Health as human capital: synthesis and extensions1
Department of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, USA, and Hoover Institution; e-mail: gbecker{at}uchicago.edu
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1. Introduction |
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 TOP 1. Introduction 2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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The first major collection of articles on human capital (Schultz, 1962
) contained discussions of education, on the job training, migration, and health. While literally thousands of articles and books followed on human capital dimensions of education and training, there have been many fewer discussions of health as human capital. This is partly because the contribution on health in this volume is not particularly insightful, but it is also because the concept of health as human capital relies on somewhat different concepts than does education or training.
A major step forward occurred with Grossman's work (Grossman, 1972) that modeled optimal investment in increasing longevity. This article stimulated a large literature, but nevertheless, articles on health as human capital have been only a small fraction of those on education and training. In fact, most of the economics literature on health discusses ways to improve the delivery of health care services, such as HMO's or health savings accounts. Health care delivery is an important topic that interacts with considerations of health as human capital, but it is mainly a different topic.
The emerging field of health as human capital builds on three interrelated developments to create a dynamic and evolving field. These developments are (i) The analysis of optimal investments in health by individuals, drug companies, and to a lesser extent by governments that follows on Grossman's analysis, and also on the discussions in the insurance literature of self protection (see Ehrlich and Becker, 1972; and Ehrlich, 2000), and the literature on investments by pharmaceuticals. (ii) The value of life literature that analyzes how much people are willing to pay for improvements in their probabilities of surviving different ages (see, especially, Usher 1973; Rosen, 1988; and Murphy and Topel, 2006). (iii) The importance of complementarities in linking health to education and other types of human capital investments, and in linking investments in health to discount rates, to progress in fighting different diseases (see the discussion of competing risks in Dow et al., 1999; and Murphy and Topel, 2006) and to other sources of overall changes in survivorship rates.
This paper will present the resulting emerging theory of health as human capital. It builds on and integrates various contributions, but it adds to the literature by incorporating a few relevant ideas that have been ignored. The paper also refers to some of the evidence that demonstrates the empirical importance of health as human capital. This evidence covers general advances in health for the past several decades, and advances in treating a few specific diseases.
Table 1 presents a strong motivation for an interest in the economic value of improved life expectancy. It shows that life expectancy at birth hovered around 40 years in the 19th century, even in countries like the United Kingdom and the United States that were among the very richest. Life expectancy improved only by a few years during that century, but really took off during the 20th century. Life expectancy at birth hit the mid-sixties by 1950, and was close to 80 years by the beginning of the twenty first century. I believe the decline in mortality at all ages was among the most significant economic and social developments of the twentieth century.
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2. Theory |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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2.1 The statistical value of life
The fundamental equation for determining the so-called ‘statistical value of life’ for any individual is the following:
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| (1) |
S (see Becker et al., 2005, that extends Usher, 1973). The left hand side gives the ‘compensating variation’ in wealth; by that is meant the increment in wealth from Wo that makes the person just as well off with So as he would be with the initial wealth and the improved survivorship.
Since V is rising in both W and S, v = W/S is positive, and measures the ‘statistical value of life’ for this person who has wealth Wo, is faced with an improvement in survivorship of S from a level of So, is age A, and faces interest rate r. Given the usual properties of utility functions, it follows that v is rising in initial wealth Wo, falls as r increases, and generally falls with age, although not necessarily monotonically. It is crucial to recognize that the statistical value of life considered by economists is not a constant even for the same person, but varies with his wealth, age, level of survivorship, magnitude of the changes in survivorship, and perhaps other variables.
For example, suppose So = 0, so that the initial survivorship condition is to die with certainty, while S > 0, so that the change makes it possible to live with perhaps a significant probability. The change may be due to a life-saving drug, surgery, etc. How large would W be then? At one extreme, W may equal Wo for any S > 0, so that a person would be willing to give all his wealth to avoid dying. Or W may be less than Wo because utility is received from bequests. The large amounts that might be spent to delay death would help explain why a large fraction of health expenditures are made at older ages when people spend on surgeries or other medical interventions to avoid dying (see Becker et al., 2006, in progress).
2.2 Expected utility
The common approach is take a more specialized formulation than eq. (1), and to specify the utility function to a discounted value of expected utilities at different ages, as in
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| (2) |
The unconditional probability of surviving to age i is the product of the conditional probabilities of surviving various ages:
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| (3) |
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| (4) |
For expositional simplicity I mainly use a two period formulation, where S0 = 1, and S1(h), where h is the number of units of health purchased to raise probability of surviving the second period.
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| (5) |
The function u1 gives the utility at age 1 if alive at that age. Hence this approach is essentially normalizing the utility from death to zero. This is fine if the utility of death is fixed and independent of events, age, probability of dying, and many other variables. However, there is an old literature on the ‘fear of death’, which implies that this fear might depend on age, the likelihood of dying, and other variables. If the fear of death changed with advancing age, with the probability of dying, or other variables, the normalization of the utility from death at 0 would not be tenable, and this fear could help explain the large expenditures on health at older ages (Becker et al., 2006). I will neglect such considerations in this paper, although they are important in understanding how the value of life can vary in interesting ways with age (implications of the fear of terrorist attacks are considered by Becker and Rubinstein, 2006).
To simplify the budget constraint, I assume a full and fair annuity market that protects a person against the risk of running out of resources because he lives longer than expected, and against having unspent resources if he dies prematurely. If the capital market were also perfect, so that individuals can borrow and lend expected wealth at a fixed interest rate, the budget constraint with full insurance would be
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| (6) |
0. I am assuming that all health spending occurs in the initial period, and that the price of x is 1.
If the utility function in eq. (5) is maximized with respect to the x's and l's subject to the constraint in eq. (6), one gets the usual first order conditions:
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| (7) |
In this case, full and fair insurance eliminates any effects of survival probabilities on savings, and the rates of substitution between present consumption and leisure time, and future consumption and leisure time. However, it is often failed to realize that this independence result with full insurance also requires separability over time in the utility function. With recursive utility due say to habits or addictions—that is, with consumption stocks—then even with fair market insurance, the probability of surviving in the future does affect the expected utility from consuming such goods in the present.
In particular, the greater the probability of surviving in the future, the smaller the incentive to consume harmfully addictive goods and the greater the incentive to consume beneficially addictive goods. So individuals with lower life expectancies should be more likely to be addicted to drugs, smoking, and other harmful goods even if they had full annuity insurance. I discuss such goods in Section 4.
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| (8) |
The marginal cost of health spending obviously depends on g' and uox, where the latter measures the opportunity cost of spending on health. The marginal cost of spending on health is also greater when future spending exceeds future income. The reason is that an increase in future survivorship would in expectation reduce the resources available for consumption in the present when future spending exceeds future income.
By using the FOC for x, eq. (8) can be written as
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| (9) |
). That is, up to a point, an increase in spending on health raises the expected value of full wealth, net of health expenditures g', since longer life adds an endowment of additional time. The other reason is due to the difference between average and marginal utility because u is concave. The intuition is that spending more in a given year only adds marginal utility, while additional years adds average utility. See Rosen, 1988, and Usher, 1973 for an emphasis on this difference, but with no discussion of the importance of the gain in time. Both reasons are considered by Murphy and Topel, 2006.
To show how these forces combine in eq. (9), assume that utility is homogeneous of degree 
in x and l, where 
1. (This functional form is used by Murphy and Topel, 2006). Then u1 can be written as
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| (10) |
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| (11) |
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| (12) |
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| (13) |
= 1, then eq. (12) reduces to eq. (13). This is to be expected, for if = 1, so that u is CRS, there is no difference between marginal and average utility, and the only gain from spending on improving life expectancy is the gain from increasing the endowment of time.
On the other hand, if < 1, so that utility is a concave function of goods and leisure, there is an additional gain from improving life expectancy because then average utility exceeds marginal utility. In that case, the lhs of eq. (12) is strictly positive, so then the rhs of eq. (12) must also be strictly positive in equilibrium. Since the rhs of eq. (12) is 0 if net full wealth alone is maximized, and since g is a convex function of h, the equilibrium positive value for the rhs of eq. (12) implies that expenditure on health must exceed its wealth maximizing level when utility is concave, and average utility exceeds marginal utility. The difference between the optimal expenditure on health and its net full wealth maximizing level depends on the degree of convexity of the health expenditure function, the degree of concavity in the period utility function, interest rates, and other variables.
2.3 The statistical value of life
Many studies have estimated how much different people need to be paid to take on additional risks to their lives, such as risky construction jobs, enlistments in the military during wartime, driving faster (which raises the probability of a deadly accident), and other risks (see the survey by Viscusi and Aldy, 2003). If they require say $500 (WTP) to accept an additional probability of dying equal to 1/10,000 (dS = –1/10,000), the statistical value of life v = WTP/dS = $500/1/10,000 = $5,000,000.
Equation (9) basically contains dS and WTP, where the marginal WTP is given by the rhs of the equation, and dS is explicitly on the left hand side. Then v = (1/1 + r)u1/u1x. If u is homogeneous of degree , then
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| (14) |
Empirical estimates of the statistical value of life range from about $2 million to $9 million for a young person in the United States, with a central tendency of $3–5 million. We can interpret eq. (14) as giving in a fuller life cycle framework an estimate of the statistical value of life that basically equals full wealth, adjusted upwards for the degree of concavity in the single period utility function (1/). Does that give an estimate for the typical young American in the $3–5 million range?
Assume an average income of $40,000 per year from 1900 annual working hours. For every hour worked, about 1.80 hours are spent in the household sector, where I ignore 68 hours per week for sleep and maintenance. Full income is then about $110,000 per year if an hour of household time is valued at the hourly earnings implied by the data on annual earnings and hours. If is taken at about 
(Becker et al., 2005, use a smaller number), adjusted full income is then $220,000 per year. When discounted at 5%, this gives a value of life of about $4.4 million, which is smack in the middle of the empirical estimates.
A value of life for the average American of over $4 million seems to many to be grossly too large, but that is because there still is a tendency to think in terms of earnings alone. The first book on the economic cost of early death, Dublin and Lotka's The Money Value of Man (1930), estimated this cost from the loss in future earnings due to early death. But the amount that a person is willing to pay to reduce the chances of dying considers not just lost earnings, but lost utility that also includes the value of leisure time, and the differences between average and marginal utilities. At $40,000 earnings per year and a 5% annual discount rate, the present value of the lost earnings from early death is about $1 million, less than a fourth of my back of the envelope estimate of $4.4 million. Therefore, the vast majority of statistical value of life comes not from foregone earnings, but from the loss of leisure time, and differences between average and marginal utilities.
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3. Complementarities |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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The health field is brimming with complementarities: by different diseases, across ages, between health, education and training, and even between the discount rate on future utilities and health. This section demonstrates a few of the more important complementarities, and the next one discusses their implications for economic inequality.
3.1 Between diseases
The probability of surviving to any age can be interpreted as the product of the probability of surviving different diseases, where these probabilities need not be independent. In the 2 period utility function we get
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| (15) |
The term di gives the probability of surviving disease I from the initial period through the second period, and hi gives health inputs into reducing di, where d’i 0.
The full annuity budget constraint is then
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| (16) |
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| (17) |
; and Murphy and Topel, 2006). An increase in the probability of surviving disease j raises the marginal utility from spending more resources on reducing the probabilities of dying from other diseases. Unless the increase in the probability of surviving j sufficiently lowers the marginal product of spending on other diseases (
g/hi), increases in the probability of surviving some diseases would tend to raise the amount spent on fighting other diseases.
During the past 30 years there has been a major improvement in fighting cardiovascular diseases among older persons (see, e.g., Cutler and Kadiyala, 2003). This analysis suggests that more attention would then be paid to fighting other diseases of old age, such as cancer, diabetes, Alzheimer's, and so forth. In fact, the ‘war on cancer’ started after advances began in reducing deaths from heart attacks, and increasing attention began to be paid to these other diseases of old age. Similarly, as deaths from Aids in some African countries have risen dramatically in the past 20 years, many young Africans have become less concerned about preventing other diseases since the probability of dying from Aids is so high in these countries (see Oster, 2006).
A partly related determinant of the increased spending on diseases of old age is that more individuals have reached old age, in good part because deaths from various diseases at younger ages have gone down dramatically. The increased number of persons reaching older ages has raised the productivity of investing in learning more about how to reduce the death rate from diseases at old age; see Section 8.
3.2 Between ages
To show the complementarity between investments in health at different ages, assume that the probability of surviving the initial age is s0(h0), and the conditional probability of surviving age 1 is s1(h1,h2), where h0 is the real resources spent in the initial period on surviving that period, and h1 and h2 are the resources spent in periods 0 and 1 respectively to raise survivorship in period 1. Then the expected utility function becomes
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| (18) |
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| (19) |
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| (20) |
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| (21) |
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| (22) |
). For it states that an exogenous increase in s1 induces greater spending on raising the survivorship during the initial period because it raises the expected utility of surviving the next period. This result is simply saying that if a person knows he is more likely to survive some future periods, he has more incentive to try to survive to these periods. Young adults in Africa have responded less to the risk of getting Aids than have adults in the United States in part because the risk of dying from other causes is so much higher in Africa (see Oster, 2006). Equation (21) shows a similar complementarity between the probability of surviving the initial period and spending in the initial period on raising survivorship in the later period. For an increase in s0 raises the marginal utility of spending to raise s1 before the realization of s0 is known. So if the probability of surviving childhood is low, it hardly pays to spend resources in childhood that would raise survivorship at older ages.
On the other hand, spending at later ages to raise survivorship at later ages does not directly depend on the probability of surviving to these ages. For survivorship is given when a person reaches the later age, and then his behavior would only depend on variables as he looks forward- this is clearly shown by the absence of s0 in eq. (22). What is relevant then are variables in period 1-whatever happened in period 0 is a given. However, there could be an indirect effect if the productivity of h2 depends on h0 or h1, or on variables other than h0 that affect s0 and also h2's productivity.
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4. Health and addictions |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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Full annuity insurance eliminates the risk of dying in the FOC's for goods and time in eq. (7) because of the assumption that the utility function is fully separable over time. When full separability is abandoned through assuming habits and addictions, or other links between present consumption and future utility, full market insurance does not eliminate the probability of surviving from the FOC's for addictive and habitual goods. For assume two goods, x and y, where x is addictive with consumption capital, or for any other reason x depends on the accumulation of consumption capital, as in
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| (23) |
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| (24) |
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| (25) |
u1/x0 < or > 0. The greater the probability of surviving through the second period, the greater the weight placed on the habitual aspect of the consumption of x. This means that dV/dx0, the total marginal utility from x0, is greater for beneficial addictions when the probability of surviving the second period, S1, is greater, while the total marginal utility of x0 is smaller for harmful addictions when S1 is greater.
For example, there is less harm from becoming addicted to activities that lower utility at older ages, such as heavy drinking, hard drugs, smoking, or fatty foods, if the probability of surviving to older ages is relatively low. Conversely, there is greater benefit from becoming addicted to activities, such as regular exercise, religion, or a spouse if the probability of surviving to older ages is high. For example, use of hard drugs by soldiers in Vietnam increased dramatically because probabilities of surviving to old age were significantly reduced by the war there. After they returned to a more normal life, addiction rates went down dramatically among the vast majority of soldiers who had been using these drugs (see, e.g., Robbins, 1993).
The foundation of this paper is that S1 is not simply given but is affected by expenditures on health, h. An increase in h raises S1, which in turn by the previous discussion would raise the demand for x0 if x is beneficially addictive. It lowers the demand for x0 if x is harmfully addictive. This means in essence that good health is complementary with good habits and addictions, and bad health is complementary with bad habits and addictions. Consumption capital in the form of good or bad habits and addictions, or in other forms, cannot be insured against by annuity markets.
It has long been observed that healthier people are less likely to smoke, to be overweight, to be on drugs, to neglect breakfast, and even to be more religious (see, e.g., Grossman and Kaestner, 1997). The explanation typically involves causation from these habits to better health, and such causation is probably of importance. However, the analysis in this section shows there is also causation from better health to better habits since the cost of bad habits is greater for persons in good health with better life expectancy.
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5. Education and health |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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Complementarities are also important between expenditures on different forms of human capital. I will consider only the complementarities between health and schooling since these have received a lot of attention, although the same analysis applies to health and training, and health and migration. An increase in survivorship at later ages raises the returns from investments in education because educational costs come at earlier ages and returns at later ages. (see Becker, 1964
, 1993: Meltzer, 1992; Ehrlich, 2000).
To show this with the two period example I have been using, suppose the cost of education, equal to E, occurs in the initial period, and that the investor is certain to survive this period. If the return next period is in the form of a higher wage rate, then w1(E) with 
.
For several reasons, an increase in education also tends to increase survival rates. Abundant evidence indicates that more educated persons take better care of their health even with a given spending on medical care by visiting better doctors, taking their prescribed medicines more regularly, eating more nutritious diets, and in many other ways. This suggests that S1(h, E), with S1/E > 0, so that an increase in schooling raises life expectancy.The utility function now becomes
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| (26) |
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| (27) |
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| (28) |
The rhs gives the cost of increasing education. The first term is the marginal dollar spent on education, and the second term is the increase in the expected resources that would be taken from the first period to finance greater spending during the second period relative to earnings in that period. Of course, spending in the second period could be lower than income in that period if ß were sufficiently low relative to r, if w1 were high relative to w0, etc.
Equation (28) implies that an increase in education raises life expectancy in two ways. Greater education raises expected wealth net of the spending on education. That produces a wealth effect that would increase spending on health, and thereby would raise survivorship in later years. Education also directly raises survivorship by making a person more productive at investing in health through better information about doctors, healthy lifestyles, and other information related to better health.
This analysis implies that greater education raises health, but not necessarily spending on health. The wealth effect from higher earnings does raise spending on health, but the effect on survivorship probabilities can imply better survivorship with lower equilibrium spending. It all depends on the elasticity of the derived demand for h, and how E affects the marginal product of h in raising S1.
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6. Health and the discount rate |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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There is debate in the health area about the interpretation of the empirical positive relation between health and education. Some argue the causation is from schooling to achieve better health, while others claim reverse causation from better health to more schooling. Both those forces appear in eq. (28). However, still another interpretation is that neither of these views is the most important, and the main causation is from persons with lower discount rates on future utility to both greater investment in schooling and greater investment in health by these people (see Fuchs, 1982
). This last interpretation takes the distribution of discount rates as given, and argues that persons with relatively low discount rates get selected into investing more in both education and health. Such selection happens, but it is a mistake to take discount rates as given. Discount rates reflect attitudes toward the future, and they are affected by spending of time, money, and energy on ‘imagination capital’ that helps to reduce how much future utilities are discounted in decision-making.
Becker and Mulligan (1997) show that the incentive to invest in raising discount rates is positively related to the magnitude of expected future utilities. Their analysis is applicable to the interaction with health and education that we are considering. Modify the utility function in eq. (23) to allow ß to depend on C, imagination capital, with ß’0, ß’<0, and let 
(C) be the cost of C, with ' > 0, '' > 0. Then the budget constraint that includes spending on C would be:
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| (29) |
Then the FOC for optimal investment in C is
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| (30) |
). Since education raises wealth, given the FOC in eq. (28), this equation implies that more educated persons discount the future less. It is possible that E also directly enters ß, with ß/E > 0, which would be another reason why educated persons have lower discount rates. The positive relation between the marginal utility of investing in imagination capital and the probability of surviving in the future implies that healthier persons will also have lower discount rates because they have greater incentive to invest in reducing the discount on future utilities. So healthier people would have lower discount rates not only because people with lower discount rates invest more in health, but also because healthier people invest more in lowering their discount rates.
This latter force has been entirely neglected in the health literature on the relation between health and the discount rate. Moreover, the analysis implies that healthier persons both invest more in education and in lowering their discount on future utilities. With lower discount rates, healthier people will invest more not only in human capital, but also in savings in the form of financial, housing, and other assets. Lower discount rates on future utilities reinforce the earlier conclusion that healthier persons are more likely than more sickly persons to develop beneficial habits and less likely to develop harmful habits.
In the expected utility formulation, since both ß and S1 are discount rates on future utilities, it is no surprise that the marginal utility of spending to increase S1 has the same form as the marginal utility to increase ß. However, while it is generally accepted that individuals can affect their probabilities of surviving different ages by how they spend their money, time, or effort, it is controversial whether individuals can affect their discount rates by how they spend their resources (see, e.g., the negative view on this by Elster, 1997).
That is just a legacy, I believe, of the tradition in economics of taking discount rates as given, and outside the control of individuals. It was once a tradition in economics too of taking death rates as given, and not affected by individual choices. That tradition has vanished as economists realized the many ways that individuals can affect their mortality rates. I expect the tradition of taking discount rates as ‘exogenous’ to also vanish as economists appreciate the many ways that parents affect their children's discount rates, and adults affect their own rates.
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7. Inequality, health, and other human capital |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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One of the pioneering articles on human capital (Mincer, 1958
) assumes that the present value of earnings, net of education costs, are the same at all levels of education when earnings are discounted by the market interest rate. In essence, this assumption eliminates all inequality in the present value of earnings by education level, net of education costs, including foregone earnings.
Further analysis and formulation of investments in human capital incorporated the older idea of non-competing groups into education and other human capital choices (see Becker, 1967, reprinted in Becker, 1993). Moreover, Becker and Chiswick (1966) showed that under certain strict conditions, actual rates of return on investments in education, which greatly exceed market interest rates, could be estimated from data on differences in wages across education classes.
These formulations established the foundation for the conclusion that education is a significant source of inequality in earnings, even after netting out all education costs. Further research demonstrated that individuals differed significantly also in other determinants of wellbeing, such as health, savings, habits, and marital stability. If these characteristics were negatively correlated with each other and with education, then the overall degree of true inequality might be much less than that in earnings due to differences in education.
However, research has demonstrated that virtually all valued characteristics are positively rather than negatively correlated with education (see Elias, 2005). These include health and life expectancy, marital stability, achievements of children, law-abidingness, beneficial as opposed to harmful habits, savings rates, propensities to vote and other activities in elections, and still additional traits. Why this is so has not been answered in anything approaching proper generality.
I propose that the answer comes from the general complementarities among different classes of human capital. I have built the analysis in this paper around health as measured by the probabilities of surviving to later ages, and I expanded out from that to consider the relation between health and education, and other forms of human capital investment. The analysis shows that different forms of human capital investments are complements, not substitutes. These forms include not only education and health-which is well known- but also health and good habits, health and lower discount rates on future utilities, and improvements in life expectancy from different diseases and at different ages.
This means that people who have better life expectancies also have higher earnings and greater education, save a larger fraction of their permanent incomes, have ‘better’ habits, and also have greater conditional life expectancies, given that they reach any age. So characteristics like earnings, habits, discount rates, and savings do not offset inequality in life expectancy, but reinforce that inequality to contribute to a still widen inequality in overall welfare. Groups are ‘non-competing’ not in a single determinant of wellbeing but in a whole series of interrelated determinants.
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8. Investment in R&D and population |
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TOP1. Introduction2. Theory3. Complementarities4. Health and addictions5. Education and health6. Health and the...7. Inequality, health, and...8. Investment in R&D...9. National income accounts...10. ConclusionsAcknowledgementsReferences |
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8.1 A model of drug innovation
Much of health economics is concerned with health delivery systems. These include issues related to government financing of medical care, co-payment arrangements, health savings accounts, moral hazard and self selection in both private and public health coverage systems, incentives to economize on spending by hospitals and doctors, and many other issues. These are all important, and they deserve the attention they receive. Basically they all affect in different ways the costs of medical care to consumers, and their incentives to take various actions.
I will neglect them to concentrate on another issue: the role of population in affecting the incentives to spend on medical R&D. To generate this analysis, consider a pharmaceutical company that can spend R to produce a medical innovation also, I, in the form of pills that raise the probability of surviving particular ages A. Once innovated, the cost of producing each pill equals c. The demand for these pills is given by
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| (31) |
is the elasticity of demand for these pills.
To determine the market price, I assume that the innovator is given a patent that provides effective monopoly power in the market for I for T years. After T years pass, generics enter and push the market price of each pill to c, its cost of production. So the innovator earns monopoly profits for T years and zero profits after that. Since profits on the pills related to I equal 
= pD – cD, given eq. (31), profits are maximized when price equals
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| (32) |
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| (33) |
, the bigger is T, and the larger is N. For the innovation to be worth the investment, it is necessary that
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| (34) |
This innovation is more likely to be profitable when N is larger, where N can be interpreted as the number of people demanding the pill when p = 1. This measure of demand depends on the number of people in the age groups (A) where the disease attacked by the pill is more prevalent, the fraction of this number that contract the disease, the income of the individuals getting the disease, public subsidies to buying the pill, and other variables. Obviously, incomes matter since as we have seen, willingness to pay to improve survivorship depends positively on incomes. Previous sections also show that the demand for this innovation would be greater, the greater the probabilities of surviving other diseases at these and later ages.
8.2 Population and the demand for innovations
I want to emphasize now the importance of sheer numbers, as measured by the number of persons in vulnerable ages groups for the particular diseases treated by innovations (for some of the theory on this applied to medical research, see Becker, 2000; Cerda, 2003; and Acemoglu and Linn, 2004). For the larger this population, the greater is the profitability of medical innovations, holding incomes, the probabilities of surviving other diseases, government subsidies, and other relevant variables constant. This is not surprising since innovations, and medical innovations in particular, have increasing returns to scale because the cost of producing innovations is independent of the scale of the demand for the innovated product or service.
The importance of the size of the market is recognized in The Orphan Drug Act of 1982. This Act stipulates that drugs which are innovated to treat diseases which have a market with fewer than 200,000 sufferers receive seven years of guaranteed market exclusivity. Studies have shown that this longer patent protection has stimulated additional expenditures on research on orphan drugs by pharmaceutical companies (see Grabowski, 2005). Still, the most profitable drugs are those that cater to large markets, such as drugs that treat high blood pressure, or erectile dysfunction, or high cholesterol.
Additional evidence on the size of the market comes from studies of the number of new chemical entities to treat diseases of different ages as a function of the relative number of persons at these ages. Figures 1 and 2 from Cerda (2003; also see Acemoglu and Linn, 2004) give the number of new chemical entities introduced in the United States to treat diseases contracted by persons at particular ages as a function of the number of persons at these ages. As the number of persons aged 45–64 increased, Fig. 1 shows that the number of drugs introduced to treat diseases that mainly affect people of these ages also increases. Figure 2 shows similar results for drugs introduced to treat diseases that mainly affect people aged 65 and older.
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8.3 Population and the aggregate value of declines in mortality
Table 2 gives results from the well-known study by Murphy and Topel (2006
) on the value of the gains to Americans from the decline in death rates between 1970 and 2000. They assume a statistical value of life to a young person of $5 million, and make various reductions to this number for older persons. They subtract all the growth in spending on medical care during this 30 year period. They then add together the value of the improvements in death rates at each age over the number of individuals at each age, with separate calculations for men and women. Their main results are in the fifth column of Table 2.
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These numbers are huge: combined for men and women they add up to about $60 trillion net of medical expenditures. Compare this to a GDP for the US of about $11 trillion, and one can appreciate how large is the aggregate value of the gain to the American people from the sizable improvements in their life expectancy since 1970. Their assumption of a $5 million statistical value of life for a young adult can be questioned, and perhaps a lower number is more appropriate (see Section 1.3). But their estimates of the gain from reduced death rates would be huge even if $3 million or $2 million rather than $5 million were used to measure the value of life.
For it is the scale of the US population of about 300 million persons that is the main driver of these aggregate gains, not the numbers used to measure the value of life to individuals of different ages. Even small gains to the average individual become large when multiplied by 300 million. This shows the power of increasing returns as population grows to magnify the gains from improvement in survivorship rates.
The full effects of improved mortality are even larger than this since the market for drugs is worldwide, and not restricted to the US. Suppose we consider just the OECD countries that excluding the US has a combined population of 867 million. To shortcut the calculation I assume a representative person in each country with a value of life that equals $5 million times the ratio of the per capita income in that country to the US per capita income. I value the change in life expectancy between 1970 and 2000 in each country by its calculated value of life, and multiply that result by the country's population. The results are summed over all OECD countries, including the US, where the increase in medical spending is subtracted to give the ‘net’ results in the bottom half of Table 2.
These values are about three times as big as the values for the US alone. Combined over men and women they total to about $190 trillion, an amazing sum. This means that enormous value is placed on the declines in mortality in OECD nations during the past couple of decades. Moreover, this value is increasing in the size of their populations, given their per capita incomes.
From the view point of public and private spending on medical R&D, the size of the market in rich countries alone is enormous for innovations that can treat reasonably common diseases. A larger and more densely packed population also has some negative effects because that makes it easier to transmit infectious diseases. At the same time, however, a larger population also provides stronger incentives to find ways of treating infectious and all other diseases. So the negative attitudes toward larger world populations (even for a given per capita income) that is common among demographers, scientists, and many others is largely wrong with regard to the determinants of innovations in basic and applied medical research.
8.4 The cost of medical innovations and population
I have assumed so far that the cost of a medical innovation is fixed at R, and I have compared that cost to the revenue from innovations. This cost may vary positively (diminishing returns) or negatively (increasing returns) with the stock of past medical innovations, and negatively with advances in scientific knowledge, such as the mapping of the human genome. The cost of innovation may also vary with mortality rates since it may be more difficult to lower mortality rates further when they are already low.
Suppose that the cost of medical innovations increases as the stock of these innovations grows and mortality rates fall. Studies do in fact indicate that there has been a significant increase over time in the cost of producing a medical innovation that receives FDA approval (see DiMasi, et al. 2003). At the same time, however, these innovations result in a larger number of persons surviving to different ages. The increased numbers and also the lower mortality rates raise the demand for additional innovations. Which force dominates, cost or demand, then depends on how sensitive each is to changes in innovation stocks and mortality rates.
To discuss this in terms of our maximizing condition eq. (31), express this equation as
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As I mentioned earlier, in recent years the cost of pharmaceutical R&D has been growing, but so too has demand because of the growth in incomes and population. Since expenditures on medical R&D have been growing, any growth in the cost of R&D for a representative innovation relative to discounted profits per pill (R/g in eq. (35)) may have been compensated by a growth in N, due in good measure to population growth, especially at older ages where medical interventions are more needed.
8.5 Explaining the declines over time in age-specific death rates
The analysis in previous sections can be applied to understanding the pattern of the decline over time in age-specific death rates. Figures 3 and 4 show the improvements in life expectancy at birth during the 20th century, and also improvements in life expectancy at ages 45 and 65. Life expectancy at birth incorporates life expectancies at all later ages. This life expectancy rose rather steadily, and remarkably, during this past century. Life expectancies at ages 45 and 65, however, changed little during the first half of that century, but both increased rapidly during the second half. This means that mortality declined mainly at younger ages during the first half of the century, and mainly at older ages during the second half.
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This pattern of changes in mortality appears to fit the pattern that would be predicted by the theory developed in this paper. At the beginning of the 20th century, there were many young people because of high birth rates and a young adult population. At the same time, infant and child death rates were quite high. For both reasons, the market for vaccinations and other innovations that would reduce deaths from childhood was large. Effort was concentrated on diseases of young age, such as diphtheria and scarlet fever, greater cleanliness in delivering children, and in the quality of children's diet. The return to investments in reducing mortality among young persons was very high at that time because there were lots of young people, and their survival of childhood would bring benefits at later ages.
By 1950 deaths in childhood were reduced so much that a small fraction of persons died before age fifty. As a result of this and the growth in the number of older persons, the emphasis in medical research shifted toward diseases that struck primarily at older ages, such as cardiovascular disease, strokes, and various forms of cancer. During the second half of the 20th century, substantial progress was made in fighting these and other diseases of older ages. For the first time, new drugs began to play a major role in fighting diseases, such as antibiotics, drugs to lower blood pressure and cholesterol, drugs to fight breast and prostate cancer, drugs against lymphoma, and so on for other diseases- for the reasons discussed in the previous section. The result was the substantial progress shown in Figure 4 in reducing death rates after age 45, and increasingly after age 65.
We have seen in Section 3.1 that a decline in death rates from one disease increases how much individuals are willing to spend to reduce the probability of dying from other diseases that strike at the same or later ages. This means that the development of a drug that reduces the probability of dying from one disease increases the demand for drugs to fight ‘competing’ diseases. The price to consumers of drugs that fight disease J that affects people at age A would enter into the demand function for disease I in eq. (31) with a negative exponent. The negative exponent indicates that diseases I and J are complements:
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| (37) |
and Cerda, 2003). That is, N in eqs (31) and (34) is affected by the number of survivors to age A, which increases when treatments improve for diseases like heart conditions and strokes that affect people at age A. So returns to population scale, in addition to complementarities between diseases at the individual level, produce complementarities in the aggregate demand functions to treat different diseases that strike at related ages. One of the paradoxes of behavior in recent decades in the United States is that on the one hand, people are greatly concerned about their health, as shown by the increase in exercise clubs and the popularity of exercise videos, and from almost daily reports on drugs, diets, and other health factors in major newspapers and TV news programs. On the other hand, a considerable amount of behavior appears to contribute to worse health, such as the consumption of boutique ice creams with high fat content, the reduction of exercise by teenagers and their substitution toward sedentary activities like computer games and chat rooms, and of course the resulting significant increase in weight of adults, and even more so of teenagers.
This unusual combination of heightened concern about health, and behavior that appears to reduce health can be understood by recognizing both that people are forward looking, and that they expect new drugs to be developed in the future. Consider the sizable increase in average weight (see Philipson and Posner, 2003), including a significant growth in obesity, during the past 25 years. This has been a source of concern by health officials and others since being overweight, and particularly being obese, is currently associated with increased risk of diabetes, high blood pressure, heart problems, and other diseases. I say ‘currently’ because it is reasonable to expect that drugs to prevent many of the negative consequences of obesity will be developed in the future. After all, the past several decades have seen the development of much better drugs to control high blood pressure, drugs to reduce cholesterol levels, and drugs to make diabetes itself more manageable.
To show how the anticipation of medical progress can affect current behavior, consider the utility function
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| (38) |
S1/x0 < 0, and that drugs raise health, so that S1/I1 > 0. I make the important assumption that an increase in I1 reduces the negative effect of x0 on S1; that is, 2S1/x0