# The V alue of Life and the Rise in Health Spending

Robert E. Hall Hoo ver Institution and Department of Economics, Stanfor d Univer sity and NBER E-mail: rehall@stanford.edu http://stanford.edu/ ̃rehall and Charles I. Jones * Department of Economics, U .C. Berkele

y and NBER E-mail: chad@econ.berkeley.edu http://elsa.berkeley.edu/ ̃chad
October 18, 2005

— V

ersion 3.0

Health care extends life. Over the past half century, Americans spent a rising share of total economic resources on health and enjo yed substantially longer lives as a result. Debate on health policy often focuses on limiting the growth of health spending. W

e in vestig ate an issue central to this debate: Is the growth of health spending the rational response to changing economic conditions—notably the growth of income per person? W e develop a model based on standard economic assumptions and argue that this is indeed the case. Standard preferences—of the kind used widely in economics to study consumption, asset pricing, and labor supply—imply that health spending is a superior good with an income elasticity well abo ve one. As people get richer and consumption rises, the marginal utility of consumption f

alls rapidly

. Spending on health to extend life allows individuals to purchase additional periods of utility

. The marginal utility of life extension does not decline. As a result, the optimal composition of total spending shifts tow ard health, and the health share grows along with income. This ef fect exists despite sharp diminishing returns in the technology of life extension. In projections based on the quantitative analysis of our model, the optimal health share of spending seems likely to exceed 30 percent by the middle of the century
.
* W e are grateful to Da

vid Cutler, Amy Finkelstein, V ictor Fuchs, Alan Garber, Michael Grossman, Emmett K

eeler, Ron Lee, T

omas Philipson, Da

vid Romer, and participants at 12 HALL AND JONES 1. INTRODUCTION The United States devotes a rising share of its total resources to health care. The share w as 5.2 percent in 1950, 9.4 percent in 1975, and 15.4 percent in 2000. Over the same period, health has impro v ed. The life expectancy of an American born in 1950 w as 68.2 years, of one born in 1975, 72.6 years, and of one born in 2000, 76.9 years. Why has this health share been rising, and what is the likely time path of the health share for the rest of the century? W e present a framew ork for answering these questions. In the model, the key decision is the division of total resources between health care and non-health consumption. Utility depends on quantity of life—life expectancy—and quality of life—consumption. People v alue health spending because it allows them to liv e longer and to enjo y better liv es. In our analysis, the rise in the health share occurs because health is a superior good with an income elasticity well abo ve one. Income growth leads consumption and health spending to rise, and the marginal utilities of both consumption and health spending to f all. But saturation occurs f aster in consumption than in health spending. As people grow richer, consumption rises but they devote an increasing share of resources to health care. In our approach, standard preferences—of the kind economists use to study issues ranging from consumption to asset pricing to labor supply—are able to explain the rising share of health spending. As consumption increases, the marginal utility of consumption f alls quickly

. In contrast, extending life does not run into the same kind of diminishing returns. Living an additional year allows a person to enjo y the same new flow of utility as from previous extensions of lifetime. Many of the important questions related to health in volve the institutional arrangements that go v ern its financing—especially Medicare and emplo yernumerous seminars and NBER meetings for helpful comments. Jones thanks the Center for Economic Demography and Aging at Berkeley for financial support.THE V ALUE OF LIFE AND HEAL TH SPENDING 3
pro vided health insurance. One approach w ould be to introduce these institutions into our model and to examine the allocation of resources that results. W e take an alternative approach. W e examine the allocation of resources that maximizes social welf are in our model. W

e abstract from the complicated institutions that shape spending in the United States and ask a more basic question: from a social welf are standpoint, how much should the nation spend on health care, and what is the time path of optimal health spending? W e look at these issues from tw o points of view, first under the hypothesis that historical lev els of health care were optimal and second under the hypothesis that they were not. In the second case, we make progress by drawing on the results of a large body of existing research on the v alue of a statistical life. The recent health literature has emphasized the importance of technological change as an explanation for the rising health share—for example, see Newhouse (1992). According to this explanation, the in v ention of new and expensive medical technologies causes health spending to rise o v er time. Although the development of new technologies unquestionably plays a role in the rise of health spending, the technological explanation is incomplete for at least tw o reasons. First, expensive health technologies do not need to be used just because they are in

v ented. Although distortions in health insurance in the United States might result in o ver-use of expensiv e new technologies, health shares of GDP hav e risen in virtually every adv anced country in the w orld, despite wide v ariation in systems for allocating health care (Jones 2003). W e in v estig ate whether the social payof f associated with the use of new technologies is in line with the cost. Second, the in v ention of the new technologies is itself endogenous: Why is the U.S. in v esting so much in order to in v ent these expensiv e technologies? By focusing explicitly on the social v alue of extending life and how this v alue changes o ver time, we shed light on these questions.4 HALL AND JONES W e be gin by documenting the f acts about aggreg ate health spending and life expectancy, the tw o key v ariables in our model. W e then present a simple stylized model that makes some strong assumptions but that delivers our basic results. From this foundation, we consider a richer and more realistic framew ork and develop a full dynamic model of health spending. The remainder of the paper estimates the parameters of the model and discusses a number of projections of future health spending deriv ed from the model. Our approach is closest in spirit to the theoretical papers of Grossman (1972) and Ehrlich and Chuma (1990), who consider the optimal choice of consumption and health spending in the presence of a quality-quantity tradeof f. Our w ork is also related to a large literature on the v alue of life and the willingness of people to pay to reduce mortality risk. Classic references include Schelling (1968) and Usher (1973). Arthur (1981), Shepard and Zeckhauser (1984), Murphy and T

opel (2003), and Ehrlich and Y in (2004) are more recent examples that include simulations of the willingness to pay to reduce mortality risk and calculations of the v alue of life. Nordhaus (2003) and Becker, Philipson and Soares (2005) conclude that increases in longevity hav e been roughly as important to welf

are as increases in non-health consumption, both for the United States and for the w orld as a whole. W e build on this literature in tw o w ays. First and foremost, the focus of our paper is on understanding the determinants of the aggreg ate health share. The existing literature focuses on individual-lev el spending and willingness to pay to reduce mortality

. Second, we consider a broader class of preferences for longevity and consumption. Many earlier papers specialize for their numerical results to constant relativ e risk av ersion utility, with an elasticity of marginal utility that is between zero and one. In part, this restriction occurs because these papers do not consider a constant term in flow utility

. As we show below, careful attention to the constant is crucial to understanding the rising health share. In particular, when a constant is included, standard utilityTHE V ALUE OF LIFE AND HEAL TH SPENDING 5 functions that exhibit a rapidly declining marginal utility of consumption are admissible. This is the key to the rising health share in the model.
2. BASIC F

ACTS W e will be concerned with the allocation of total resources to health and other uses. W

e believ e that the most appropriate measure of total resources is consumption plus go vernment purchases of goods and services. That is, we treat in

vestment and net imports as intermediate products. Similarly, we measure spending on health as the delivery of health services to the public and do not include in

v estment in medical f

acilities. Thus we dif

fer conceptually (but hardly at all quantitativ ely) from other measures that include in v estment in both the numerator and denominator

. When we speak of consumption of goods and services, we include go v ernment purchases of non-health goods and services. Figure 1 shows the fraction of total spending devoted to health care, according to the U.S. National Income and Product Accounts. The numerator is consumption of health services plus go v ernment purchases of health services and the denominator is consumption plus total go v ernment purchases of goods and services. The fraction has a sharp upw ard trend, but growth is irregular

. In particular, the fraction grew rapidly in the early 1990s, flattened in the late 1990s, and resumed growth after 2000. Figure 2 shows life expectancy at birth for the United States. Following the tradition in demography, this life expectancy measure is not expected remaining years of life (which depends on unknown future mortality rates), but is life expectancy for a hypothetical individual who f aces the cross-section of mortality rates from a given year. Life expectancy has grown about 1.7 years per decade. It shows no sign of slowing o v er the 50 years reported in the figure. In the first half of the 20th century, howev er, life expectancy grew at about twice this rate, so a longer times series w ould show some curv ature.6 HALL AND JONES FIGURE 1. The Health Share in the United States

1950 1960 1970 1980 1990 2000 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Year Health Share

Note: The numerator of the health share is consumption of health services plus go vernment purchases of health services and the denominator is consumption plus total go vernment purchases of goods and services. For further information on sources, see Section 5.

3. BASIC MODEL W e be gin with a model based on the simple but unrealistic assumption that mortality is the same in all age groups. W e also assume that preferences are unchanging o v er time, and income and productivity are constant. This model sets the stage for our full model where we incorporate age-specific mortality and productivity growth. As we will show in Section 4, the stark assumptions we make in this section lead the full dynamic model to collapse to the simple static problem considered here. The economy consists of a collection of people of dif ferent ages who are otherwise identical, allowing us to focus on a representativ e person. Let x denote the person’ s state of health, which we will call health status . The mortality rate of an individual is the in v erse of her health status, 1 /x

. Since peopleTHE V ALUE OF LIFE AND HEAL TH SPENDING 7

FIGURE 2. Life Expectancy in the United States

1950 1960 1970 1980 1990 2000 68 70 72 74 76 78

Year Life Expectancy

Note: Life expectancy at birth data are from T able 12 of National V

ital Statistics Report V olume 51, Number 3 “United States Life T ables, 2000", December 19, 2002. Center for Disease Control.8 HALL AND JONES of all ages f

ace this same mortality rate, x is also equal to life expectancy. For simplicity at this stage, we assume zero time preference. Expected lifetime utility for the representativ e individual is

U ( c, x ) = Z ∞

0 e − (1 /x ) t u ( c ) dt = xu ( c )

. (1) That is, lifetime utility is the present v alue of her per-period utility u ( c ) discounted for mortality at rate 1 /x

. In this stationary en vironment, consumption is constant so that expected utility is the number of years an individual expects to live multiplied by per-period utility

. W e assume for now that period utility depends only on consumption; in the next section, we will introduce a quality-of-life term associated with health. Here and throughout the paper, we normalize utility after death at zero. Rosen (1988) pointed out the following important implication of a specification of utility in

volving life expectancy: When lifetime utility is per-period utility, u , multiplied by life expectancy, the lev el of u matters a great deal. In many other settings, adding a constant to u has no ef fect on consumer choice. Here, adding a constant raises the v alue the consumer places on longevity relativ e to consumption of goods. Ne g ative utility also creates an anomaly— indif ference curves have the wrong curv ature and the first-order conditions do not maximize utility

. As long as u is positiv e, preferences are well behaved. 1 The representativ e individual receiv es a constant flow of resources y that can be spent on consumption or health:

c + h = y . (2)

1 Rosen also discussed the following issue: If the elasticity of utility rises abo ve one for low v alues of consumption—as it can for the preferences we estimate in this paper —mortality becomes a good rather than a bad. A consumer w ould achieve a higher expected utility by accepting higher mortality and the correspondingly higher level of later consumption. Thus one cannot take expected utility for a given mortality rate as an indicator of the welf are of an individual who can choose a lower rate. This issue does not arise in our w ork, because we consider explicit optimization o ver the mortality rate. An opportunity for impro vement of the type Rosen identified w ould mean that we had not maximized expected utility

.THE V ALUE OF LIFE AND HEAL TH SPENDING 9 The economy has no physical capital or foreign trade that permits shifting resources from one period to another

. Finally, a health production function go verns the individual’ s state of health:

x = f ( h )

. (3) The social planner chooses consumption and health spending to maximize the utility of the individual in (1) subject to the resource constraint (2) and the production function for health status (3). That is, the optimal allocation solves

max c,h f ( h ) u ( c ) s.t. c + h = y . (4) The optimal allocation equates the ratio of health spending to consumption to the ratio of the elasticities of the health production function and the flow utility function. W

ith s ≡ h/y , the optimum is

s ∗ 1

− s ∗ = h ∗

c ∗ = η h η c , (5) where η h ≡ f 0 ( h ) h x , and η c ≡ u 0 ( c ) c

u . Now suppose we ignore the f act that income and life expectancy are taken as constant in this static model and instead consider what happens if income grows. The short-cut of using a static model to answer a dynamic question anticipates the findings of our full dynamic model quite well. The response of the health share to rising income depends on the mo v ements of the tw o elasticities in equation (5). The crux of our argument is that the consumption elasticity f

alls relativ e to the health elasticity as income rises, causing the health share to rise. Health is a superior good because satiation occurs more rapidly in non-health consumption. Why is η c decreasing in consumption? In most branches of applied economics, only marginal utility matters. For questions of life and death, howev er, this is not the case. W e hav e normalized the utility associated with death10 HALL AND JONES
at zero in our framew ork, and how much a person will pay to liv e an extra year hinges on the level of utility associated with life. In our application, adding a constant to the flow of utility u ( c ) has a material ef fect—it permits the elasticity of utility to v ary with consumption. Thus our approach is to take the standard constant-elastic specification for marginal utility but to add a constant to the level of utility

. In this w ay, we stay close to the approach of many branches of applied economics that make good use of a utility function with constant elasticity for marginal utility

. In finance, it has constant relativ e risk av ersion. In dynamic macroeconomics, it has constant elasticity of intertemporal substitution. In the economics of the household, it has constant elasticity of substitution between pairs of goods. What matters for the choice of health spending, however, is not just the elasticity of marginal utility, but also the elasticity of the flow utility function itself. W

ith the constant term added to a utility function with constantelastic marginal utility, the utility elasticity declines with consumption for con v entional parameter v alues. The resulting specification is then capable of explaining the rising share of health spending. W e specify flow utility as:

u ( c ) = b + c 1

− γ 1

− γ . (6)
Based on evidence discussed later in the paper, we consider γ > 1 to be likely

. In this case, the base level of utility, b , needs to be positiv e and large enough to ensure that flow utility is alw ays positiv e. The flow of utility u ( c ) is then bounded because the exponent on consumption is ne g ative. This means the elasticity η c is decreasing in consumption. More generally, any bounded utility function u ( c ) will deliver a declining elasticity, at least eventually, as will the unbounded u ( c ) = α + β log c . Thus the key to our explanation of the rising health share

— a marginal utility of consumption that f alls suf ficientlyTHE V ALUE OF LIFE AND HEAL TH SPENDING 11 quickly

— is obtained by adding a constant to a standard class of utility functions. An alternativ e interpretation of the first-order condition is also informativ e. Let L ( c, x ) ≡ U ( c, x ) /u 0 ( c ) denote the v alue of a life in units of output. Then, the optimal allocation of resources can also be characterized as

s ∗ = η h · L ( c ∗ , x ∗ ) /x ∗

y . (7) The optimal health share is proportional to the v alue of a year of life L/x divided by per-capita income. If the flow of utility is giv en as in equation (6), it is straightforw ard to show that the v alue of a year of life satisfies L ( c, x ) x = bcγ

− c

γ
− 1 . (8) For γ > 1 , the growth rate of the v alue of a life year approaches γ times the growth rate of consumption from abo ve. Therefore, the v alue of a year of life will grow f

aster than consumption (and income) if γ is larger than 1. According to equation (7), this is one of the key ingredients needed for the model to generate a rising health share. A rapidly-declining marginal utility of consumption leads to a rising health share pro vided the health production elasticity η h does not itself f all too rapidly

. For example, if the marginal product of health spending in extending life were to f

all to zero
— say it w as technologically impossible to live beyond the age of 100

— then health spending w ould cease to rise at that point. As we discuss later, for the kind of health production functions that match the data, the production elasticity declines v ery gradually, and the declining marginal utility of consumption does indeed dominate, producing a rising health share. Finally, we can also generalize the utility function to U ( c, x ) in place of
xu ( c ) , so that lifetime satisf action is not necessarily proportional to the length of the lifetime. The solution for this case is s ∗ / (1

− s ∗ ) = η h η x /η c , where η x ≡ U

x x/U is the elasticity of utility with respect to life expectancy. Our12 HALL AND JONES result, then, is that the health share rises when the consumption elasticity f alls f aster than the product of the production and life expectancy elasticities. As just one example U ( c, x ) = x α u ( c ) deliv ers a constant η x ev en with sharply diminishing returns to life expectancy (that is, α close to zero), so our main results are unchanged in this case. The simple model dev elops intuition, but it f alls short on a number of dimensions. Most importantly, the model assumes constant total resources and constant health productivity

. This means it is inappropriate to use this model to study how a growing income leads to a rising health share, the comparativ e static results not withstanding. Still, the basic intuition for a rising health share emerges clearly

. The health share rises o ver time as income grows if the jo y associated with living an extra year does not diminish as quickly as the marginal utility of consumption.

4. THE FULL DYNAMIC MODEL W e turn now to the full dynamic model, allowing age-specific mortality and the associated heterogeneity, as well as growth in total resources and productivity growth in the health sector
. This model also incorporates a quality-of-life component associated with health spending. An individual of age a in period t has an age-specific state of health, x a,t . As in the basic model, the mortality rate for an individual is the in v erse of her health status. Therefore, 1

− 1 /x a,t is the per-period surviv al probability of an individual with health x a,t . An individual’ s state of health is produced by spending on health h a,t :

x a,t = f ( h a,t ; a, t )

. (9) In this production function, health status depends on both age and time. Forces outside the model that v ary with age and time may also influence health status; examples include technological change and education.THE V ALUE OF LIFE AND HEAL TH SPENDING 13 The starting point for our specification of preferences is the flow utility of the individual, u a,t ( c a,t , x a,t )

. In addition to depending on consumption, flow utility depends on health status, x a,t . Spending on health therefore af fects utility in tw o w ays, by increasing the quantity of life through a mortality reduction and by increasing the quality of life. For reasons that will become clear in the empirical section, we also allow flow utility to depend on both time and age. For simplicity, we assume the time and age ef

fects are additiv e, so that

u a,t ( c a,t , x a,t ) = b a,t + u ( c a,t , x a,t ) (10) Here b a,t is the base v alue of flow utility for a person of age a and u ( c a,t , x a,t ) is the part that v aries with the current consumption and health status. Further

more, we assume the in

v ariant part of the utility function takes the following form:

u ( c a,t , x a,t ) = c 1

− γ a,t 1

− γ + α x 1

− σ

a,t 1

− σ , (11) where γ , α , and σ are all positiv e. The first part of this function is the standard constant-elastic specification for consumption. W e assume further that health status and consumption are additiv ely separable in utility and that quality of life is a constant-elasticity function of health status. In this en vironment, we consider the allocation of resources that w ould be chosen by a social planner who places equal weights on each person aliv e at a point in time and who discounts future flows of utility at rate β

. Let N a,t denote the number of people of age a alive at time t
. Then social welf are is

∞ X t =0 ∞ X
a =0 N a,t β t u a,t ( c a,t , x a,t )

. (12)
The optimal allocation of resources is a choice of consumption and health spending at each age that maximizes social welf are subject to the production14 HALL AND JONES function for health in (9) and subject to a resource constraint we will specify momentarily

. It is con v enient to express this problem in the form of a Bellman equation. Let V t ( N t ) denote the social planner’ s v alue function when the age distribution of the population is the v ector N t ≡ ( N

1 ,t , N 2 ,t , ..., N

a,t , ... )

. Then the Bellman equation for the planner’ s problem is V t ( N t ) = max { h a,t ,c a,t } ∞

X
a =0 N a,t u a,t ( c a,t , x a,t ) + β V t +1 ( N t +1 ) (13) subject to

∞

X
a =0 N a,t ( yt − c a,t − h a,t ) = 0 , (14)

N a +1 ,t +1 = 1

− 1

x a,t ! N

a,t , (15) N 0 ,t = N

0 , (16)

x a,t = f ( h a,t ; a, t )

. (17) yt +1 = e g y yt , (18) The first constraint is the economy-wide resource constraint. Note that we assume that people of all ages contribute the same flow of resources, y t . The second is the law of motion for the population. W e assume a large enough population so that the number of people aged a + 1 next period can be taken equal to the number aged a today multiplied by the surviv al probability

. The third constraint specifies that births are exogenous and constant at N 0 . The final tw o constraints are the production function for health and the law of motion for resources, which grow exogenously at rate g y . Let λ t denote the Lagrange multiplier on the resource constraint. The optimal allocation satisfies the following first order conditions for all a :

u c ( c a,t , x a,t ) = λ t , (19)THE V ALUE OF LIFE AND HEAL TH SPENDING 15 β ∂ V t +1 ∂ N a +1 ,t +1 · f 0 ( h a,t )

x 2 a,t + u x ( c a,t , x a,t ) f 0 ( h a,t ) = λ t , (20)
where we use f 0 ( h a,t ) to represent ∂ f ( h a,t ; a, t ) /∂ h a,t . That is, the marginal utility of consumption and the marginal utility of health spending are equated across people and to each other at all times. This condition together with the additive separability of flow utility implies that people of all ages hav e the same consumption c t at each point in time, but they have dif ferent health expenditures h a,t depending on age. Let v a,t ≡ ∂ V t ∂ N a,t denote the change in social welf are associated with having an additional person of age a alive. That is, v a,t is the social v alue of life at age a in units of utility

. Combining the tw o first-order conditions, we get:

β v a +1 ,t +1 u c + u x x 2

a,t u c = x 2

a,t f 0 ( h a,t ) , (21)
The optimal allocation sets health spending at each age to equate the marginal benefit of saving a life to its marginal cost. The marginal benefit is the sum of tw o terms. The first is the social v alue of life β v a +1 ,t +1 /u c . The second is the additional quality of life enjo yed by people as a result of the increase in health status. The marginal cost of saving a life is dh/dm , where dh is the increase in resources devoted to health care and dm is the reduction in the mortality rate. For example, if reducing the mortality rate by .001 costs $2000, then saving a statistical life requires 1 /. 001 = 1000 people to undertake this change, at a total cost of $2 million. Our model contains health status x as an intermediate v ariable, so it is useful to write the marginal cost as dh dm = dh/dx dm/dx . Since health status is defined as in

v erse mortality, m = 1 /x so that dm = dx/x 2

. In the previous example, we required 1 /dm people to reduce their mortality rate by dm to sav e a life. Equiv alently, setting dx = 1 , we require x 2 people to increase their health status by one unit in order to sav e a statistical life. Since16 HALL AND JONES the cost of increasing x is dh/dx = 1 /f 0 ( h ) , the marginal cost of saving a life is therefore x 2 /f 0 ( h )

. By taking the deriv ativ e of the v alue function, we find that the social v alue of life satisfies the recursive equation:

v a,t = u a,t ( ct , x a,t ) + β 1

− 1 x a,t ! v a +1 ,t +1 + λ t ( yt − ct − h a,t )

. (22)

The additional social welf are associated with having an extra person aliv e at age a is the sum of three terms. The first is the lev el of flow utility enjo yed by that person. The second is the expected social welf are associated with having a person of age a + 1 aliv e next period, where the expectation emplo ys the surviv al probability 1

− 1 /x a,t . Finally, the last term is the net social resource contribution from a person of age a , her production less her consumption and health spending. The literature on competing risks of mortality suggests that a decline in mortality from one cause may increase the optimal lev el of spending on other causes, as discussed by Dow, Philipson and Sala-i-Martin (1999). This property holds in our model as well. Declines in future mortality will increase the v alue of life, v a,t , raising the marginal benefit of health spending at age a .

4.1. Relation to the Static Model It is w orth pausing for a moment to relate this full dynamic model to the simple static framew ork. W ith constant income y , a time- and age-in v ariant health production function f ( h ) , β = 1 , and a flow utility function that depends only on consumption, the Bellman equation for a representative agent can be written as
V ( y ) = max

c,h u ( c ) + (1

− 1 /f ( h )) V ( y ) s.t. c + h = y . (23)THE V ALUE OF LIFE AND HEAL TH SPENDING 17 Given the stationarity of this en vironment, it is straightforw ard to see that the v alue function is V ( y ) = max c,h f ( h ) u ( c ) s.t. c + h = y , (24) the static model we developed earlier, restated in discrete time.

5. Q U ANTIT A TIVE ANAL

YSIS In the remainder of the paper, we estimate the parameters of our model and pro vide a quantitative analysis of its predictions. W e are conscious of uncertainty in the literature reg arding the v alues of many of the parameters in our model. The calculations that follow should be viewed as illustrativ e and suggestive, and we hav e done our best to indicate the range of outcomes one w ould obtain with other plausible v alues of the parameters. W e be gin by describing the data we use, then proceed to estimating the parameter v alues, and finally conclude with solving the model. W e assume a period in the model is fiv e years in the data. W e org anize the data into 20 five-year age groups, starting at 0–4 and ending at 95–99. W e consider 11 time periods in the historical period, running from 1950 through 2000. W e obtained data on age-specific mortality rates from T able 35 of National V ital Statistics Report V

olume 51, Number 3 United States Life T ables, 2000 , December 19, 2002, Center for Disease Control. This source reports mortality rates every 10 years, with age breakdowns generally in 10-year interv als. W e interpolated by time and age groups to produce estimates for 5-year time interv als and age categories. W e also obtained data on age-specific mortality rates due to accidents and homicides from Health, United States 2004 and from v arious issues of V

ital Statistics of the United States
. Data on age-specific health spending is taken from Meara, White and Cutler (2004). These data are for 1963, 1970, 1977, 1987, 1996, and 2000. Using18 HALL AND JONES the age breakdowns for these years, we distributed national totals for health spending across age categories, interpolated to our 5-year time interv als. National totals for health spending are from T able 2.5.5 of the revised National Income and Product Accounts of the Bureau of Economic Analysis, accessed at bea.go v on February 13, 2004 (for priv ate spending) and T able 3.15 of the previous NIP As, accessed December 2, 2003 (for go v ernment spending). Data on go vernment purchases of health services are no longer reported in the accounts. The empirical counterpart for our measure, y , of total resources is total priv ate consumption plus total go v ernment purchases of goods and services, from the sources described abo v e.
6. ESTIMA

TING THE HEAL

TH PRODUCTION FUNCTION W e be gin by assuming a functional form for the production function for health status. Our main approach treats mortality from accidents and homicides as exogenous and assumes health inputs af fect non-accident mortality

. The distinction between the tw o categories is especially important for older children and young adults, where health-related mortality is so low that declines in accidents account for a substantial part of the o verall trend in mortality

. W

e assume that the in

v erse of the non-accident mortality rate

—

 ̃ x a,t ≡ 1 /m non

a,t — is a Cobb-Douglas function of health inputs:

 ̃ x a,t = A a ( zt h a,t w a,t ) θ a
. (25)
In this production function, A a and θ a are parameters that are allowed to depend on age. z t is the ef ficiency of a unit of output devoted to health care, taken as an exogenous trend; it is the additional impro vement in the productivity of health care on top of the general trend in the productivity of goods production. The unobserved v ariable w a,t captures the ef fect of all other determinants of mortality, including education and pollution.THE V ALUE OF LIFE AND HEAL TH SPENDING 19 The production function for o verall health is therefore:

x a,t = f

a,t ( h a,t ) = 1

m acc a,t + m non

a,t = 1 m acc a,t + 1 /  ̃ x a,t , (26) where m acc is the exogenous mortality rate from accidents and homicides. 6.1. Identification T o explain our approach to identifying the parameters of this production function— A a and θ a — we introduce a new v ariable, s a,t ≡ h a,t /yt , the ratio of age-specific health spending to income per capita. W e rewrite our health production function as

 ̃ x a,t = A a ( zt yt · s a,t · w a,t ) θ a
. (27) The o v erall trend decline in age-specific mortality between 1950 and 2000 can then be decomposed into the three terms in parentheses. First is a trend due to technological change, z t yt . In our benchmark scenario, we assume technical change in the health sector occurs at the same rate as in the rest of the economy, so that z t = 1 is constant. Because y t rises in our data at 2.31 percent per year, this is the rate of technical change assumed to apply in the health sector
. In a robustness check, we assume technical change is f aster in the health sector, allowing z t to grow at one percent per year so that technical change in the health sector is 3.31 percent. The second cause of a trend decline in age-specific mortality is resource allocation: as the economy allocates an increasing share of per capita income to health spending at age a , mortality declines. This ef fect is captured by s a,t . Third, unobserved mo v ements of w a,t cause age-specific mortality to decline. W e hav e already remo v ed accidents and homicides from our mortality measure, but increases in the education of the population, declines in pollution, and declines in smoking may all contribute to declines in mortality

. The key assumption that allows us to identify θ a econometrically is that our observed trends

— technological change and resource allocation

— account20 HALL AND JONES for a known fraction μ of the trend decline in age-specific mortality

. For example, in our benchmark case, we assume that technical change and the increased allocation of resources to health together account for μ = 2 / 3 of the decline in non-accident mortality, leaving 1/3 to be explained by other f actors. As a robustness check, we also consider the case where these percentages are 50-50, so that μ = 1 / 2
. W e first discuss why this is a plausible identifying assumption and then explain exactly how it allows us to estimate θ a . A large body of research seeks to understand the causes of declines in mortality

. Newhouse and Friedlander (1980) is one of the early cross-sectional studies documenting a low correlation between medical resources and health outcomes. Subsequent w ork designed to solve the dif ficult identification problem (more resources are needed where people are sicker) hav e generally supported this finding (Newhouse 1993, McClellan, McNeil and Newhouse 1994, Skinner, Fisher and W ennberg 2001, Card, Dobkin and Maestas 2004, Finkelstein and McKnight 2005). This w ork often refers to “flat of the curve” medicine and emphasizes the low marginal benefit of additional spending. On the other hand, ev en this literature emphasizes that certain kinds of spending

— for example the “ef fectiv e care” category of W ennberg, Fisher and Skinner (2002) that includes flu v accines, screening for breast and colon cancer, and drug treatments for heart attack victims

— can hav e important effects on health. Goldman and Cook (1984) attribute 40 percent of the decline in mortality from heart disease between 1968 and 1976 to specific medical treatments; Heidenreich and McClellan (2001) take this one step further and conclude that the main reason for the decline in early mortality from heart attacks during the last 20 years is the increased use of medical treatments. Of course, a substantial part of “medical treatments” may include impro vements in technology (Cutler, McClellan, Newhouse and Remler 1998). Skinner et al. (2001) emphasize that technological adv ances hav e been responsible for “large av erage health benefits” in the U.S. population. Nevertheless, otherTHE V ALUE OF LIFE AND HEAL TH SPENDING 21 f actors including behavioral changes, increased education, and declines in pollution have certainly contributed to the decline in mortality (Chay and Greenstone 2003, Grossman 2005). While it w ould be a stretch to say there is a consensus, this literature is generally consistent with the identifying assumption made here. In particular, our identifying assumption leads to the following decomposition of the sources of age-specific mortality decline. A v eraged across our age groups, 35 percent is due to technological change, 32 percent to increased resource allocation to health, and 33 percent (by assumption) to other f actors. In our robustness check that assigns 50 percent to other f actors, the split is 26 percent to technological change and 24 percent to increased resource allocation. When we allow technical change to be a percentage point f aster in the health sector, 40 percent of the mortality decline is due to technical change, 27 percent to resource allocation, and 33 percent (by assumption) to unobserved f actors. Our assumption about the fraction μ implies that there is a trend in the unobserv able w a,t that accounts for a fraction 1

− μ of the impro v ement in mortality in age group a . The ratio of the trend in the unobserved component to the trend in the observed component based on technical change and rising resources is 1

− μ

μ . The remaining part of w a,t lacks any trend. W e call it  a,t . Thus log w a,t = 1

− μ μ (log zt yt + log s a,t ) +  a,t , (28) W e also normalize  a,t to hav e a zero mean. Mo v ements in  a,t may be correlated with technical change and resource allocation, an issue we address below. Using this equation to remo v e the unobserved w a,t from the production function, we have

log  ̃ x a,t = log A a + θ a

μ (log zt yt + log s a,t ) +  ̃  a,t , (29)22 HALL AND JONES where  ̃  a,t ≡ θ a  a,t is a mean-zero, trendless disturbance term in the production function. The absence of a trend in  ̃  a,t allows us to estimate θ a . W e use a linear time trend as an instrument in equation (29) to estimate the coef ficient on the second term, θ a μ . W e reco ver the true elasticity θ a from this coef ficient by multiplying by the known proportion μ . This adjustment remo ves the omitted-v ariable bias that w ould otherwise cause us to o v erstate the elasticity

. Following this discussion, we use GMM to estimate A a and θ a in equation (29). Our tw o orthogonality conditions are that  ̃  a,t has zero mean and that is has zero co v ariance with a linear time trend. Because h a,t is strongly trending, the trend instrument is strong and the resulting estimator has small standard errors. Figure 3 shows the GMM estimates of θ a , the elasticity of adjusted health status,  ̃ x , with respect to health inputs, by age category. The groups with the largest impro vements in health status o v er the 50-year period, the v ery young and the middle-aged, have the highest elasticities, ranging from 0.25 to 0.40. The f act that the estimates of θ a generally decline with age, particularly at the older ages, constitutes an additional source of diminishing returns to health spending as life expectancy rises. For the oldest age groups, the elasticity of health status with respect to health inputs is only 0.042. Figure 4 shows the actual and fitted v alues for tw o representativ e age groups. Because the health technology has tw o parameters for each age—intercept and slope—the equations are successful in matching the lev el and trend of health status. The same is true in the other age categories. From these estimates, we can calculate the marginal cost of saving a life at each age. Before turning to these calculations, we pro vide a summary of the empirical literature on the v alue of a statistical life (VSL), an alternativ e measure of the same concept, from the benefit side.THE V ALUE OF LIFE AND HEAL TH SPENDING 23 FIGURE 3. Estimates of the elasticity of health status with respect to health inputs

0 20 40 60 80 100

0 0.1 0.2 0.3 0.4 0.5

Age Note: The height of each bar reports our estimate of θ a , the elasticity of adjusted health status with respect to health inputs. The ranges at the top of the bars indicate ± tw o standard errors.

6.2. Evidence on the V

alue of a Statistical Life In ev aluating our results, three dimensions of the VSL literature are relev ant. W e are interested in (i) the lev el of the VSL, (ii) the rate at which the VSL changes o ver time, and (iii) how the VSL v aries with age. Most estimates of the lev el of the v alue of a statistical life are obtained by measuring the compensating dif ferential that w orkers receiv e in more dangerous jobs. V

iscusi and Aldy (2003) pro vide the most recent survey of this evidence and find estimates of the v alue of a statistical life that range from $4 million to $9 million, in year 2000 prices.24 HALL AND JONES

FIGURE 4. Estimation of the parameters of the health technology

50 60 70 80 90 100 110Health status, 35 − 39

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 2 4 6 8 10 12

Health inputHealth status, 65 − 69 Age 35 − 39, left scale

Age 65 − 69,

right scale

Note: The solid lines show actual health inputs h on the horizontal axis and health status, x , on the vertical axis, for tw o age groups, 35-39 and 65-69, for the period 1950 through 2000. The dashed lines show the fitted v alues from the estimated production function for health.THE V ALUE OF LIFE AND HEAL TH SPENDING 25 Ashenfelter and Greenstone (2004) pro vide an alternativ e approach to estimating the VSL. Their research design exploits the f act that states took dif ferential adv antage of the relaxation of federal mandatory speed limits that occurred in 1987. They find that a much lower number of $1.5 million (in 1997 prices) represents an upper bound on the VSL, suggesting that v arious problems including omitted v ariable bias and selection problems account for the higher estimates in the labor market literature. How does the v alue of life change o ver time? Recall that a rising v alue of life is crucial in this model to understanding the rising health share. Unfortunately, there is relatively little empirical evidence on changes in the v alue of life o ver time. Costa and Kahn (2003) appear to pro vide the first estimates from a consistent set of data on changes in the v alue of life in the United States. They use decennial census data from 1940 to 1980 and estimate the v alue of a statistical life in 1980 of $5.5 million (in 1990 dollars). Moreo ver, they find that this v alue has been rising o v er time at a rate equal to between 1.5 and 1.7 times the growth rate of per capita GDP

. Hammitt, Liu and Liu (2000) made a similar study for T

aiw an, combining a time series of cross-sections, and they estimate an elasticity of the v alue of a statistical life with respect to per capita GDP of between 2 and 3. Because life expectancy itself grows relativ ely slowly, these studies therefore support the key requirement in this paper that the v alue of a year of life as a ratio to per capita income is rising o v er time, and pro vide an estimate of how rapidly the rise occurs. A dif ferent approach to estimating changes in the v alue of life finds the opposite result, howev er. In addition to surveying the existing literature that estimates the v alue of life at a point in time, V iscusi and Aldy (2003) also conduct a “meta-analysis” to estimate the elasticity of this v alue with respect to income. Looking across some 60 studies from 10 countries, they re gress the av erage v alue of life estimates from each study on a measure of average26 HALL AND JONES income from each study and obtain an estimate of the elasticity of the dollar v alue of life with respect to income of about 0.5 or 0.6, with a 95 percent confidence interv al that is typically about 0.2 to 0.8. This finding appears to be consistent with several other estimates from dif ferent meta-analysis studies that are also summarized by V iscusi and Aldy

. Some additional insight on this issue comes from looking back at our model. Recall that equation (8) in the simple model suggests that the v alue of life as a ratio to life expectancy is roughly proportional to consumption raised to the power γ . That is, in units of output, the v alue of a year of life grows with
cγ

. One w ay of thinking about γ is that it is the in v erse of the intertemporal elasticity of substitution, which recent empirical w ork estimates to be less than one. This suggests that γ > 1 , and in f act the v alues that Costa and Kahn (about 1.6) and Hammitt, Liu, and Liu (about 2 or 3) find accord well with this interpretation. Kaplow (2003) puzzles o v er the low income elasticity estimates from the meta-analysis literature for a similar reason; the recent empirical w ork by Costa and Kahn and Hammitt, Liu, and Liu helps to resolve this puzzle, we think. Finally, we turn to evidence on v ariation in the v alue of a statistical life by age. Aldy and V

iscusi (2003) summarize the existing empirical literature, which primarily consists of contingent v aluation studies. They go on to pro vide their own age-specific estimates using the hedonic w age re gression approach. Qualitativ ely, they support the contingent v aluation literature in finding an in

verted-U shape for VSL by age. Quantitativ ely, their main finding is that the v alue of life for a 30 to 40-year old is about $5.5 million while the v alue of life for a 60-year old is about $2.5 to $3.0 million, a gradient of about 1/2 across these age groups. T o summarize, we take the following stylized f acts from the VSL literature. First, there is substantial uncertainty re g arding the level of the VSL: it could be as low as 1.25 million in the late 1980s, but could range much higher toTHE V ALUE OF LIFE AND HEAL TH SPENDING 27 numbers like 5 million or more. These numbers are plausibly interpreted as the v alue of life at some av erage age, which we will take to be the 35 to 39-year olds. Second, recent estimates suggest that the VSL grows o v er time, at a rate something like 1.6 or 2 times the growth rate of income. Finally, it appears that the VSL v aries with age in an in verted-U pattern, with a relatively gentle slope, f

alling by about 1/2 between the ages of 35 and 60. 6.3. The Marginal Cost of Saving a Life Our estimates of the health production function allow us to calculate the marginal cost of saving a life, given the observed allocation of resources. Recall, from the discussion surrounding equation (21), that this marginal cost is x 2 /f 0 ( h )

. W

ith our functional form for the health technology, the marginal cost of saving a life is h  ̃ x/θ

. This expression has a nice interpretation:  ̃ x is the in v erse of the non-accident mortality rate, so it can be thought of as the number of living people per non-accident death. h is health spending per person, so h  ̃

x is the total amount of health spending per death. The division by θ adjusts for the f

act that we are interested in the marginal cost of saving a life, not the average. Importantly, this calculation only in volves the health production function. For this part of the paper, the preference side of the model is irrelev ant. T able 1 shows this marginal cost of saving a life for v arious age groups. W e can interpret these results in terms of the three findings from the empirical VSL literature. First, the marginal cost of saving the life of a 40-year old in the year 2000 w as about $1.9 million. In our robustness checks, this marginal cost reached as high as $2.5 million (in the case where θ a is identified with the assumption that only 1/2 rather than 2/3 of declines in mortality are due to technical change and resource allocation). These numbers are at the lower end of the lev el estimates of the VSL from the literature. If one believ es the lower numbers, this suggests that health spending w as at approximately the right28 HALL AND JONES

T ABLE 1. The Marginal Cost of Saving a Life (thousands of 2000 dollars) Robust Per Y

ear of Growth Maximum Life Sa

ved Rate Age 1950 1980 2000 2000 2000 1950–2000 0-4 10 160 590 (790) 8 7.8 10-14 270 2,320 9,830 (13,110) 150 7.2 20-24 1,170 3,840 8,520 (11,360) 153 4.0 30-34 500 2,120 4,910 (6,540) 107 4.6 40-44 160 740 1,890 (2,520) 52 4.9 50-54 70 330 1,050 (1,400) 38 5.4 60-64 50 280 880 (1,180) 47 5.9 70-74 40 280 790 (1,050) 66 6.2 80-84 40 340 750 (1,000) 123 6.1 90-94 50 420 820 (1,090) 373 5.6 Note: The middle columns of the table report estimates of the marginal cost of sa ving a life for v arious age groups. These estimates are calculated as h  ̃ x /θ , using the estimates of θ given in Figure 4 and using actual data on health spending and mortality by age. Standard errors for these v alues based on the standard errors of θ a are small. The Robust Maximum column shows the maximum marginal cost we obtained in the v arious robustness checks described in the text. These v alues turned out to occur in the case where we assume that 50% of the decline in age-specific mortality is due to the unobserved trend in w a,t . The “Per Y ear of Life Sa

ved” column divides the cost of sa ving a life by life expectancy at that age. The “Growth Rate” column reports the a verage annual growth rate between 1950 and 2000.

lev el as a whole for this age group in 2000. Alternativ ely, of course, if one believ es the higher estimates of the VSL from the literature, the calculation from T able 1 w ould suggest that health spending for this group w as too low. The second-to-last column of the table pro vides an alternative view of the marginal cost of saving a life by stating the cost per year of life sav ed. It shows the cost of saving a statistical life in the year 2000, divided by life expectancy at each age. For example, the marginal cost of saving an extra year of life at age 50 is about $38,000. Interestingly, the cost of saving a life year in the youngest age category is only about $8,000, while the cost forTHE V ALUE OF LIFE AND HEAL TH SPENDING 29

saving a life year for the oldest ages rises to well abo ve $100,000. W e discuss the implications of this finding in a later section. Finally, the last column of the table shows the growth rate for the marginal cost of saving a life. Since the marginal cost is h  ̃ x /θ and θ is constant o v er time, these growth rates do not depend at all on the estimates of θ

. The growth rates are very high, on the order of 5 percent per year. By comparison, the empirical VSL literature finds significantly lower growth rates. T aking income growth to be about 2 percent per year, for example, the income elasticity from Costa and Kahn (2003) of about 1.6 suggests that the VSL grows at a rate of 2 × 1

. 6 = 3

. 2 percent per year. This implies that the v alue of life in 1950 or 1960 w ould hav e been much higher than the marginal cost of saving a life. Therefore, the U.S. may hav e been spending too little on health prior to the most recent decade, ev en taking the lev el of the VSL from the lower end of the estimates.
7. ESTIMA

TING THE PREFERENCE P ARAMETERS W e present results for tw o approaches. The first treats the observed lev els of health spending as optimal and estimates the preference parameters. The second estimates preference parameters from the evidence in the empirical VSL literature; it implies that health spending w as inef ficiently low until the end of the 20th century
. 7.1. T reating Observed Health Spending as Optimal Our model contains the following preference parameters: the discount factor β , the base lev els of flow utility b a,t , the consumption parameter γ , the quality-of-life parameter σ , and the weighting parameter α . For the moment, we consider the case where health status does not af fect flow utility so that
α = 0
. W e will reintroduce quality-of-life shortly

.30 HALL AND JONES W e hav e explored a v ariety of parametric restrictions on the base utility, b a,t . These include making it a constant for all ages and years, making it v ary by age, and giving it a trend o v er time. The evidence in f avor of age ef fects is strong. There is evidence of trends in base utility, but not at the same rate for dif

ferent age groups. W

e hav e not found a useful parametric restriction—candidates such as a set of age ef fects and a set of time ef

fects result in suf ficiently large residuals that the other parameters take on improbable v alues. Accordingly, we treat the v alues of b a,t as parameters themselv es, without imposing any restriction. Because there is one of these parameters for each data point, estimation is a matter of solving for the v alues, not minimizing a GMM norm or other criterion. Further, this means there are not enough equations to estimate the other tw o parameters, β and γ . W e use outside evidence on these parameters before solving for the v alues of b a,t . For the curv ature parameter of the utility function, γ , we look to other circumstances where curv ature af fects choice. Large literatures on intertemporal choice (Hall 1988), asset pricing (Lucas 1994), and labor supply (Chetty 2005) each suggest that γ = 2 is a reasonable v alue. Recall that higher v alues of γ lead to f

aster growth in the v alue of life and therefore w ould deliv er ev en more rapid growth in the health share than what follows. W ith respect to the discount f actor, β , we choose a v alue that is consistent with this choice of γ and with a 5 percent real return to saving. T aking consumption growth from the data of 2.08 percent per year, a standard Euler equation gives an annual discount f actor of 0.992, or, for the 5-year interv als in our model, 0.963. Given the data and the v alues of θ a , β , and γ , we first calculate the implied v alue of life from equation (21) and then reco ver the base levels of utility from a rearranged version of equation (22). Figure 5 shows the results of the calculations. Each line portrays the base lev el of utility for ev ery age group in a particular year, for the 11 years at 5-year interv als from 1950 though 2000.THE V ALUE OF LIFE AND HEAL TH SPENDING 31 FIGURE 5. Estimates of base flow utility, b a,t

0 20 40 60 80 100 −400 −200 0 200 400 600

Solved

Age Note: Each line shows the cross section of base levels of utility in a given year. The periods co ver 5 years each from 1950 through 2000. The hea vy dashed line labeled “Solved” reports the set of base v alues inferred from the v alue of life in 2000 according to the model.

The lines share a common pattern—neg ativ e flow utility in the youngest group and usually in the second-youngest group, and also ne g ativ e flow utility for teenagers. Ne g ativity of flow utility does not contradict any principles of the model. The motiv ation for continuing to liv e is to capture next period’ s v alue of life. Ne g ative flow utility marks a dif ficult period of life that people choose to liv e through so that they can enjo y later periods with positiv e utility

. For older people, flow utility stabilizes at a common, lower positiv e lev el o v er all periods. Flow utility rises somewhat for the very elderly

. Alternatively, we can interpret the solved v alues for b a,t as the residuals from the first-order condition in a model with a constant b

. Economically, they arise because the marginal cost of saving a life—the right side of equation (21),32 HALL AND JONES with v alues shown in T

able 1—v aries considerably more across ages than the v alue of life on the preference side w ould in the absence of v ariation in b

. That is, with a constant b across ages, the v alue of life on the preference side—the left side of equation (21)—turns out to be relativ ely flat across ages. For example, consider the marginal cost of saving a life reported in T able 1. The w ay the model interprets the f act that we spend so little on health care for children from 0 to 4 and so much on those between 5 and 9, relativ e to the marginal mortality reductions from spending, is by having a substantially lower b for the younger group. In the alternative interpretation of the estimates of b a,t as residuals, they could be seen as measures, at dif ferent times, of the misallocation of health care. The institutions that go vern the deliv ery of health care may systematically f ail to take adv antage of opportunities to reduce inf ant mortality, for example. This interpretation helps explain our finding that the b a,t do v ary o ver time, contrary to the hypothesis that they are unchanging parameters of preferences. These calculations pro vide estimates of the base lev el of utility during the historical period. For our projections for the next 50 years, we need future v alues of the base utility parameters. For this purpose, we make use of additional information, namely the lev el of the v alue of life in utility units from equation (21) in the last historical year, 2000. This level information is not used in the calculation of the historical v alues of b a,t from equation (22), which is in dif

ference form. T

o make use of the lev el information, we hypothesize that b a,t will not change o v er the future from its v alues in 2000. This hypothesis makes sense, because there is no systematic trend in the historical v alues in Figure 5. Then we proceed in the following w ay: When we solve the model for the years 2000 through 2095, we treat b a as a set of unknowns to solve and then require that the model solution match the v alue of life in 2000. The heavy dashed line in Figure 5 shows these solv ed v alues.THE V ALUE OF LIFE AND HEAL TH SPENDING 33 Except for the more erratic v alues for the younger groups, the match is quite good. 7.2. Matching the Earlier V alue of Life Estimates As an alternative approach to estimating the preference parameters, we drop the assumption that the observed data are generated by maximizing social welf

are giv en our estimated health technology

. Instead, we take the age-specific spending data and the consumption data as giv en and compute the v alue of life at each age, β v a +1 ,t +1 /u c , from these data. For future v alues of health spending by age, we project the existing data forw ard at a constant growth rate. Until the year 2020, this growth rate is the av erage across the age-specific spending growth rates. After 2020 we assume spending grows at the rate of income growth. The rate must slow at some point; otherwise the health share rises abo ve one. Our results are similar if we delay the date of the slowdown to 2050. W e then estimate a constant and common v alue b a,t = b and the curv ature parameter γ to match some estimates from the VSL literature. Our baseline scenario features a v alue of life for 35–39 year olds of $2 million in 1987. W e project this back to 1950 and forw ard to 2000, using a growth rate of 1

. 6 × 2

. 31 = 3

. 70 percent per year, based on the Costa and Kahn income elasticity

. By matching the v alue of life for this age group in 1950 and 2000, we obtain b = 21

. 637 and γ = 1

. 624 for the case where health status does not af fect flow utility (i.e. α = 0 ). Finally, we recalibrate the time discount f actor β to an interest rate of 5 percent based on this new v alue of γ .

7.3. The Quality-of-Life Parameters T o calibrate the quality-of-life parameters σ and α , we draw upon the extensiv e literature on quality-adjusted life years (QAL Ys), elicited by surv eying sick and healthy people, medical experts, and others—see Fryback et al.34 HALL AND JONES (1993) and Cutler and Richardson (1997). This w ork focuses on the QAL Y weight, the flow utility lev el of a person with a particular disease as a fraction of the flow utility lev el of a similar person in perfect health. Surveys ask people what probability p of perfect health with probability 1

− p of certain death w ould make them indif ferent to their current health or what fraction of a year of future perfect health w ould make them indif ferent to a year in their current health status. Both of these measures correspond to the relativ e flow utility in our framew ork. Cutler and Richardson (1997) estimate QAL Y weights by age. W

ith newborns normalized to have a weight of unity, they find QAL Y weights of 0.94, 0.73, and 0.62 for people of ages 20, 65, and 85, in the year 1990. W e use these figures to estimate α and σ . For the case where b a,t is constant across age and time, we use the tw o equations,

u ( ct , x 20 ,t )

. 94 = u ( ct , x 65 ,t )

. 73 = u ( ct , x 85 ,t )

. 62 , for t = 1990 to solve for α and σ . Because the v alue of life itself depends on these parameters, we estimate the utility parameters b and γ at the same time. The resulting v alues are α = 1

. 922 , σ = 1

. 051 , b = 54

. 17 , and γ = 1

. 593
. W ith four equations and four unknowns, estimation is a matter of solving for the v alues, so there are no standard errors. In addition to the QAL

Y interpretation, these numbers can be judged in another w ay. They imply that a 65 year-old w ould giv e up 88 percent of her consumption, and an 85 year-old w ould give up 93 percent of her consumption to hav e the health status of a 20 year-old. The intuition behind these large numbers is the sharp diminishing returns to consumption measured by γ . T o explain what may seem to be a small dif ference in relativ e utilities of .94 v ersus .73 requires large dif ferences in consumption. Health is extremely v aluable.THE V ALUE OF LIFE AND HEAL TH SPENDING 35 8. SOL VING THE MODEL W e now solve the model o v er the sample period 1950 through 2000 and also project the economy out to the year 2050. W e solve the model using both of our approaches to calibrating the preferences parameters (the b a,t and γ ) and using tw o approaches to the quality of life ( α = 0 and α > 0 ). For the historical period, we take resources per person, y , at its actual v alue. For the projections, we use the historical growth rate for the sample period, 2.31 percent per year. The details for the numerical solution of the model are av ailable from either author’ s website. Figure 6 shows the calculated share of health spending o v er the period 1950 through 2050. A rising health share is a robust feature of the optimal allocation of resources in the health model. The key force at w ork in the model behind this result is that the marginal utility of consumption in a giv en period f alls rapidly

. As the U.S. gets richer and richer, the most v aluable thing people can purchase is more time to liv e. The figure shows a substantial dif ference between projected health shares for the tw o approaches. Our first approach matches the actual health share between 1950 and 2000 exactly

. The projection based on that approach implies a rapidly growing health share in the future, reaching 33 percent in 2050. The second approach, based on the VSL estimates in the literature, produces a much flatter health share. Fundamentally, this slower rate of increase is driv en by the lower v alue of γ (1.6 v ersus 2.0) used in our second approach; recall that γ go v erns the growth rate of the v alue of life and therefore determines the growth rate of the health share. This second approach suggests underspending on health for the last 50 years and generates a health share that reaches 29 percent in 2050. W e conduct a number of robustness checks to illustrate how our benchmark results change when key parameter v alues are v aried in plausible w ays. For example, Figure 6 also shows what happens when health quality af fects utility

.36 HALL AND JONES FIGURE 6. Simulation Results: The Health Share of Spending

1950 2000 2050 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Matching the value of life

( α =0) ( α >0)
Treating observed health spending as optimal

Actual

Year Health Share, s

Circles “o” show actual data for the health share. The steeply-sloping line for the period 2005-2050 show the projected health share assuming γ = 2 , where the b a preference parameters are inferred from treating the historical data as if it were generated by the model. The gently sloping lines for the period 1950-2050 show the hypothetical historical and projected share for preferences inferred from the VSL literature (e.g. γ = 1
. 62 ). W ithin these tw o approaches, the upper line corresponds to the case that includes a quality-of-life term ( α > 0 ), while the lower line does not ( α = 0 ).

Our calibration of α and σ for this case implies a high willingness to pay for health quality

. As a result, o v erall health spending is higher in this scenario. While our benchmark case leads to an optimal health share of just under 20 percent in 2000, allowing for quality of life in utility raises the share to 28 percent. The o verall trend in the health share is quite similar

. Figure 7 shows several other robustness checks. Allowing for technical change in the health sector to be one percentage point f aster than in the rest of the economy or reducing the share of mortality decline explained by technical change and resource allocation from 2/3 to 1/2 deliv er relativelyTHE V ALUE OF LIFE AND HEAL TH SPENDING 37 FIGURE 7. Robustness Checks: The Health Share of Spending

1950 2000 2050 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Actual

Matching VSL: Faster Tech. Matching VSL: 50% health Matching VSL: $3 million Optimal: 50% and Tech.

Year Health Share, s

Circles “o” show actual data for the health share. This graph shows five alternative simulations. For tw o of these alternatives, “Faster T ech.” and “50% health,” we consider both the approach that treats observed spending as optimal and the approach that matches a v alue of life in 1987 of $2 million. “Faster T ech.” assumes that technical change in the health sector is 1 percentage point f aster than in the rest of the economy. “50% health” assumes that 1/2 of the decline in age-specific mortality (rather than our baseline v alue of 2/3) is due to technological change and increased resource allocation. Finally, the fifth case matches a v alue of life in 1987 for the 35–39 year olds of $3 million.

similar results. In both of these cases, less of the decline in age-specific mortality is due to health spending, so the estimates of θ a in the production function are smaller

. Since health spending runs into sharper diminishing returns, the o v erall health share of spending is lower. These simulations suggest that the observed share in the year 2000 w as roughly optimal. Alternativ ely, another robustness check in the figure assumes the v alue of a stastical life is $3 million dollars in 1987 rather than $2 million. This means the marginal benefit of health spending is higher, so the simulation deliv ers38 HALL AND JONES FIGURE 8. Health Spending by Age

0 20 40 60 80 100 400 1000 3000 8000 20000 60000

1950 2000 2050

Age Constant 2000 dollars

Note: Circles denote actual data and solid lines show simulation results for the baselines scenario in which b and γ are chosen to match estimates from the VSL literature.

a substantially higher health share. In 2000, for example, the optimal health share is 26 percent, and it rises to 36 percent by 2050. Figure 8 examines the micro data underlying the health share. This figure shows actual and simulated health spending by age, for 1950, 2000, and 2050 for our second approach in the baseline scenario (in the first approach, actual and simulated spending are equated by construction). A comparison of the results for the year 2000 shows that actual and optimal spending are f airly similar for most ages, with tw o exceptions. Optimal health spending on the youngest age group is substantially higher than actual spending: given the high mortality rate in this group, the marginal benefit of health spending is v ery high, as w as shown earlier

. Similarly, while optimal health spending generally rises until age 80, it declines after that point. It is w orth noting inTHE V ALUE OF LIFE AND HEAL TH SPENDING 39 FIGURE 9. Simulation Results: Life Expectancy at Birth

1950 2000 2050 68 70 72 74 76 78 80 82

Matching the value of life Treating observed health spending as optimal

Actual

Year Life Expectancy

See notes to Figures 6 and 7. Life expectancy is calculated using the cross-section distribution of mortality rates at each point in time.

this respect that the underlying micro data we use for health spending groups all ages abo ve 75 together, so we do not know what the actual pattern of spending looks like abo ve the age of 75. Figure 9 shows the actual and projected levels of life expectancy at birth for all eight of our simulation runs. The first thing to note in the figure is the o verall similarity of the life expectancy numbers. Because there are such sharp diminishing returns to health spending in our health production function, relativ ely large dif ferences in health spending lead to relatively small dif ferences in life expectancy. A second thing to note is that the projected path does not grow quite as f ast as historical life expectancy. The reason is ag ain related to the relativ ely sharp diminishing returns to health spending that we estimate.40 HALL AND JONES 9. CONCLUDING REMARKS A model based on standard economic assumptions yields a strong prediction for the health share. Pro vided the marginal utility of consumption f alls suf ficiently rapidly—as it does for an intertemporal elasticity of substitution well under one—the optimal health share rises o ver time. The rising health share occurs as consumption continues to rise, but consumption grows more slowly than income. The intuition for this result is that in any given period, people become saturated in non-health consumption, driving its marginal utility to low lev els. As people get richer, the most v aluable channel for spending is to purchase additional years of life. This fundamental mechanism in the model is supported empirically in a number of dif

ferent w ays. First, as discussed earlier, it is consistent with con v entional estimates of the intertemporal elasticity of substitution. Second, the mechanism predicts that the v alue of a statistical life should rise f aster than income o ver time; Costa and Kahn (2003) and Hammitt et al. (2000) find this to be the case. Cross-country evidence also suggests that health spending rises more than one-for-one with income; this evidence is summarized by Gerdtham and Jonsson (2000). One source of evidence that runs counter to our prediction is the micro evidence on health spending and income. At the individual level within the United States, for example, income elasticities appear to be substantially less than one, as discussed by Newhouse (1992). A serious problem with this existing evidence, however, is that health insurance limits the choices f acing individuals, potentially explaining the absence of income ef fects. Our model makes a strong prediction that if one looks hard enough and carefully enough, one ought to be able to see income ef fects in the micro data. Future empirical w ork will be needed to judge this prediction. A suggestiv e informal piece of evidence is that exercise seems to be a luxury good: among people withTHE V ALUE OF LIFE AND HEAL TH SPENDING 41 sedentary jobs, high w age people seem to spend more time exercising than low w age people, despite the higher opportunity cost of their time. As mentioned in the introduction, the recent health literature has emphasized the importance of technological change as an explanation for the rising health share. In our view, this is a proximate rather than a fundamental explanation. The dev elopment of new and expensiv e medical technologies is surely part of the process of rising health spending, as the literature suggests; Jones (2003) pro vides a model along these lines with exogenous technical change. However, a more fundamental analysis looks at the reasons that new technologies are developed. Distortions associated with health insurance in the United States are probably part of the answer, as suggested by W eisbrod (1991). But the f

act that the health share is rising in virtually ev ery adv anced country in the w orld—despite wide v ariation in systems for allocating health care—suggests that deeper forces are at w ork. A fully-w orked out technological story will need an analysis on the preference side to explain why it is useful to in

vent and use new and expensiv e medical technologies. The most ob vious explanation is the model we propose in this paper: new and expensive technologies are v alued because of the rising v alue of life. V iewed from every angle, our results support the proposition that both historical and future increases in the health spending share are desirable. The magnitude of the future increase depends on parameters whose v alues are known with relativ ely low precision, including the v alue of life, how rapidly that v alue has grown o v er time, and the fraction of the decline in agespecific mortality that is due to technical change and the increased allocation of resources to health care. Nev ertheless, we believ e it likely that maximizing social welf

are in the United States will require the development of institutions that are consistent with spending 30 percent or more of GDP on health by the middle of the century
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