𝑚𝑡 = 𝑚
̅ 𝑡.
The laissez-faire allocation features low health spending, labor productivity, longevity,
savings and income but high fertility as observed in developing countries.
5.2. Employer-based health insurance and public policies
This subsection considers private health subsidies through employer-based health insurance(EHI) and a set of taxes and subsidies to explore their roles in internalizing the healthexternalities to achieve the social optimum.
5.2.1. Employer-based health insurance
The EHI has the advantage of group insurance over individual health insurance to ease theconcern of private information on an individual’s health status or health spending.9 We explore
the EHI chosen by firms for profit maximization and its role in mitigating the efficiency lossof the health externalities. Let 𝜋𝑡 ∈ (0,1) and 𝜋𝑡𝑜 ∈ (0,1) denote the portions of young-age andold-age health spending covered by the EHI, respectively. In equilibrium with a competitive
insurance market, health insurance premium per young or old worker 𝐼𝑡 or 𝐼𝑡𝑜 equals theinsurance coverage of health spending per young worker, 𝐼𝑡 = ℎ̅𝑡 𝜋𝑡 , or per old worker, 𝐼𝑡𝑜 =𝑇̅(ℎ̅𝑡−1 , 𝑚
̅ 𝑡 )𝑚̅ 𝑡 𝜋𝑡𝑜 .
With the EHI, we rewrite the production function as̅𝑡
𝐼 𝑜 +𝑀̅ (𝐼𝑡 + 𝐻
̅𝑡 , 𝑡(19) 𝑦𝑡 = 𝑓 (𝑘𝑡 , Ω
̅
𝑇̅ (ℎ

̅ 𝑡)
𝑡−1 ,𝑚)𝑙𝑡 ),

̅𝑡 ≡ ℎ̅𝑡 (1 − 𝜋𝑡 ) is average out-of-pocket health spending per young worker, and 𝑀
̅𝑡 ≡where 𝐻𝑇̅(ℎ̅𝑡−1 , 𝑚
̅ 𝑡 )𝑚̅ 𝑡 (1 − 𝜋𝑡𝑜 ) is average out-of-pocket health spending per old worker. Firms

8

Otherwise, the marginal product of capital would be too low to compensate for capital depreciation, thus causing
dynamic inefficiency, a scenario coined as over-savings, which can be ruled out by binding solvency in thedynastic model.
9
Based on this advantage of group insurance, we focus on the EHI and abstract from individual health insurance.
In practice, the EHI is much more popular than individual health insurance. As indicated by data from the US
Bureau of Statistics, the EHI covers almost 85% of the population with private health insurance.

11

observe health insurance premium (𝐼𝑡 , 𝐼𝑡𝑜 ) but do not observe individual health spending (ℎ𝑡 ,𝑚𝑡 ) and old-age longevity 𝑇(ℎ𝑡−1 , 𝑚𝑡 ).
Firms’ profit function becomes̅𝑡
𝐼 𝑜 +𝑀̅ (𝐼𝑡 + 𝐻
̅ 𝑡 , ̅𝑡𝑓 (𝑘 𝑡 , Ω
𝑇̅ (ℎ

̅ 𝑡)
𝑡−1 ,𝑚) 𝑙𝑡 ) − 𝑅𝑡 𝑘𝑡 − 𝑤𝑡 𝑙𝑡 − 𝜆𝑡 𝐼𝑡 − 𝜆𝑜𝑡 𝐼𝑡𝑜 /𝑛𝑡−1 ,where 𝜆𝑡 ∈ (0,1) and 𝜆𝑜𝑡 ∈ (0,1) are the firms’ subsidy rates on young-age insurance 𝐼𝑡 andold-age insurance per young worker 𝐼𝑡𝑜 /𝑛𝑡−1 , respectively. Profit maximization yields
̅𝑡
𝐼 𝑜 +𝑀̅ (𝐼𝑡 + 𝐻
̅𝑡 , 𝑡(20) 𝑤𝑡 = 𝑓𝑙 (𝑘𝑡 , Ω
̅
𝑇̅ (ℎ

̅ 𝑡)
𝑡−1 ,𝑚̅ (ℎ̅𝑡 , 𝑚) 𝑙𝑡 ) = 𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ),̅𝑡
𝐼 𝑜 +𝑀̅ (𝐼𝑡 + 𝐻
̅𝑡 , 𝑡(21) 𝑅𝑡 = 𝑓𝑘 (𝑘𝑡 , Ω
̅
𝑇̅ (ℎ

̅ 𝑡)
𝑡−1 ,𝑚̅ (ℎ̅𝑡 , 𝑚) 𝑙𝑡 ) = 𝑓𝑘 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ),𝑜̅̅ (𝐼𝑡 + 𝐻
̅𝑡 , 𝐼𝑡 +𝑀𝑡 ) 𝑙𝑡 ) = 𝑓ℎ̅ (𝑘𝑡 , Ω̅ (ℎ̅𝑡 , 𝑚(22) 𝜆𝑡 = 𝑓𝐼 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ),
̅
̅ )
𝑇̅ (ℎ
,𝑚
𝑡−1

𝜆𝑜
𝑡

(23) 𝑛𝑡

̅𝑡
𝐼𝑡𝑜 +𝑀𝑡−1

̅ (𝐼𝑡 + 𝐻
̅𝑡 ,= 𝑓𝐼𝑜 (𝑘𝑡 , Ω
̅
𝑇̅ (ℎ

𝑡−1

)𝑙 ) =
̅ ) 𝑡
,𝑚
𝑡

̅ (ℎ
̅𝑡 ,𝑚
̅ 𝑡 )𝑙𝑡 )𝑓𝑚
̅̅̅ (𝑘𝑡 ,Ω.
̅𝑡−1 ,𝑚
̅ 𝑡)
𝑇̅ (ℎIn (20) and (21), the marginal products or rental prices depend on the EHI. In (22) and (23),firms’ subsidies on health insurance for young and old workers depend positively on themarginal product of average health spending by young and old workers, respectively.
5.2.2. Public policies
Government spending includes lump-sum transfers to the young and elderly, 𝑃𝑡 and 𝑃𝑡𝑜 ,respectively, and subsidies on savings at rate 𝜉𝑡𝑠 and on health spending at young and old age
at respective rates 𝜉𝑡 and 𝜉𝑡𝑜 . Government revenue is from taxes on young-age and old-age
labor income at respective rates 𝜏𝑡 and 𝜏𝑡𝑜 , on consumption at rate 𝜏𝑡𝑐 , and on capital income
𝑟𝑡 = 𝑅𝑡 − 1 at rate 𝜏𝑡𝑠 , where 𝑟𝑡 > 0 under the binding solvency for dynamic efficiency. The
government balances its budget in every period:
̅𝑡−1 ,𝑚
̅ 𝑡 )𝑚̅ 𝑡 +𝑃̅𝑡𝑜𝜉 𝑜 𝑇̅ (ℎ𝜉𝑡 ℎ̅𝑡 + 𝜉𝑡𝑠 𝑠̅𝑡 + 𝑃̅𝑡 + 𝑡
𝜏𝑡𝑐 𝑐̅𝑡 +𝑛̅𝑡−1= 𝜏𝑡 (1 − 𝑣𝑛̅𝑡 )𝑤𝑡 +̅𝑡−1 ,𝑚(𝜏𝑡𝑜 𝑤𝑡 +𝜏𝑡𝑐 𝑑̅𝑡 )𝑇̅ (ℎ̅ 𝑡 )+𝜏𝑡𝑠 𝑠̅𝑡−1 (𝑅𝑡 −1)

.

𝑛̅𝑡−1

5.2.3. Equilibrium allocations with EHI and public policies
With the EHI and public policies, household budget constraints become𝑐𝑡 (1 + 𝜏𝑡𝑐 ) = 𝑏𝑡 + (1 − 𝑣𝑛𝑡 )𝑤𝑡 (1 − 𝜏𝑡 ) + 𝑃𝑡 − 𝑠𝑡 (1 − 𝜉𝑡𝑠 ) −ℎ𝑡 (1 − 𝜋𝑡 − 𝜉𝑡 ) − 𝐼𝑡 (1 − 𝜆𝑡 ),𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑑𝑡 (1 + 𝜏𝑡𝑐 ) = 𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑤𝑡 (1 − 𝜏𝑡𝑜 ) + [𝑅𝑡 − (𝑅𝑡 − 1)𝜏𝑡𝑠 ]𝑠𝑡−1 + 𝑃𝑡𝑜 −𝑇(ℎ𝑡−1 , 𝑚𝑡 )𝑚𝑡 (1 − 𝜋𝑡𝑜 − 𝜉𝑡𝑜 ) − 𝐼𝑡𝑜 (1 − 𝜆𝑜𝑡 ) − 𝑏𝑡 𝑛𝑡−1 .
Combining these budget constraints into a single constraint for the dynasty yields
12

(24)

𝑐𝑡 =

(1−𝑣𝑛𝑡 )𝑤𝑡 (1−𝜏𝑡 )+𝑃𝑡 −𝑠𝑡 (1−𝜉𝑡𝑠 )−ℎ𝑡 (1−𝜋𝑡 −𝜉𝑡 )−𝐼𝑡 (1−𝜆𝑡 )(1+𝜏𝑡𝑐 )+𝑇(ℎ𝑡−1 ,𝑚𝑡 )[𝑤𝑡 (1−𝜏𝑡𝑜 )−𝑚𝑡 (1−𝜋𝑡𝑜 −𝜉𝑡𝑜 )−𝑑𝑡 (1+𝜏𝑡𝑐 )]+𝑃𝑡𝑜 +[𝑅𝑡 −(𝑅𝑡 −1)𝜏𝑡𝑠 ]𝑠𝑡−1 −𝐼𝑡𝑜 (1−𝜆𝑜𝑡).

𝑛𝑡−1 (1+𝜏𝑡𝑐 )The dynasty maximizes utility subject to household budget constraints, taking prices, EHI
coverage rates, private health subsidies and public policies as given. The first-order condition
with respect to 𝑑𝑡 is similar to (8) in the social optimum or (14) in the laissez faire; and thosewith respect to 𝑠𝑡 , ℎ𝑡 , 𝑚𝑡 and 𝑛𝑡 are given as follows:
(25)

(26)

𝑠
𝑈 ′ (𝑐𝑡 )(1+𝜏𝑡𝑐 )[𝑅𝑡+1 −(𝑅𝑡+1 −1)𝜏𝑡+1
]
=
,
𝑠
𝑐
′
𝛼𝑈 (𝑐𝑡+1 )𝑛𝑡 (1−𝜉𝑡 )(1+𝜏𝑡+1 )𝑇ℎ (ℎ𝑡 ,𝑚𝑡+1 )𝛽𝑈 ′ (𝑇(ℎ𝑡 ,𝑚𝑡+1 )𝑑𝑡+1 )𝑑𝑡+1
𝑈′ (𝑐𝑡 )(1−𝜋𝑡 −𝜉𝑡 )
(1+𝜏𝑐
𝑡)

𝑇ℎ (ℎ𝑡 ,𝑚𝑡+1 )[

+

𝑐
𝑜
𝑜
𝛼𝑈′ (𝑐𝑡+1 )(𝑤𝑡+1 (1−𝜏𝑜𝑡+1 )−𝑑𝑡+1 (1+𝜏𝑡+1 )−𝑚𝑡+1 (1−𝜋𝑡+1 −𝜉𝑡+1 ))]𝑛𝑡 (1+𝜏𝑐)
𝑡+1
𝑈′ (𝑐𝑡 )(1−𝜋𝑡 −𝜉𝑡 )
(1+𝜏𝑐
𝑡)

= 1,

𝑐
𝑜 𝑜
𝛼𝑈′ (𝑐𝑡 )(𝑤𝑡 (1−𝜏𝑜𝑡 )−𝑑𝑡(1+𝜏𝑡 )−𝑚𝑡 (1−𝜋𝑡 −𝜉𝑡 ))]𝑐𝑛𝑡−1 (1+𝜏𝑡 )𝑜𝛼𝑈′ (𝑐𝑡 )𝑇(ℎ𝑡−1 ,𝑚𝑡 )(1−𝜋𝑜𝑡 −𝜉𝑡 )
𝑛𝑡−1 (1+𝜏𝑐𝑡)

𝑇𝑚 (ℎ𝑡−1 ,𝑚𝑡 )[𝛽𝑈 ′ (𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑑𝑡 )𝑑𝑡 +

(27)

(28)

𝜌𝐺′ (𝑛𝑡 )=𝑈 ′ (𝑐 )𝑡𝑣𝑤𝑡 (1−𝜏𝑡 )(1+𝜏𝑡𝑐 )𝑠 (1−𝜉 𝑠 ))𝑃 𝑜𝛼𝑈 ′ (𝑐= 1,)𝐼 𝑜 𝜆𝑜𝛼𝑈 ′ (𝑐𝑡+1 𝑡+1 𝑡+1
𝑡+1+ 𝑛𝑡 (1+𝜏𝑡𝑐) + 𝑈 ′ (𝑐 )𝑛𝑡+1
2 (1+𝜏 𝑐 ) + 𝑈 ′ (𝑐 )𝑛2 (1+𝜏 𝑐 ) +
𝑡

𝑡

𝑡

𝑡

𝑡+1

𝑡

𝑡

𝑡+1

𝑜 )
𝑐
𝑜 )]
𝛼𝑈 ′ (𝑐𝑡+1 )𝑇(ℎ𝑡 ,𝑚𝑡+1 )[𝑤𝑡+1 (1−𝜏𝑡+1−𝑑𝑡+1 (1+𝜏𝑡+1)−𝑚𝑡+1 (1−𝜉𝑡+1𝑐2
𝑈 ′ (𝑐𝑡 )𝑛𝑡 (1+𝜏𝑡+1 ),̅ (ℎ̅𝑡 , 𝑚̅ (ℎ̅𝑡 , 𝑚where 𝑤𝑡 = 𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) and 𝑅𝑡 = 𝑓𝑘 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ).
In (25), the marginal rate of substitution between young-age consumption acrossgenerations equals the after-tax private return to savings (net of the subsidy) over the number𝑠of children. A rise in capital income tax 𝜏𝑡+1
reduces the relative price of current consumption,

thus generating a negative substitution effect on savings. Conversely, a rise in the savings
subsidy rate 𝜉𝑡𝑠 creates a positive substitution effect on savings. If consumption tax rates are𝑐constant 𝜏𝑡𝑐 = 𝜏𝑡+1
to cancel out their wedge in the intertemporal substitution, then capital𝑠
income tax 𝜏𝑡+1
and savings subsidy 𝜉𝑡𝑠 move in the same direction for optimal savings.

In (26) and (27), public health subsidies on young-age and old-age health spending𝑜𝑜(𝜉𝑡 , 𝜉𝑡𝑜 , 𝜉𝑡+1
) and EHI coverage rates on young-age and old-age health spending (𝜋𝑡 , 𝜋𝑡𝑜 , 𝜋𝑡+1)

increase health spending by reducing the marginal costs of private health spending. Thissuggests that public health subsidies or private health subsidies through the EHI are useful to𝑜internalize the health externalities. Since public health subsidies ( 𝜉𝑡 , 𝜉𝑡𝑜 , 𝜉𝑡+1
) and EHI𝑜coverage rates on health spending (𝜋𝑡 , 𝜋𝑡𝑜 , 𝜋𝑡+1
) are substitutable, there is a key role of the EHI

13

in easing the financial pressure of funding public healthcare. By lowering the after-tax earnings𝑜at old age, old-age labor income taxes (𝜏𝑡𝑜 , 𝜏𝑡+1
) reduce the marginal benefits of health spending,𝑐thus lowering health spending. Consumption taxes (𝜏𝑡𝑐 , 𝜏𝑡+1
) reduce the costs and benefits of𝑐health spending. If consumption tax rates are constant 𝜏𝑡𝑐 = 𝜏𝑡+1
, then their effects on healthspending fully cancel out when substituting (14) into (26) and (27).
In (28), young-age labor income tax 𝜏𝑡 reduces the time cost of childrearing, while old-age𝑜labor income tax 𝜏𝑡+1
reduces after-tax earnings at old age and the transfer cost of childrearing.

Consumption tax 𝜏𝑡𝑐 reduces the time cost, bequest cost and transfer cost of childrearing. A risein savings subsidy 𝜉𝑡𝑠 lowers the bequest cost of childrearing, whereas a rise in public transfer𝑜to the elderly 𝑃𝑡+1
raises the transfer cost by increasing parental transfers to children. Moreover,𝑜𝑜public and private health subsidies for the elderly (𝜉𝑡+1, 𝐼𝑡+1𝜆𝑜𝑡+1 ) decrease old-age health costs

and increase inter-vivos transfers to children (the transfer cost of childrearing). As a result,
without appropriate taxes that generate positive effects on fertility, the EHI tend to reduce
fertility from its socially optimal level in (12), and thus the EHI alone cannot achieve the social
optimum when individuals choose the number of children. Since public and private healthsubsidies are substitutable, appropriate health subsidies along with appropriate taxes canachieve the social optimum as shown by the next subsection.
5.2.4. Optimal EHI, optimal public health subsidies and optimal taxes̅ (ℎ̅𝑡 , 𝑚̅ (ℎ̅𝑡 , 𝑚
For notational ease, we denote 𝑓𝑘 (𝑡) ≡ 𝑓𝑘 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) , 𝑓𝑙 (𝑡) ≡ 𝑓𝑙 (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) ,̅ (ℎ̅𝑡 , 𝑚̅ (ℎ̅𝑡 , 𝑚𝑓ℎ̅ (𝑡) ≡ 𝑓ℎ̅ (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ) and 𝑓𝑚̅ (𝑡) ≡ 𝑓𝑚̅ (𝑘𝑡 , Ω̅ 𝑡 )𝑙𝑡 ). Among the policy instruments
commonly used in developed countries, we treat some instruments as given and use the other
instruments to achieve the optimal outcome. Equating the equilibrium conditions derived in
Subsection 5.2.3 with the optimal conditions of the social planner yields the following optimal
policies that eliminate the efficiency loss of the health externalities and achieve the social
optimum in an economy with both the EHI and public health subsidies.
Proposition 1. Given the socially optimal allocation with ℎ𝑡 = ℎ̅𝑡 and 𝑚𝑡 = 𝑚
̅ 𝑡 and public
policies {𝑃𝑡 , 𝑃𝑡𝑜 , 𝜉𝑡 , 𝜉𝑡𝑜 , 𝜉𝑡𝑠 , 𝜏𝑡𝑐 }∞𝑡=0 , the optimal taxes on capital income and labor income to the
young

and

elderly {𝜏𝑡𝑠 , 𝜏𝑡 , 𝜏𝑡𝑜 }∞𝑡=0 andoptimalemployer-based

healthinsurance{𝜋𝑡 , 𝜋𝑡𝑜 , 𝐼𝑡 , 𝐼𝑡𝑜 , 𝜆𝑡 , 𝜆𝑜𝑡 }∞𝑡=0 from the conditions for utility and profit maximization under feasibility
and a balanced government budget are determined as follows
(i)
(ii)

𝑣𝑓𝑙 (𝑡)(𝜏𝑡 +𝜏𝑡𝑐 )+𝑘𝑡+1 (𝜏𝑡𝑐 +𝜉𝑡𝑠 )(1+𝜏𝑡𝑐 )

=

𝑜
𝑐
𝑜
𝑐
𝑜
𝑜
𝑇(ℎ𝑡 ,𝑚𝑡+1 )[𝑚𝑡+1 (𝜉𝑡+1+𝜏𝑡+1)−𝑓𝑙 (𝑡+1)(𝜏𝑡+1+𝜏𝑡+1)]+𝑃𝑡+1
+𝐼𝑡+1
𝜆𝑜
𝑡+1
,
𝑐
𝑛𝑡 𝑓𝑘 (𝑡+1)(1+𝜏𝑡+1 )[1−(𝜉 +𝜋𝑡 )](1+𝜏𝑐𝑡+1 )[𝑓𝑙 (𝑡+1)−𝑚𝑡+1 ]𝑓̅ℎ (𝑡) = 1 − 1+𝜏𝑐 { 1−𝜏𝑡 𝑜
(

𝑡) (

,

𝑜
𝑜
𝑡+1 )𝑓𝑙 (𝑡+1)−[1−(𝜉𝑡+1 +𝜋𝑡+1 )]𝑚𝑡+1 }14

𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑓𝑙 (𝑡)[(𝜉𝑡𝑜 +𝜋𝑡𝑜 )−𝜏𝑡𝑜 ]𝑜𝑜𝑜 ,𝑡−1 {𝑓𝑙 (𝑡)(1−𝜏𝑡 )−𝑚𝑡 [1−(𝜉𝑡 +𝜋𝑡 )]}(iii)

𝑓𝑚̅ (𝑡) = 𝑛(iv)

𝑓𝑘 (𝑡 + 1) = (𝜏𝑐(v)

𝜉𝑡 ℎ𝑡 + 𝜉𝑡𝑠 𝑘𝑡+1 𝑛𝑡 + 𝑃𝑡 + 𝑡

𝑠
𝜏𝑡+1(1+𝜏𝑡𝑐 )𝑠𝑐𝑠𝑐𝑐
𝑡+1 +𝜏𝑡+1 )(1+𝜏𝑡 )−(𝜉𝑡 +𝜏𝑡 )(1+𝜏𝑡+1 ),𝜉 𝑜 𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝑚𝑡 +𝑃𝑡𝑜
𝑛𝑡−1

𝜏𝑜 𝑇(ℎ𝑡−1 ,𝑚𝑡 )= 𝑓𝑙 (𝑡) [𝜏𝑡 (1 − 𝑣𝑛𝑡 ) + 𝑡

𝑛𝑡−1

]+

𝑑 𝑇(ℎ𝑡−1 ,𝑚𝑡 )𝜏𝑡𝑐 [𝑐𝑡 + 𝑡

𝑛𝑡−1] + 𝜏𝑡𝑠 𝑘𝑡 [𝑓𝑘 (𝑡) − 1],(vi)

𝐼𝑡 = ℎ̅𝑡 𝜋𝑡 ,(vii)

𝐼𝑡𝑜 = 𝑇̅(ℎ̅𝑡−1 , 𝑚
̅ 𝑡 )𝑚̅ 𝑡 𝜋𝑡𝑜 ,(viii) 𝜆𝑡 = 𝑓ℎ̅ (𝑡),(ix)𝑛𝑡−1 𝑓𝑚̅ (𝑡)𝜆𝑜𝑡 = ̅ ̅

̅ 𝑡)
𝑇 (ℎ𝑡−1 ,𝑚.

Proof. See Appendix A.
𝑜
In condition (i), the positive effects of public transfers to the elderly 𝑃𝑡+1
, old-age public𝑜𝑜health subsidy 𝜉𝑡+1
and old-age private health subsidy 𝐼𝑡+1
𝜆𝑜𝑡+1 on the costs of childrearing𝑜counteract the negative effects of young-age and old-age labor income taxes (𝜏𝑡 , 𝜏𝑡+1
) andsavings subsidy 𝜉𝑡𝑠 on the costs of childrearing.
Condition (ii) internalizes the marginal product of average young-age health spending
through young-age public health subsidy 𝜉𝑡 , young-age EHI coverage 𝜋𝑡 , old-age public health𝑜𝑜subsidy 𝜉𝑡+1
and old-age EHI coverage 𝜋𝑡+1
. When the health externalities are stronger, 𝑓̅ℎ (𝑡)is higher, thus requiring a higher public health subsidy or a higher EHI coverage to internalize
the marginal product of average young-age health spending to achieve the social optimum.
Condition (iii) internalizes the marginal product of average old-age health spending throughold-age public health subsidy 𝜉𝑡𝑜 and old-age EHI coverage 𝜋𝑡𝑜 net of old-age labor income𝑐taxes 𝜏𝑡𝑜 . Under a stationary consumption tax rate 𝜏𝑡𝑐 = 𝜏𝑡+1
, conditions (ii) and (iii) suggestthat age-specific health subsidies either from the government or firms are essential tointernalize the health externalities.
When consumption tax rates are constant, condition (iv) implies the same sign for savings𝑠subsidy 𝜉𝑡𝑠 and capital income tax 𝜏𝑡+1
to remove their wedge in intertemporal substitution.

Condition (v) balances the government budget. Conditions (vi) and (vii) show an actuariallyfair EHI policy. Conditions (viii) and (ix) internalize the health externalities by equating private
health subsidies to young workers and old workers (𝜆, 𝜆𝑜 ) with the marginal products of average
young-age health spending and average old-age health spending, respectively.

15

Proposition 1 suggests that with appropriate taxes, the mixture of public and private healthsubsidies can co-exist to achieve the social optimum. This result is relevant to countries likethe US where young workers are primarily covered by the EHI, while old workers are coveredby the public health program, such as the Medicare.
Consistent with our earlier discussion for (28), firms’ subsidies on old-age health insurance𝑜𝐼𝑡+1
𝜆𝑜𝑡+1 increase the transfer cost of childrearing, tending to reduce fertility from its socially

optimal level. As a result, without having appropriate taxes, private health subsidies throughthe EHI alone cannot fully internalize health externalities to labor productivity when
individuals choose the number of children. However, if the EHI is absent, all the conditions in
Proposition 1 can still hold. These results are summarized in Corollary 1, which derives optimal
public policies to eliminate the efficiency loss of health externalities in an economy byintroducing universal public health in the healthcare system without the EHI. This case isrelevant for many OECD countries such as Australia that have only universal public health.
Corollary 1. Given the socially optimal allocation with ℎ𝑡 = ℎ̅𝑡 and 𝑚𝑡 = 𝑚
̅ 𝑡 and public
policies {𝑃𝑡 , 𝑃𝑡𝑜 , 𝜉𝑡𝑠 , 𝜏𝑡𝑐 , }∞𝑡=0 , the government can determine optimal public health subsidies
and optimal taxes on capital and labor income {𝜉𝑡 , 𝜉𝑡𝑜 , 𝜏𝑡𝑠 , 𝜏𝑡 , 𝜏𝑡𝑜 }∞𝑡=0 in the absence of
employer-based health insurance.
When universal public health is absent, we can also derive the optimal EHI and optimal
taxes to eliminate the efficiency loss of health externalities in an economy. This case is relevant
for health reform that replaces the universal public health with the universal EHI to ease thefinancial burden of universal public health.
Corollary 2. Given the socially optimal allocation with ℎ𝑡 = ℎ̅𝑡 and 𝑚𝑡 = 𝑚
̅ 𝑡 and public
policies {𝑃𝑡 , 𝑃𝑡𝑜 , 𝜉𝑡𝑠 , 𝜏𝑡𝑐 , }∞𝑡=0 , the government can determine optimal taxes on capital and labor
income{𝜏𝑡𝑠 , 𝜏𝑡 , 𝜏𝑡𝑜 }∞𝑡=0

tosupportoptimalemployer-based

healthinsurance{𝜋𝑡 , 𝜋𝑡𝑜 , 𝐼𝑡 , 𝐼𝑡𝑜 , 𝜆𝑡 , 𝜆𝑜𝑡 }∞𝑡=0 in the absence of universal public health.
The next section shows qualitative results in the steady state with specific functional forms.

6. Steady state with specific functions
We assume a Cobb-Douglas production function as
(29)

𝜃

𝜙

𝑦 = 𝐴𝑘 [(ℎ 𝑚

1−𝜙

𝜇

1−𝜃

) 𝑙]

,

16

𝜙

1−𝜙 𝜇

with 𝐴 > 0, 0 < 𝜃, 𝜇 < 1 , Ω(ℎ, 𝑚) = (ℎ 𝑚

) , and 𝜙 ∈ (0,1) , where 𝜇 measures thedegree of health externalities to labor productivity and 𝜙 measures the relative role of average
young-age health spending in determining the health externalities.
The utility function has a constant elasticity of intertemporal substitution for consumption
and fertility, measured by 1⁄𝜎1 > 0 and 1⁄𝜎2 > 0, respectively, as follows:(30)

𝑈(𝑥) =

𝑥 1−𝜎1 −1
1−𝜎1, 𝐺(𝑛) =𝑛1−𝜎2 −1,1−𝜎2where 𝑥 = 𝑐, 𝑇(ℎ, 𝑚)𝑑. The longevity function is:(31)𝜓𝑧(ℎ,𝑚)𝑇(ℎ, 𝑚) = 𝐷 (𝛿+𝜖𝑧(ℎ,𝑚)) ,which increases with health spending at young and old age at diminishing rates underrestrictions 𝐷 ∈ (0,1], 𝛿 > 0, 𝜖 ≥ 1, 𝜓 ∈ (0,1), and 𝑧(ℎ, 𝑚) = ℎ𝜙 𝑚1−𝜙 . Appendix B gives
the socially optimal allocation at the steady state.
The next subsection presents qualitative results for the optimal EHI and optimal public
policies using the specific functions in the steady state.
6.1. Optimal EHI, optimal public health subsidies and optimal taxes
We define the following expressions for optimal policies:
Λ1 ≡
Λ2 ≡

𝑇(ℎ,𝑚)𝑑𝑛+ 𝑐 + (1 − 𝑣𝑛) [(1−𝑣𝑛)𝛼𝑚𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)𝑣𝑛

Λ 3 ≡ 𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) [+𝑇(ℎ,𝑚)𝛼(𝑚−𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙))𝑣𝑛2𝑘− 𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − 𝑣 ],𝑚𝑇(ℎ,𝑚)[𝑣𝑛−𝛼(1−𝑣𝑛)],𝑣𝑛2

𝑇(ℎ,𝑚)(𝑣𝑛−𝛼(1−𝑣𝑛))𝑣𝑛2+𝛼𝑚(1−𝑣𝑛)𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)𝑇𝑚 (ℎ,𝑚)].
𝑣𝑛(𝑇𝑚 (ℎ,𝑚)𝑚+𝑇)

The optimal public health subsidies and optimal EHI supported by optimal taxes with thespecific functions in the steady state are given as follows:
Proposition 2. Given the social planner’s allocation with the specific functions, ℎ = ℎ̅ and𝑚=𝑚
̅ and public policies (𝑃, 𝑃𝑜 , 𝜉, 𝜉 𝑜 , 𝜉 𝑠 , 𝜏 𝑐 ), optimal taxes on capital income and labor

income of the young and elderly (𝜏 𝑠 , 𝜏, 𝜏 𝑜 ) and optimal employer-based health insurance(𝜋, 𝜋 𝑜 , 𝐼, 𝐼𝑜 , 𝜆, 𝜆𝑜 ) in the steady state from the conditions for utility and profit maximization
under feasibility and a balanced government budget are determined as follows:
(i)
(ii)
(iii)

𝜉𝑠 𝑛

𝜏 𝑠 = (𝑛−𝛼),𝑜𝜏 =ℎ𝜉+𝑃−Λ1 𝜏𝑐 +Λ2 𝜉 𝑜 +𝑘(𝛼−𝑣𝑛)𝜉𝑠 (𝑣𝑛−𝛼(1−𝑣𝑛))𝑃𝑜 𝛼𝑚(1−𝑣𝑛)𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)2+−𝛼𝑣
𝑣(𝑇𝑚 (ℎ,𝑚)𝑚+𝑇)
𝑣𝑛2

,

Λ3
𝑛𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)[(1−𝜏𝑜 )𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)−𝑚]+𝜏𝑜 𝑇(ℎ,𝑚)𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)
𝑜
𝜋 =
− 𝜉𝑜,
𝑇(ℎ,𝑚)𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)−𝑚𝑛𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)17

𝛼[𝑇(ℎ,𝑚)(𝑚𝜉 𝑜 −𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)𝜏𝑜 )+𝑃𝑜 +𝑚𝑛𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)𝜋 𝑜 ]1(iv) 𝜏 = (𝑣𝑓 (𝑘,Ω(ℎ,𝑚)𝑙)) {

𝑛2

𝑙

𝜏𝑐 [
(v)

𝑇(ℎ,𝑚)𝛼(𝑚−𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙))𝜋 = 1−𝜉−𝑛2+− 𝑣𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − 𝑘] − 𝑘𝜉 𝑠 },(1−𝑓ℎ (𝑘,Ω(ℎ,𝑚)𝑙))[(1−𝜏𝑜 )𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)−(1−𝜋 𝑜 −𝜉 𝑜 )𝑚]𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)−𝑚,(vi) 𝐼 = ℎ𝜋,(vii) 𝐼𝑜 = 𝑚𝑇(ℎ, 𝑚)𝜋 𝑜(viii) 𝜆 = 𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙),(ix) 𝜆𝑜 =𝑛𝑓𝑚 (𝑘,Ω(ℎ,𝑚)𝑙)𝑇(ℎ,𝑚).

Proof. See Appendix C.
In condition (i), the optimal capital income tax 𝜏 𝑠 and savings subsidy 𝜉 𝑠 share the same
sign because dynamic efficiency or binding solvency requires 𝑓𝑘 (𝑘, Ω(ℎ, 𝑚)𝑙) = 𝑛/𝛼 > 1 (seeAppendix B). The condition for dynamic efficiency or binding solvency implies that fertility
must exceed the subjective discount factor 𝑛 > 𝛼. Intuitively, a high subjective discount factor
leads to high savings (hence a low marginal product of capital), whereas high fertility leads toa high marginal product of capital. A high subjective discount factor may also lead to low
fertility as it attaches a high current value to future welfare in the dynasty. This lower boundon fertility is novel in its own right.
Without health externalities (𝜇 = 0), 𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙) = 0 and 𝑓𝑚 (𝑘, Ω(ℎ, 𝑚)𝑙) = 0 (seeAppendix B). Then, conditions (iii), (v), (viii) and (ix) in Proposition 2 imply that 𝜋 𝑜 = 𝜏 𝑜 −𝜉 𝑜 , 𝜋 = 𝜏 𝑜 − 𝜉 and 𝜆 = 𝜆𝑜 = 0, respectively. In this sense, setting all taxes, public subsidies,public transfers and private health subsidies at zero yields the socially optimal outcome. In
other words, when health externalities are absent, the laissez-faire allocation without the EHI
would be the same as the planner’s allocation in this lifecycle-dynastic model (the First Welfare
Theorem). This feature is absent in lifecycle models with overlapping generations that abstractfrom intergenerational transfers.
By contrast, in the presence of health externalities to labor productivity 𝜇 > 0, the marginal
products of health spending at young and old ages are positive, 𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙) > 0 and
𝑓𝑚 (𝑘, Ω(ℎ, 𝑚)𝑙) > 0, and increase with 𝜇 (see Appendix B). In this sense, age-specific labor
income taxes and age-specific health subsidies are necessary to internalize the healthexternalities. Specifically, taking public transfers, public health subsidies, savings subsidies
and consumption taxes (𝑃, 𝑃𝑜 , 𝜉, 𝜉 𝑜 , 𝜉 𝑠 , 𝜏 𝑐 ) as given, stronger health externalities may increaseor decrease optimal young-age and old-age labor income taxes in conditions (ii) and (iv), and18

may raise optimal private health subsidies for young and old workers through the EHI( 𝜋, 𝜋 𝑜 , 𝐼, 𝐼 𝑜 , 𝜆, 𝜆𝑜 ) at different rates in conditions (iii), and (v) to (ix). Consistent with

Proposition 1, optimal EHI coverage rates (𝜋, 𝜋 𝑜 ) and optimal public health subsidies (𝜉, 𝜉 𝑜 )are substitutable as reflected in conditions (iii) and (v).
In the absence of the EHI in the economy, optimal universal public health financed by
optimal taxes is a special case of Proposition 2 with 𝜋 = 𝜋 𝑜 = 𝐼 = 𝐼𝑜 = 𝜆 = 𝜆𝑜 = 0 as follows.
We define the following expressions for optimal public policies:𝑃𝑜𝑓 (𝑘,Ω(ℎ,𝑚)𝑙)Λ 4 ≡ 𝛼 [ 𝑛 + 𝑚 𝑇 (ℎ,𝑚)𝑚𝛼𝑣𝑛ℎ𝑇 (ℎ,𝑚)ℎ(𝑇(ℎ, 𝑚) + 𝛼(1−𝑣𝑛)−𝑣𝑛)−

𝑣𝑛(ℎ𝑓ℎ (𝑘,Ω(ℎ,𝑚)𝑙)+𝑃)𝛼(1−𝑣𝑛)−𝑣𝑛],(𝑛−𝛼)(𝛼−𝑣𝑛)𝑘Λ 5 ≡ − 𝛼(1−𝑣𝑛)−𝑣𝑛 ,Λ6 ≡ −

𝑇(ℎ,𝑚)[𝑣𝑛𝑑+𝛼(1−𝑣𝑛)(𝑚−𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)]−𝑣𝑛𝑐}𝑛𝛼{(1−𝑣𝑛)𝑛[𝑣𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙)+𝑘]−𝛼𝑇(ℎ,𝑚)Λ7 ≡ (,

𝛼(1−𝑣𝑛)−𝑣𝑛

𝑛𝑇𝑚 (ℎ,𝑚)) [𝑇(ℎ, 𝑚) +𝛼𝑣𝑛ℎ𝑇ℎ (ℎ,𝑚)𝛼(1−𝑣𝑛)−𝑣𝑛Λ 8 ≡ 𝑛[𝑣𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) + 𝑘] − 𝛼 [𝑐 +],𝑇(ℎ,𝑚)(𝑑+𝑚−𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙))𝑛].

Corollary 3. Given the social planner’s allocation with the specific functions, ℎ = ℎ̅ and 𝑚 =𝑚̅ and public policies (𝑃, 𝑃𝑜 , 𝜉 𝑠 , 𝜏 𝑐 ), the government can determine optimal public healthsubsidies and taxes on capital and labor income (𝜉, 𝜉 𝑜 , 𝜏 𝑠 , 𝜏, 𝜏 𝑜 ) in the steady state in theabsence of the EHI as follows
𝜉𝑠 𝑛

(i)

𝜏 𝑠 = (𝑛−𝛼),

(ii)

𝜉𝑜 =

(iii)

𝑜

𝜏 =

(iv)

𝜉 = (1 − 𝛼)𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙) +(v)

8
𝜏 = 𝑓 (𝑘,Ω(ℎ,𝑚)𝑙)[𝛼(1−𝑣𝑛)−𝑣𝑛.
]Λ4 +Λ5 𝜏𝑠 +Λ6 𝜏𝑐Λ7𝜉 𝑜 (𝑚+,𝑇(ℎ,𝑚)
𝑛𝑓 (𝑘,Ω(ℎ,𝑚)𝑙))− 𝑚 (ℎ,𝑚)𝑇𝑚 (ℎ,𝑚)𝑇𝑚

𝑓𝑙 (𝑘,Ω(ℎ,𝑚)𝑙),𝛼𝑇ℎ (ℎ,𝑚)𝑇(ℎ,𝑚)𝜉 𝑜,
𝑇𝑚 (ℎ,𝑚)𝑛𝛼[ℎ𝜉+𝑃+(𝑛−𝛼)𝑘𝜏𝑠 ]+Λ 𝜏𝑐𝑙When universal public health is absent in the economy, the optimal EHI and optimal taxes
are a special case of Proposition 2 with 𝜉 = 𝜉 𝑜 = 0 as follows:
Corollary 4. Given the socially optimal allocation with the specific functions, ℎ = ℎ̅ and 𝑚 =𝑚̅ and public policies (𝑃, 𝑃𝑜 , 𝜉 𝑠 , 𝜏 𝑐 ), the government can determine optimal taxes on capitaland labor income (𝜏 𝑠 , 𝜏, 𝜏 𝑜 ) to support optimal employer-based health insurance(𝜋, 𝜋𝑜 , 𝐼, 𝐼𝑜 , 𝜆, 𝜆𝑜 ) in the steady state in the absence of universal public health.
19The theoretical results in Sections 5 and 6 illustrate the importance of health subsidies, i.e.,
public health subsidies or private health subsidies through the EHI, in internalizing the healthexternalities. The economy can fully mitigate the efficiency loss of the health externalities
through appropriate health subsidies and taxes. The next section calibrates the model in Section
6 to provide a quantitative assessment of the effects of health externalities to labor productivity
for policy improvements.

7. Quantitative analysis in steady state with specific functions
Public health subsidies are available worldwide. Almost all the OECD countries have universal
public healthcare such as Australia where the EHI is not popular. By contrast, the health systemin the US largely consists of private providers and private health insurance, especially for theworking population. Most workers in the US receive their health care through the EHI, whilemost seniors (as well as low-income families and children) receive their health care via publichealth programs.
To compare the quantitative results for policy improvements between a country with
universal public health to a country with both the EHI and public health subsidies, we calibrate
key parameters of the model in Section 6 to Australian and US data. Specifically, we calibrate
key parameters of the model (𝛼, 𝛽, 𝛿/𝑦, 𝐷, 𝜌, 𝑃/𝑦, 𝑃𝑜 /𝑦, 𝜏, 𝜏 𝑜 , 𝜏 𝑐 , 𝜏 𝑠 , 𝜉 𝑠 , 𝜉, 𝜉 𝑜 , 𝜋, 𝜋 𝑜 ) to match
certain targets in each country and assume that both Australia and the US share the same
parameter values for 𝜎1 , 𝜎2 , 𝑣, 𝜃, 𝐴, 𝜇, 𝜖, 𝜓 and 𝜙 in preferences and technologies.10 One periodin our model is 30 years. We assume that individuals in this model are born when they are 5
years old and live to be at most 94 years old.11
7.1. Calibration, data moments and parameter values
This subsection describes the calibration procedure, the targeted data moments used in thecalibration and the parameter values obtained from the literature or from the calibration. The
targeted moments and parameter values are reported in Tables 1 and 2, respectively.
[Tables 1 and 2 go here]
7.1.1. Parameter values for both countries
To express the optimal and equilibrium conditions for 𝑐, 𝑑, 𝑠, 𝑘, ℎ, 𝑚 in Section 6 as a fractionof 𝑦 for calibration, 12 we assume log utility for consumption ( 𝜎1 = 1 ). Since a lower

10This assumption is reasonable as Australia and the US share similar levels of economic development.
Hence, agents in our model enter old age at 65 years old, the earliest age where old agents are eligible for the
public health program, Medicare, in the US.
12
We normalize all the quantity variables by output per young worker, 𝑦, to detrend the model.
1120intertemporal elasticity of substitution for fertility than for consumption can account for thesecular decline in fertility when income rises (e.g., Greenwood et al., 2005), we set 1⁄𝜎2 =1⁄4.5. The fixed time rearing a child at 𝑣 = 0.1 is reasonable for the parental time cost in theOECD countries.13 Capital’s income share takes a standard value 𝜃 = 0.33. We normalize thetotal factor productivity parameter in the final good production technology 𝐴 to unity. The
degree of the health externalities at 𝜇 = 9% follows the estimate of Bloom et al. (2022) thatmarginally better health of the labor force raises labor productivity by 6% to 12% in 133
countries when 𝜃 = 0.33. We set the coefficient of the health function 𝜖 at unity, the return
factor on health spending 𝜓 at 0.8 and the share of young-age health spending in the healthtechnology 𝜙 at 0.5 for both Australia and the US.
7.1.2. Country-specific parameter values
We can rewrite the longevity function as follows:
𝑇 = 𝐷 [𝛿𝑦(ℎ/𝑦 𝜙 𝑚)𝑚/𝑦𝑦ℎ/𝑦 𝜙 𝑚

+𝜖(

𝑚/𝑦

)

𝜓

] .

𝑦

For 𝜖 at unity, 𝜓 at 0.8 and 𝜙 at 0.5, we calibrate country-specific coefficients 𝛿/𝑦 and 𝐷 in
the longevity function to match country-specific health spending (% of GDP) and old-age life
𝑇 in Australian and US data (see Table 1) using the first-order conditions with respect to youngand old-age health spending.14 This results in 𝛿/𝑦 = 0.81% and 𝐷 = 0.6832 in Australia, and𝛿/𝑦 = 6.05% and 𝐷 = 0.7836 in the US, given that old-age longevity is lower but healthspending is higher in the US than in Australia.
By using the first-order condition for savings, the country-specific intergenerational
discount factor 𝛼 follows the country-specific investment to GDP ratio in Table 1. The countryspecific taste for the old parent’s consumption 𝛽 is half of 𝛼 because old-age life is shorter than
the young age. From the first-order condition for fertility, the country-specific taste for thenumber of children 𝜌 matches the country-specific fertility rate per young parent and otherobservations in Table 1. Doing so gives 𝛼 = 0.7103, 𝛽 = 0.3552 and 𝜌 = 0.3069 in Australia,
whereas 𝛼 = 0.6317, 𝛽 = 0.3159 and 𝜌 = 0.3205 in the US, given that the investment to
GDP ratio is lower and fertility is higher in the US than in Australia.

13

For example, the fraction of time that a young parent allocated to a child aged 0-17 is about 0.08 in the US(Yew et al., 2024).
14
As one period corresponds to 30 years in our model, longevity (𝑇) is calculated from life expectancy at age 35(𝐿𝐸) i.e., 𝑇 = 𝐿𝐸/30 − 1.

21We set 𝜋 = 𝜋 𝑜 = 0% in Australia that has no EHI. From total health spending (8.8% ofGDP) and public health spending (6% of GDP) in Australia, the flat rate of health subsidies(𝜉 = 𝜉 𝑜 ) is 68.18%. From the OECD tax system, the average labor income tax rates at young
and old age (𝜏 = 𝜏 𝑜 ) are 28%; the tax rate on returns to savings (𝜏 𝑠 ) follows the 30% corporate
income tax; and the consumption tax rate (𝜏 𝑐 ) is 10% in Australia. As compulsory retirementsavings (superannuation) are exemptible from labor income taxes, we set the savings subsidy
rate (𝜉 𝑠 ) at 2%.15 Public transfers to the elderly at 4% of output per young worker (𝑃𝑜 /𝑦)follow public pension spending (% of GDP). We then set public transfers per young worker(𝑃/𝑦) at 19.14% to balance the government budget.
For the policy instruments in the US, we follow Zhao (2017) to set the old-age public healthsubsidy rate (𝜉 𝑜 ), or the Medicare subsidy rate, at 0.5 and the fraction of young-age medical
expenses covered by the EHI (𝜋) at 0.65. We set 𝜉 = 0% in the US as most young workers
below age 65 are covered by the EHI rather than public health subsidies. We set 𝜋 𝑜 = 0% in
the US as old workers (aged 65 or above) receive old-age public health subsidy 𝜉 𝑜 rather thanthe EHI.16 From the OECD tax system for the US, average labor income tax rates at young andold ages (𝜏 = 𝜏 𝑜 ) are 27% and the tax rate on returns to savings (𝜏 𝑠 ) follows the 33% corporate
income tax. Following Conesa and Dominguez (2020), the consumption tax rate (𝜏 𝑐 ) is 6% in
the US. We set 𝜉 𝑠 at 0.8% to match the cost of tax subsidies for retirement savings at $146
billion in 2014 (Friedman, 2017) that is about 0.8% of US GDP in 2014. In the same way wecompute Australian public transfers, US public transfers to the elderly at 6% of output peryoung worker (𝑃𝑜 /𝑦) follow public pension spending (% of GDP), and public transfers per
young worker (𝑃/𝑦) in the US at 18.09% balance the government budget.
7.2. Quantitative results
To compare different equilibrium outcomes, we use the consumption equivalent variation tomeasure the welfare change in economy i relative to the welfare level in benchmark competitive
equilibrium (BCE). Specifically, it is the percentage change in both young-age and old-age
consumption in every period Δ𝑖 such that the welfare level in the BCE (𝑈0𝐵𝐶𝐸 ) reaches thewelfare level in economy 𝑖 (𝑈0𝑖 ). Using logarithmic preference for consumption in Eq. (1), it isstraightforward to show that15

The compulsory superannuation contributions are 1.5% of GDP in Australia. From the labor income tax rate at
28%, subsidized savings equal 0.42% of GDP. Then, the roundup savings subsidy is 2% as the national savingrate (or equivalently the national investment rate in our model) is 22% in Australia (World Bank).
16
Our quantitative analysis abstracts from means-tested health subsidies in the US such as Medicaid thatsubsidizes healthcare to low-income families and children who may be out of the US labor force.

22

𝑈0𝑖 = 𝑈0𝐵𝐶𝐸 +(𝛽+𝛼)ln(1+Δ𝑖 )

,

(1−𝛼)

and thus
Δ𝑖 = 𝑒𝑥𝑝 [(𝑈0𝑖 −𝑈0𝐵𝐶𝐸 )(1−𝛼)𝛽+𝛼] − 1.

7.2.1. Comparisons of equilibrium outcomes
Tables 3a and 3b compare the equilibrium outcomes at steady states for Australia and the US
in ten cases, respectively. The social optimum (SO) is reported in Case (1), and the laissez faire
without the EHI (LF) is reported in Case (2). The benchmark competitive equilibrium (BCE)
is given in Case (3). Cases (4) to (10) are counterfactual experiments with a 10% variation in
one of the concerned policy instruments from the BCE, holding other policy rates unchanged.
These counterfactual experiments provide some insights into the quantitative effects of thevariation in a specific policy instrument. Case (4) in Table 3a lowers 𝜉 for Australia and Case(4) in Table 3b lowers 𝜋 for the US. We also lower 𝜉 𝑜 in Case (5), lower 𝑃𝑜 /𝑦 in Case (6), andraise 𝜏, 𝜏 𝑜 , 𝜏 𝑠 and 𝜏 𝑐 in Cases (7) to (10), respectively. In each of these counterfactual
experiments, we adjust young-age public lump-sum transfers to balance the government budget
when a specific tax or subsidy varies from the benchmark rate.
[Tables 3a and 3b go here]
Compared with the SO allocation in Case (1), the LF in Case (2) has lower old-age
longevity or old-age labor (𝑇), young-age health spending over output per young worker (ℎ/𝑦),old-age health spending over output per young worker (𝑚/𝑦), capital over output per young
worker (𝑘/𝑦), young-age labor (1 − 𝑣𝑛) and total labor per young worker (𝑙 = 1 − 𝑣𝑛 + 𝑇/𝑛).
However, the LF allocation has higher fertility (𝑛), young-age consumption over output peryoung worker (𝑐/𝑦) and old-age consumption over output per young worker (𝑑/𝑦) than the
SO allocation. These quantitative results for both Australia and the US in Tables 3a and 3b are
consistent with the theoretical results established in Subsection 5.1.
For Australia BCE in Case (3) of Table 3a, old-age longevity or old-age labor (𝑇) is
substantially higher than the LF level, owing to high young-age and old-age public healthsubsidies (𝜉, 𝜉 𝑜 ) in Australia. However, taxes on labor income and consumption cause higher
fertility (𝑛), and hence lower young-age and total labor time (1 − 𝑣𝑛, 𝑙 = 1 − 𝑣𝑛 + 𝑇/𝑛) as
well as lower capital over output per young worker (𝑘/𝑦) in the BCE than in the LF. Overall,the LF has a welfare gain equivalent to a 0.21% rise in consumption from the BCE. These
results suggest that the welfare loss of distortionary taxes outweighs the welfare gain of publichealth subsidies. Thus, the current policy rates in Australia reduce efficiency from the LF.
23

When comparing the Australia BCE with the SO in Table 3a, the BCE has lower old-age
longevity or old-age labor (𝑇), young-age health spending over output per young worker (ℎ/𝑦),capital over output per young worker (𝑘/𝑦), young-age labor (1 − 𝑣𝑛) and total labor per
young worker (𝑙 = 1 − 𝑣𝑛 + 𝑇/𝑛) than the SO. However, the BCE has higher old-age healthspending over output per young worker (𝑚/𝑦), fertility (𝑛), young-age consumption over
output per young worker (𝑐/𝑦) and old-age consumption over output per young worker (𝑑/𝑦)than the SO. The welfare improvement of the SO over the BCE in Australia is equivalent to a
6.28% rise in consumption due to the health externalities and suboptimal policy rates.
Specifically, high tax rates on consumption, capital income and labor income, especiallyold-age labor income, in Australia BCE have led to substantially lower capital over output peryoung worker (𝑘/𝑦), young-age labor (1 − 𝑣𝑛) and total labor per young worker (𝑙 = 1 −𝑣𝑛 + 𝑇/𝑛 ) than the SO levels but considerably higher fertility ( 𝑛 ) than the SO level.
Additionally, young-age health spending over output per young worker (ℎ/𝑦) is lower but oldage health spending over output per young worker (𝑚/𝑦) is higher in the BCE than the SO
levels, suggesting potential gains of increasing young-age public health subsidy ( 𝜉 ) andreducing old-age public health subsidy (𝜉 𝑜 ) to correct health spending toward the SO level.
In Cases (4) and (5) of Table 3a, the welfare loss of a 10% reduction in the young-age orold-age health subsidy is equivalent to a 0.72% or 0.53% decline in consumption. Since healthsubsidies reduce the marginal cost of health spending but increase the marginal cost ofchildrearing, lower health subsidies lead to lower longevity (𝑇) and higher fertility (𝑛) than the
BCE. Young-age labor and total labor also fall when fertility rises. Thus, health subsidies are
essential to mitigate the efficiency loss of the health externalities. Subsidizing young-age healthspending brings greater welfare improvements than subsidizing old-age health spending
because young-age health spending improves labor productivity and labor supply at both young
and old ages.
In Case (6) of Table 3a, a 10% reduction in old-age public transfers leads to a moderate
welfare loss equivalent to a 0.25% decline in consumption. Lower old-age public transfers
reduce the transfer cost of childrearing, causing fertility to rise and labor to fall from the BCE.
In Case (7) of Table 3a, a 10% rise in the young-age labor income tax yields a relatively small
welfare loss equivalent to a 0.09% decline in consumption.
However, in Case (8) or (9) in Table 3a, a 10% increase in the old-age labor income tax orcapital income tax causes a large welfare loss equivalent to a 0.66% or 0.62% decline in
consumption from the BCE level. In this regard, the negative effects of old-age labor income
24taxes on the transfer cost of childrearing and on the marginal benefits of health spending aresubstantial. Hence, the old-age labor income tax imposes larger distortions on individual
decisions on fertility, young-age and total labor and old-age longevity or labor than the youngage labor income tax. The negative effect of the capital income tax on the marginal benefit ofsavings is also substantial, causing capital over output per young worker to fall substantially.
A 10% rise in the consumption tax in Case (10) results in a moderate welfare loss equivalent
to a 0.3% decline in consumption.
For the US in Table 3b, the BCE in Case (3) has higher young-old and old-age healthspending, and hence higher old-age longevity or labor, than their counterparts in the LF owing
to a high old-age public health subsidy (𝜉 𝑜 ) and high young-age EHI coverage (𝜋) in the BCE.
However, high distortionary taxes, especially high capital income taxes, cause much higher
fertility and much lower capital over output per young worker than their counterparts in the LF.
The welfare cost of the distortionary taxes on fertility outweighs the welfare gain of healthsubsidies in the BCE. Thus, the elimination of the taxes and subsidies in the BCE leads to a
welfare gain equivalent to a 0.97% rise in consumption in the LF over the BCE. In other words,
the current policies in the US reduce efficiency from the LF. Hence, there is a large welfare
gain of the SO over the US BCE equivalent to a 7.37% rise in consumption.
Specifically, the US BCE has lower old-age longevity or labor, young-age and old-age
health spending over output per young worker, capital over output per young worker, youngage labor and total labor per young worker than the SO levels. The BCE also has much higher
fertility, young-age consumption over output per young worker and old-age consumption over
output per young worker than those in the SO. Moreover, the welfare loss of policies is greaterin the US BCE than in the Australia BCE.
In Cases (4) and (5) of Table 3b, the welfare loss of a 10% reduction in the young-age EHI
coverage 𝜋 is equivalent to a 1.01% fall in consumption, whereas the welfare loss of a 10%
reduction in the old-age health subsidy 𝜉 𝑜 is equivalent to a 0.5% fall in consumption, muchsmaller than that from reducing the young-age EHI coverage. As in Australia, a 10% reduction
in US old-age public transfers in Case (6) of Table 3b yields a moderate welfare loss equivalent
to a 0.36% decline in consumption.
In Case (7) of Table 3b, a 10% rise in the young-age labor income tax in the US leads to a
relatively small welfare loss equivalent to a 0.11% fall in consumption. In Case (8) or (9) of
Table 3b, a 10% rise in the old-age labor income tax or capital income tax in the US engenders
a large welfare loss equivalent to a 0.72% or 1.21% decline in consumption. In Case (10) of

25

Table 3b, a 10% rise in the consumption tax in the US yields a small welfare loss equivalent toa 0.18% fall in consumption.
Notice that the higher capital income tax in Case (9) of Table 3b leads to the largest welfare
loss relative to other counterfactual experiments for the US BCE. These quantitative results are
consistent with the conventional wisdom that the capital income tax is harmful for savings (e.g.,Judd, 1985; Chamley, 1986). However, the efficiency loss of the capital income tax in theprevious models is smaller than that in this model that treats fertility, labor supply and longevity
as endogenous variables. For instance, a higher capital income tax in the US not only
substantially decreases capital over output per young worker, young-age labor and total labor
but also considerably increases fertility from their BCE levels.
In general, the quantitative effects of the counterfactual experiments for the US BCE in
Table 3b are consistent with those for Australia BCE in Table 3a.17 The main difference is thatthe welfare loss of a reduction in the young-age health subsidy or a rise in the tax on labor
income or capital income is much higher in the US than in Australia. This is mainly because
the US has lower longevity and a lower investment rate but higher fertility than Australia.
Moreover, a rise in the young-age labor income tax in both countries in Case (7) has thesmallest welfare loss compared with an equivalent rise in any other taxes. This is in contrast
with the conventional view that regards the consumption tax as the least distortionary taxes.
Intuitively, the consumption tax has negative effects on the time cost, bequest cost and transfer
cost of childrearing, while the young-age labor income tax has a negative effect only on thetime cost of childrearing. Thus, the consumption tax yields higher fertility but lower capitalover output per young worker, young-age labor and total labor per young worker than theyoung-age labor income tax. This result is different from those that ignore intergenerationaltransfers and assume fixed fertility and old-age labor.
7.2.2. Optimal health subsidies and optimal taxes
This subsection derives optimal health subsidies at steady states quantitatively for Australia
and the US using benchmark parameterization in Table 2. Since the income tax is the largest
source of government revenue and the most important source of funding public health subsidies
in Australia and the US,18 this subsection explores optimal public health subsidies financed by

17

Fertility is responsive to changes in tax rates quantitatively in our model, in line with the empirical evidence
that finds statistically significant effects of taxes on fertility (Whittington, 1992; Whittington et al., 1990).
18
To cover the cost of public health subsidies in Australia and the US, taxpayers pay Medicare taxes. High-income
earners may need to pay additional Medicare taxes.

26

taxes on labor income and capital income. In doing so, we take the benchmark savings subsidy
as given and set all other government instruments to zero.
Given the AUS BCE savings subsidy 𝜉 𝑠 = 2%, Panel (1) of Table 4 shows that the optimal
young-age public health subsidy (𝜉) at 73.49% is much higher than the benchmark rate at
68.18%, while the optimal old-age public health subsidy (𝜉 𝑜 ) at 66.18% is slightly lower thanthe benchmark rate at 68.18%. The results suggest that age-specific health subsidies rates are
essential to achieve the SO. Particularly, young-age workers should have higher healthsubsidies than old-age workers. To fund public health subsidies optimally, we need tax rates
on young-age and old-age labor income and capital income at 𝜏 = 10.61%, 𝜏 𝑜 = 6.01% and𝜏 𝑠 = 15.82%, respectively. Thus, to achieve the SO, young workers receiving higher optimal
health subsidies should also pay higher labor income taxes than old workers. The current
Medicare levy in Australia is 2% of taxable income,19 which is uniform across ages and far
below the optimal labor income taxes computed here.
Given the US BCE savings subsidy 𝜉 𝑠 = 0.8%, Panel (2) of Table 4 shows that for the US,
the optimal young-age EHI coverage is 𝜋 = 57.98% and the optimal old-age public healthsubsidy is 𝜉 𝑜 = 46.41%, lower than their benchmark rates 65% and 50%, respectively. The
corresponding optimal private health subsidy to a young worker is 𝜆 =58.62%, which is also
lower than the benchmark rate 80% in the US (Jeske and Kitao, 2009). Therefore, ourquantitative results suggest that the benchmark health subsidy rates in the US are too highcompared with the optimal rates. Hence, to achieve the SO, young workers should receive
higher health subsidies than old workers, as in the current health policy in the US.
Funding the old-age public health subsidy (Medicare) optimally in the US requires optimal
tax rates on young-age and old-age labor income and capital income at 𝜏 = 0.47%, 𝜏 𝑜 =10.48% and 𝜏 𝑠 = 2.84%, respectively. In the US, the Medicare tax rate is 1.45% for theemployer and 1.45% for the employee. Thus, the overall contribution rate 2.9% is higher than
the aforementioned optimal young-age labor income tax rate but lower than the aforementioned
optimal old-age labor income tax rate. To achieve the SO, old workers covered by the Medicare
should pay more taxes than young workers in contrast to the result for Australia that requires
young workers receiving more health subsidies to pay more taxes than old workers to achieve
the SO. Hence, even if the young-age labor income tax is the least distortionary tax relative toother taxes to fund public health subsidies, the optimal rate of the young-age labor income tax
depends positively on how much young workers benefit from public health subsidies.
19

The Medicare Levy surcharge, an additional Medicare tax, is 1% to 1.5% depending on the income level.

27

Additionally, the optimal capital income tax rate should be low given the low benchmark
savings subsidy rate in the US.
[Table 4 goes here]
When using the benchmark consumption tax to finance health subsidies as well, optimal
public and private health subsidies (𝜉, 𝜉 𝑜 , 𝜋) and optimal labor income taxes (𝜏, 𝜏 𝑜 ) are lower
in both countries, as shown by Panels (1) and (2) of Table 5, than those in Table 4. In this case,reductions in labor income taxes are substantial for old workers who will now receive labor
income subsidies.20 This is because the consumption tax reduces the time cost, bequest cost
and transfer cost of childrearing, thereby requiring the optimal old-age labor income tax to fall
substantially such that the transfer cost of childrearing can be high enough to counteract theeffects of the consumption tax. Taking the benchmark savings subsidy 𝜉 𝑠 as given, the optimal
capital tax 𝜏 𝑠 is unaffected by the benchmark consumption tax, as shown in Proposition 2.
[Table 5 goes here]
7.2.3. Quantitative implications of health externalities to labor productivity
This subsection examines the sensitivity of the optimal allocation and optimal policy at thesteady state when the health externalities become stronger (i.e., a rise in 𝜇 from the benchmark
0.09 to 0.15), holding other policy instruments at their benchmark levels.21
Figures 1a and 1b illustrate the quantitative sensitivity of optimal fertility, optimal time
allocations and optimal proportional allocations to variations in 𝜇 for Australia and the US,
respectively. Internalizing the stronger health externality increases the optimal young-age andold-age health spending over output per young worker, old-age longevity or labor, capital over
output per young worker, young-age labor and total labor per young worker. Doing so increases
the costs of childrearing and the relative price of consumption, thereby reducing the optimal
fertility and consumption over output per young worker at young and old ages.
[Figures 1a and 1b go here]
When only using income taxes to finance health subsidies in Figures 2a and 2b at thebenchmark savings subsidy, the internalization of the stronger health externality increases
optimal health subsidies and income taxes. The results are consistent with Proposition 1 thatstronger health externalities require higher public health subsidies or higher private health20

As noted in Brewer et al. (2021), income-tax deductions for the elderly exist in the US and many OECD
countries.
21
Variations in the degree of health externalities do not affect fertility, time allocations and proportional
allocations in the competitive equilibrium as agents ignore the external effects of their health spending.

28

subsidies through the EHI to internalize the marginal products of average health spending at
young and old ages.
[Figures 2a and 2b go here]
Hence, the sensitivity analysis also suggests that universal public health becomes more
expensive to internalize stronger health externalities by increasing workers’ tax burden. Thismay motivate a health reform that expands the EHI to replace public health subsidies. We
conduct a sensitivity analysis in Figures 3a and 3b to consider only the universal EHI in
Australia and the US to fully replace public health. The quantitative results for the health reformare consistent with Proposition 1 that the EHI must be accompanied by appropriate taxes, suchas income taxes in this exercise, to correct the marginal cost of childrearing to achieve the SO.
Specifically, internalizing the stronger health externality increases the optimal young-age
and old-age EHI coverage (𝜋, 𝜋 𝑜 ), optimal old-age labor income tax (𝜏 𝑜 ) and optimal capitalincome tax (𝜏 𝑠 ). However, doing so decreases the young-age labor income tax (𝜏) or increases
the young-age labor income subsidy for 𝜏 < 0. The difference in labor income taxes between
the young and old workers increases with the degree of the health externalities. The optimal
capital income tax increases in Figures 2a, 2b, 3a and 3b because the stronger healthexternalities reduce optimal fertility in Figures 1a and 1b. The optimal young-age EHI coverage
should be higher than the optimal old-age EHI coverage as the former benefits workers’ healthlifetime. Moreover, comparing Figures 2a and 2b with Figures 3a and 3b reveals a lower tax
burden on young and old workers when replacing public health by the EHI as the young-age
and old-age labor income tax rates are lower at each degree of health externalities.22[Figures 3a and 3b go here]
The health reform that replaces public health with the universal EHI may be an extreme
health reform. In fact, the theoretical results in Propositions 1 and 2 as well as the quantitative
analysis in Subsections 7.2.2 and 7.2.3 suggest that public health subsidies and the EHI, which
co-exist in the US, can support each other by covering different age groups of workers toachieve the SO outcome. Consequently, the findings of this paper may provide new insights
for future health reforms in countries without universal public health or the EHI.

8. Conclusion
This paper developed a lifecycle-dynastic model with the choices of health spending, savings,
labor supply, longevity and fertility to explore the effects of health externalities to labor

22

For Australia, replacing public health with the EHI reduces the tax burden of old workers if the degree of healthexternalities is not sufficiently high.

29

productivity for efficiency concerns and policy implications. This paper makes the following
theoretical contributions. First, the health externalities cause low health spending, longevity,
labor productivity, labor supply, savings and output but high fertility because perceived private
returns to health spending are below the social return.
Second, appropriate public or private health subsidies or both, accompanied by appropriate
taxes, can attain the socially optimal outcome through age-specific rates of health subsidies
and labor income taxes. In other words, the key policy tool to internalize health externalities tolabor productivity is to find the respective optimal policy rates in each health system. Intuitively,
health subsidies, particularly to young workers, either through universal public health insuranceor employer-based group health insurance, increase the cost of childrearing and decrease thecost of health spending, thus increasing longevity, young-age and old-age labor and laborproductivity and lowering fertility. Conversely, taxes on young-age and old-age labor income
and consumption reduce the costs of childrearing. Taxes on young-age and old-age labor
income also increase the cost of health spending. Subsidies on savings have negative effects
on the cost of childrearing and the cost of savings which counteract the wedge of capital income
taxes. Old-age public transfers induce altruistic parents to increase transfers to children, thusincreasing the cost of childrearing. Appropriate subsidies and taxes with these opposing effects
can give rise to the socially optimal outcome.
Third, private health subsidies through the EHI can ease the financial burden of funding
public healthcare but the EHI for profit maximization alone cannot fully internalize healthexternalities to labor productivity when individuals choose the number of children. This isbecause the EHI increases the cost of childrearing excessively, causing fertility to fall below
the socially optimal level. The EHI and appropriate taxes together can achieve the socially
optimal outcome when their opposite effects on the cost of childrearing just correct theexcessive fertility that arises from the health externalities.
Our quantitative analyses provide several results that may shed light on optimal health andtax policies. From calibration results based on Australian and the US data, fertility in bothbenchmark economies is above their laissez-faire and socially optimal levels, while young-age
labor supply and capital per young worker in both benchmark economies are below the laissezfaire and socially optimal levels. Lower capital per young worker also implies lower output peryoung worker. Owing to high public and private health subsidies to young workers, young-age
health spending and old-age labor supply or longevity in both benchmark economies are above
the laissez-faire levels but below the socially optimal levels. Due to high public health subsidies
to the elderly in Australia, old-age health spending is higher but total labor supply per young
30

worker is lower in the Australia benchmark than the laissez-faire and socially optimal levels.
In contrast, public health subsidies to workers are lower in the US than in Australia, thus oldage health spending and total labor supply per young worker in the US benchmark are above
the laissez-faire levels but below the socially optimal levels. These quantitative results suggest
policy improvements on the tax and health policies.
Specifically, when financing public health subsidies only by income taxes, the optimal
policies to internalize health externalities to labor productivity depend on whether an economy
has universal public health or not. For Australia with only universal public health, optimal
public health subsidies and labor income taxes should be higher for young workers than for old
workers because optimal young-age public health subsidies are higher than optimal old-age
public health subsidies. On the contrary, for the US with public health for the elderly including
old workers and private health subsidies for young workers, the optimal old-age labor income
tax should be higher than the optimal young-age labor income tax. The optimal taxes here
depart from those in models with fixed fertility and without health externalities to laborproductivity. It allows for age-dependent health subsidies and labor income taxes in relation to
the age-dependent marginal products of health spending at young and old age.
Furthermore, the internalization of the stronger health externality increases optimal healthsubsidies and income taxes to attain optimal allocations, suggesting that public health isexpensive for non-negligible degrees of the health externalities. Replacing public health withthe universal EHI lowers the optimal labor-income tax for all workers. However, replacing
public health with the universal EHI requires old workers to pay higher labor income taxes than
young workers to achieve the social optimum, and this intergenerational tax gap increases withthe degree of the health externalities. Thus, this health reform may be extreme compared to ahealth system that allows for the co-existence of both public health subsidies and the EHI, suchas by targeting each type of health subsidies at different age groups of workers.
In practice, public health subsidies have the advantage of providing healthcare services for
those outside the labor force including young children and future workers, while the universal
EHI has the advantage of reducing public health expenditure especially for countries facing
rapid population ageing and high public debt. We leave these for future research.

31

References
Acton, R.K., Cao, W., Cook, E.E., Imberman, S.A., Lovenheim, M.F., 2022. The effect ofvaccine mandates on disease spread: evidence from college COVID-19 mandates. NBER
Working Paper Series w30303.
Aisa, R., Pueyo, F., 2006. Government health spending and growth in a model of endogenous
longevity. Economics Letters 90(2), 249-253.
Altonji, J.G., Hayashi, F., Kotlikoff, L.J., 1997. Parental altruism and inter vivos transfers:
Theory and evidence. Journal of Political Economy 105(6), 1121–66.
Australia Treasury. https://treasury.gov.au/speech/compulsory-superannuation-and-nationalsaving
Australian Government Actuary. https://aga.gov.au/
Becker, G., 1964. Human Capital. Chicago: University of Chicago Press.
Bhattacharya, J., Qiao, X., 2007. Public and private expenditures on health in a growth model.
Journal of Economic Dynamics and Control 31(8), 2519-2535.
Bloom, D.E., Canning, D., 2003. Health as human capital and its impact on economic
performance. The Geneva Papers on Risk and Insurance 28(2), 304–315.
Bloom, D.E., Canning, D., Kotschy, R., Prettner, K., Schünemann, J.J., 2022a. Health andeconomic growth: Reconciling the micro and macro evidence. NBER Working Paper
26003.
Bloom, D.E., Kuhn, M., Prettner, K., 2022b. Modern infectious diseases: Macroeconomic
impacts and policy responses. Journal of Economic Literature 60(1), 85-131.
Brewer, B., Conway, K.S., Rork, J., 2021. Do income tax breaks for the elderly affect economic
growth? Contemporary Economic Policy DOI: 10.1111/coep.12549.
Chakraborty, S., 2004. Endogenous lifetime and economic growth. Journal of Economic
Theory 116(1), 119–37.
Chen, C., Feng, Z., Gu, J., 2022. Health, health Insurance, and inequality. Working Papers
tecipa-730, University of Toronto, Department of Economics.
Conesa, J.C., Domínguez, B., 2020. Capital taxes and redistribution: the role of management
time and tax deductible investment. Review of Economic Dynamics 37, 156-172.
Cremieux, P., Ouellette, P., Pilon, C., 1999. Health care spending as determinants of healthoutcomes. Health Economics 8, 627–39.
de Courville, C., Cadarette, S.M., Wissinger, E., Alvarez, F.P., 2022. The economic burden of
influenza among adults aged 18 to 64: A systematic literature review. Influenza Other
Respi Viruses 16(3), 376–385.
Davies, J.B., Kuhn, P., 1992. Social security, longevity, and moral hazard. Journal of Public
Economics 49(1), 91-106.
Fang, H., Gavazza, A., 2011. Dynamic inefficiencies in an employment-based health insurance
system: theory and evidence. American Economic Review 101, 3047-3077.
Feng, Z., Zhao, K., 2018. Employment-based health insurance and aggregate labor supply.
Journal of Economic Behavior and Organization 154, 156–174.
Feng, Z., Villamil, A., 2022. Funding employer-based insurance: regressive taxation andpremium exclusions. Economic Theory 73, 509–540.
Frankovic, I., Kuhn, M., 2023. Health insurance, endogenous medical progress, healthexpenditure growth, and welfare. Journal of Health Economics 87, 102717.
Friedman, J.N., 2017. Tax Policy and Retirement Savings. In Auerbach, A.J., Smetters, K.
(eds), The Economics of Tax Policy (New York), online edn, Oxford Academic.
https://doi.org/10.1093/acprof:oso/9780190619725.003.0018.
Greenwood, J., Seshadri, A., Vandenbroucke, G., 2005. The baby boom and baby bust.
American Economic Review 95, 183–207.
32

Hitiris, T., Posnett, J., 1992. The determinants and effects of health expenditure in developed
countries. Journal of Health Economics 6, 173–81.
Jeske, K., Kitao, S., 2009. US tax policy and health insurance demand: can a regressive policy
improve welfare? Journal of Monetary Economics 56(2), 210–221.
Jung, J., Tran, C., 2022. Social health insurance: A quantitative exploration. Journal of
Economic Dynamics and Control 139, 104374.
Kelly, M., 2020. Medicare for all or Medicare for none? A macroeconomic analysis ofhealthcare reform. Journal of Macroeconomics 63, 103170.
Kim, T.K., Lane, S.R., 2013. Government health expenditure and public health outcomes.
American International Journal of Contemporary Research 3, 1–13.
Kuhn, M., Prettner, K., 2016. Growth and welfare effects of healthcare in knowledge-based
economies. Journal of Health Economics 46, 100–19.
Laitner, J., Juster, F.T., 1996. New evidence on altruism: A study of TIAA-CREF retirees.
American Economic Review 86(4), 893–908.
Martin, S., Rice. N., Smith, P.C., 2008. Does health care spending improve health outcomes?
Evidence from English programme budgeting data.
Journal of Health Economics 27(4), 826-842.
Centers for Disease Control and Prevention. National Center for Health Statistics.
https://www.cdc.gov/nchs/index.htm
OECD., 2016. Universal health coverage and health outcomes.
OECD., 2022. Consumption Tax Trends 2022. https://www.oecd.org/tax/consumption-taxtrends-19990979.htm
OECD data. https://data.oecd.org/
Or, Z., 2000. Determinants of health outcomes in industrialized countries: a pooled, crosscountry, time-series analysis, 53–77. Paris: OECD.
Pestieau, P., Ponthière, G., Sato, M., 2008. Longevity, health spending, and pay-as-you-go
pensions. Finanzarchiv 64 (1), 1–18.
Philipson, T.J., Becker, G.S., 1998. Old-age longevity and mortality-contingent claims. Journalof Political Economy 106, 551–73
Prettner, K., Bloom, D.E., Strulik, H., 2013. Declining fertility and economic well-being: Do
education and health ride to the rescue? Labour Economics 22, 70-79.
Pruckner G.J., Schober, T., Zocher, K., 2020. The company you keep: health behavior
among work peers. The European Journal of Health Economics 21, 251–259.
Rivera, B., Currais, L., 2004. Public Health Capital and Productivity in the Spanish Regions:
A Dynamic Panel Data Model. World Development
32(5), 871-885.
Sauermann, J., 2016. Performance measures and worker productivity. IZA World of Labor.
Suhrcke, M., Urban, D. 2010. Are cardiovascular diseases bad for economic growth? Health
Economics 19(12), 1478-1496.
Tang, K.K., Zhang, J., 2007. Health, education, and life cycle savings in the development
process. Economic Inquiry 45(3), 615-630.
Tompa, Emile. 2002. The impact of health on productivity: Empirical evidence and policy
implications. In The Review of Economic Performance and Social Progress 2002:
Towards a Social Understanding of Productivity, ed. Andrew Sharpe, France St-Hilaire,and Keith Banting, 181–202.
Montreal: Institute for Research on Public Policy.
World Bank data. https://data.worldbank.org/
Weinzierl, M., 2011. The surprising power of age-dependent taxes. Review of Economic
Studies 78, 1490–518.

33

White, C., 2021. Measuring social and externality benefits of influenza vaccination. Journal of
Human Resources 56(3), 749–85.
Whittington, L. A., 1992. Taxes and the family: the impact of the tax exemption for
dependents on marital fertility. Demography 29(2), 215-226.
Whittington, L. A., Alm, J., Peters, E., 1990. Fertility and the personal exemption: Implicit
pronatalist policy in the United States. American Economic Review 80(3), 545-556.
Wolfe, B.L., Gabay, M., 1987. Health status and medical expenditures: More evidence of alink. Social Science & Medicine 25, 883–88.
Yew, S.L., Zhang, J., 2018. Health spending, savings and fertility in a lifecycle-dynastic modelwith longevity externalities. Canadian Journal of Economics 51(1), 186–215.
Yew, S.L., Li, S.M., Moslehi, S. 2024. Optimal parental leave subsidization with endogenous
fertility and growth. Economic Inquiry 62(1), 97-125.
Zhang, J., 1995. Social security and endogenous growth. Journal of Public Economics 58, 185–
213.
Zhang, J., Zhang, J., Leung, M., 2006. Health spending, saving, and public policy. Canadian
Journal of Economics 39, 68–93.
Zhao, K., 2017. Social insurance, private health insurance and individual welfare. Journal of
Economic Dynamics and Control 78, 102-117.

34

TABLE 1. Selected observations in Australia and the US
Descriptions

Notations

Australia

US

Savings subsidy

𝜉𝑠

2%

0.8%

Consumption tax

𝜏𝑐

10%

6%

𝜏𝑜 = 𝜏

28%

27%

Capital income tax

𝜏𝑠

30%

33%

Old-age public transfers (% of GDP)

𝑃𝑜 /𝑦

4%

6%

Old-age public health subsidy

𝜉𝑜68.18%

50%

Young-age public health subsidy

𝜉

68.18%

0%Old-age EHI coverage rate

𝜋𝑜

0%

0%

Young-age EHI coverage rate

𝜋

0%

65%

(𝑚 + ℎ)/𝑦

8.8%

15.6%

𝑇

0.5913

0.4815Investment rate (% of GDP)

𝑘𝑛⁄𝑦

22%

18%

Fertility per young parent

𝑛

0.91

0.965

Old- and young-age labor income tax

Total health spending (% of GDP)
Longevity at age 35

SOURCE: Consumption tax (OECD, 2022; Conesa and Dominguez, 2020); taxes on labor and capital income(OECD data, 2000-2020); longevity (Australian Government Actuary, 2000-2017; Centers for Disease Control
and Prevention, 2000-2017); total and public health spending % of GDP (OECD data, 2000-2020); fertility (OECDdata, 2000-2020); old-age public transfers % of GDP (OECD data, 2000-2017); investment rate (World Bank data,
2000-2020); the fraction of young-age medical expenses covered by EHI and US old-age universal public healthsubsidy (Zhao, 2017); Australian savings subsidy (Australia Treasury); US savings subsidy (Friedman, 2017).

35

TABLE 2. Benchmark parameterization
a. Parameter values for both Australia (AUS) and US
𝑡 = 30

Number of years per period𝑣 = 0.1

Fixed time rearing a child

𝜎1 = 1

Reciprocal of intertemporal elasticity of substitution for consumption

𝜎2 = 4.5

Reciprocal of intertemporal elasticity of substitution for fertility

𝜃 = 0.33

Capital’s income share

𝐴=1

Total factor productivity

𝜖=1

Coefficient reducing the marginal effectiveness of health spending𝜓 = 0.8

Return factor on health spending

𝜙 = 0.5

Share of young-age health spending in the technology of health status

𝜇 = 0.09

Elasticity of output with respect to average health in final goods production

b. Country-specific parameter values
AUS: 𝛼 = 0.7103

Intergenerational discounting factor

US: 𝛼 = 0.6317
AUS: 𝛽 = 0.3552

Taste for parental old-age consumption

US: 𝛽 = 0.3159
AUS: 𝜌 = 0.3069

Taste for the number of children

US: 𝜌 = 0.3205
AUS:
US:

𝛿
𝑦

𝛿
𝑦

= 0.81%

Coefficient reducing the effectiveness of health spending= 6.05%

AUS: 𝐷 = 0.6832

Autonomous factor of longevity

US: 𝐷 = 0.7836

36

TABLE 3a. Comparisons of equilibrium outcomes at steady states: AUS
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

SO

LF

BCE

𝜉 = 0.6136

𝜉 𝑜 = 0.6136

𝑃𝑜 /𝑦 = 0.036

𝜏 = 0.308

𝜏 𝑜 = 0.308

𝜏 𝑠 = 0.33

𝜏 𝑐 = 0.11

𝑇

0.60480.54810.59130.58700.58700.59130.59130.5896

0.59130.5913ℎ/𝑦

0.04040.01750.02780.02380.02880.02770.02780.02700.02770.0277𝑚/𝑦

0.05990.03740.06030.06320.05210.06040.06030.05930.06050.0605𝑘/𝑦0.28830.28120.24180.24050.23990.24010.24120.2383

0.2366

0.2398𝑛0.81310.83360.91000.91350.91530.91480.91170.92000.91640.9157

1 − 𝑣𝑛0.91870.91660.90900.90870.90850.90850.90880.90800.90840.9084𝑙

1.66251.5741

1.55881.5512

1.54971.55491.5575

1.54881.55371.5542𝑐/𝑦0.4537

0.48240.47530.47720.47880.47570.47550.47710.47760.4758𝑑/𝑦0.30510.36690.36580.37140.37340.36800.3666

0.37230.37010.3684Δ𝑖

0.06280.0021NA-0.0072-0.0053-0.0025-0.0009-0.0066-0.0062-0.0030

VariableNote: Young-age public lump-sum transfers are used to balance government budget for cases (4) to (10).

TABLE 3b. Comparisons of equilibrium outcomes at steady states: US
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

SO

LF

BCE

𝜋 = 0.585

𝜉 𝑜 = 0.45

𝑃𝑜 /𝑦 = 0.054

𝜏 = 0.297

𝜏 𝑜 = 0.297

𝜏 𝑠 = 0.363

𝜏 𝑐 = 0.066

𝑇

0.49190.40780.48150.47070.47550.48140.48150.4767

0.48140.4815ℎ/𝑦0.05140.02550.04840.04120.0486

0.04820.04840.0466

0.0481

0.0483𝑚/𝑦

0.11410.09010.10760.11190.10000.10800.10770.10580.10820.1078𝑘/𝑦

0.23710.22880.18650.18430.18520.18460.18590.18380.17950.1856𝑛0.87930.91110.96500.97390.97050.97270.9674

0.97600.97670.9687

1 − 𝑣𝑛0.91210.90890.90350.90260.90290.90270.90330.90240.90230.9031𝑙

1.47151.35651.40251.38591.39291.39771.40101.39081.39521.4001𝑐/𝑦0.45080.48380.4786

0.48340.48170.47910.4787

0.48150.48210.4788𝑑/𝑦

0.40300.54050.47960.50030.49170.4841

0.48100.49300.4891

0.4818Δ𝑖

0.07370.0097NA-0.0101-0.0050-0.0036-0.0011-0.0072-0.0121-0.0018

VariableNote: Young-age public lump-sum transfers are used to balance government budget for cases (4) to (10).

38

TABLE 4. Optimal health subsidies and income taxes (%) at steady states
Policy variables(1)
Optimal rates at AUS BCE 𝜉 𝑠 = 0.02

(2)
Optimal rates at US BCE 𝜉 𝑠 = 0.008

𝜉

0.7349NA𝜋

NA0.5798𝜉𝑜0.66180.4641𝜏𝑠

0.15820.0284𝜏

0.10610.0047𝜏𝑜0.06010.1048

Note: 𝜏 𝑐 = 𝑃/𝑦 = 𝑃𝑜 /𝑦 = 0.

TABLE 5. Optimal health subsidies and income taxes (%) at steady states(1)
Optimal rates at AUS BCE(2)Optimal rates at US BCE

𝜉 𝑠 = 0.02, 𝜏𝑐 = 0.1

𝜉 𝑠 = 0.008, 𝜏𝑐 = 0.06

𝜉

0.6732NA𝜋

NA

0.5060𝜉𝑜0.58300.3700𝜏𝑠

0.15820.0284𝜏

0.0832-0.0109𝜏𝑜-0.1587-0.0525

Policy variables

Note: 𝑃/𝑦 = 𝑃𝑜 /𝑦 = 0.

FIGURE 1a. Sensitivity of optimal allocations to higher 𝜇 at steady state for Australia

FIGURE 1b. Sensitivity of optimal allocations to higher 𝜇 at steady state for the US

NOTE: miu is 𝜇.
40

FIGURE 2a. Sensitivity of optimal public health subsidies and optimal income taxes to higher
𝜇 at AUS benchmark 𝜉 𝑠 at steady state

FIGURE 2b. Sensitivity of optimal young-age EHI coverage, optimal old-age health subsidy
and optimal income taxes to higher 𝜇 at US benchmark 𝜉 𝑠 at steady state

NOTE: 𝜏 𝑐 = 𝑃/𝑦 = 𝑃𝑜 /𝑦 = 0, miu is 𝜇, zeta is 𝜉, pie is 𝜋, zetao is 𝜉 𝑜 , tau is 𝜏, tauo is 𝜏 𝑜 andtaus is 𝜏 𝑠 .

41

FIGURE 3a. Sensitivity of optimal EHI coverage and optimal income taxes to higher 𝜇 at
AUS benchmark 𝜉 𝑠 at steady state

FIGURE 3b. Sensitivity of optimal EHI coverage and optimal income taxes to higher 𝜇 at US
benchmark 𝜉 𝑠 at steady state

NOTE: 𝜏 𝑐 = 𝑃/𝑦 = 𝑃𝑜 /𝑦 = 0, miu is 𝜇, pie is 𝜋, pieo is 𝜋 𝑜 , tau is 𝜏, tauo is 𝜏 𝑜 and taus is 𝜏 𝑠 .

42Appendixes
A. Proof of Proposition 1. The conditions for optimal taxes and subsidies arise from equating
the equilibrium conditions with the socially optimal conditions. Substituting market clearing
conditions, consistency, zero profit conditions in production and health insurance provision,(5), (6), (v), (vi) and (vii) into the single budget constraint of the dynasty in (24) recovers
feasibility in (7). Substituting (iv) into (25) yields (9). Substituting consistency, (ii), (iii), (v)and (5) into (26) yields (10). Substituting consistency, (iii), (v), (5) and (14) into (27) attains(11). Substituting consistency, (i), (iii), (iv), (vii), (ix), (5), (25) and 𝑠𝑡 = 𝑘𝑡+1 𝑛𝑡 into (28)yields (12). Conditions (viii) and (ix) follow (22) and (23). Finally, condition (v) satisfies thegovernment balanced budget constraint.
B. Steady-state socially optimal allocation with specific functions. Using specific functions,the steady-state socially optimal allocation from conditions (7) to (12) is as follows:
(A1)

𝑐=

𝑦(1−𝛼𝜃)−ℎ−𝑚𝑇(ℎ,𝑚)/𝑛1,

1+(1/𝑛)(𝛽𝑛/𝛼)𝜎1
𝛽𝑛

1

𝑐

(A2)

𝑑 = ( 𝛼 )𝜎1 (𝑇(ℎ,𝑚)),(A3)𝑘 = 𝛼𝜃𝑦/𝑛,(A4)𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙) + (𝑛) 𝑇ℎ (ℎ, 𝑚)[𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − 𝑚] − 1 = 0,

(A5)

𝑓𝑚 (𝑘, Ω(ℎ, 𝑚)𝑙) + ( 𝑚(A6)𝜌𝑛2−𝜎2 𝑐 𝜎1 + 𝛼 [𝑐( 𝛼 )𝜎1 + 𝑚𝑇(ℎ, 𝑚)] −𝛼𝑇 (ℎ,𝑚)𝛽𝑛𝑛) [𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − 𝑚] −𝑇(ℎ,𝑚)𝑛= 0,

1

𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙)[𝑣𝑛2 + 𝛼𝑇(ℎ, 𝑚)] + 𝑦𝛼𝜃𝑛 = 0,1𝛼𝜃 𝜃 1−𝜃where 𝑦 = [𝐴 ( 𝑛 ) ]𝜇(z(ℎ, 𝑚)) 𝑙, 𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) = (1 − 𝜃)𝑦/𝑙,𝑓𝑘 (𝑘, Ω(ℎ, 𝑚)𝑙) = 𝜃𝑦/𝑘, 𝑇ℎ (ℎ, 𝑚) =𝑇(ℎ,𝑚)𝜓𝛿𝜙
𝑇(ℎ,𝑚)𝜓𝛿(1−𝜙), 𝑇𝑚 (ℎ, 𝑚) = [𝛿+𝜖𝑧(ℎ,𝑚)]𝑚 ,[𝛿+𝜖𝑧(ℎ,𝑚)]ℎ𝑓ℎ (𝑘, Ω(ℎ, 𝑚)𝑙) = 𝜇(1 − 𝜃)𝑦𝜙/ℎ, 𝑓𝑚 (𝑘, Ω(ℎ, 𝑚)𝑙) = 𝜇(1 − 𝜃)𝑦(1 − 𝜙)/𝑚.
From (A3) and the expression for 𝑓𝑘 , binding solvency requires 𝑅 = 𝑓𝑘 = 𝑛⁄𝛼 > 1.
C. Proof of Proposition 2. At the steady state, the first-order condition with respect to 𝑑 issimilar to (A2). First-order conditions (25) to (28) with specific functions are𝛼𝜃𝑦𝑛(1−𝜏𝑠 )(A7)

𝑠 = 𝑛(1−𝜉𝑠 )−𝛼𝜏𝑠,(A8)(𝑛) 𝑇ℎ (ℎ, 𝑚)[(1 − 𝜏 𝑜 )𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − (1 − 𝜋 𝑜 − 𝜉 𝑜 )𝑚] − (1 − 𝜋 − 𝜉) = 0,

𝛼

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(A9)

𝑇𝑚 (ℎ, 𝑚)[(1 − 𝜏 𝑜 )𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙) − (1 − 𝜋 𝑜 − 𝜉 𝑜 )𝑚] −(1 − 𝜋 𝑜 − 𝜉 𝑜 )𝑇(ℎ, 𝑚) = 0,

(A10) 𝜌𝑛2−𝜎2 𝑐 𝜎1 (1 + 𝜏 𝑐 ) − 𝑓𝑙 (𝑘, Ω(ℎ, 𝑚)𝑙)[(1 − 𝜏)𝑣𝑛2 + 𝛼(1 − 𝜏 𝑜 )𝑇(ℎ, 𝑚)] +𝛽𝑛1𝛼 [𝑐( 𝛼 )𝜎1 (1 + 𝜏 𝑐 ) + (1 − 𝜉 𝑜 )𝑚𝑇(ℎ, 𝑚) − 𝐼𝑜 𝜆𝑜 − 𝑃𝑜 − 𝜃𝑦𝑛 + 𝜏 𝑠 (𝜃𝑦𝑛 − 𝑠)] = 0.
Substituting (i) and 𝑠 = 𝑘𝑛 into (A7) yields (A3). Substituting (ii), (iii) and (v) into (A8) yields(A4). Substituting (ii) and (iii) into (A9) yields (A5). Substituting conditions (i) to (iv), (vii)and (ix), and 𝑠 = 𝑘𝑛 into (A10) yields (A6). Substituting market clearing conditions, the zeroprofit condition in production, the zero-profit condition for health insurance provision, thegovernment budget constraint, (vi) and (vii) into (24) can recover feasibility in (7) at the steadystate or optimal 𝑐 in (A1). Conditions (viii) and (ix) follow (22) and (23). Finally, conditions(i)-(iv) satisfy the government balanced budget constraint.

44