El valor de la longevidad en Mexico.
| Autor | Martinez, Gabriel |
The Value of Longevity in Mexico
Introduction
Life expectancy at birth in Mexico has increased significantly during the last few decades (Gomez de Leon and Rabell, 2001), and the health status of the overall population has improved. Moreover, in recent years the Mexican government has spent a significant amount of public resources to finance Seguro Popular (SP), a health insurance program created primarily for the poor, who were previously excluded from formal social insurance. But how much are the gains in longevity and improved health worth? Has sp yielded positive results in terms of benefits and costs?
Estimating the value of health and longevity is key to understand the benefits of increased spending and evaluating polices that aim to devote additional public resources to health. Assessing whether sp has resulted in positive net outcomes is not only needed to justify the continuation of the program in Mexico, but also to inform decisions about implementation of similar programs in other countries.
In the analysis we present here we apply the framework developed by Murphy and Topel (2003, 2006) to calculate willingness to pay (WTP) for improvements in health and longevity to the case of Mexico. We calculate the value of longevity during the period 1930-2010. We provide evidence on the value of longevity associated to avoiding obesity-related deaths, and we exemplify how cost benefit analysis can be performed to a subsidized health program, such as SP. As a byproduct of the model, we estimate the value of remaining years of life (VRL). We find that gains in longevity in Mexico during the period 1930-2010 have been very high, amounting to around 15 trillion dollars, (1) which is equivalent to 17 times the country's 2010 gross domestic product (GDP). We also estimate that the annual value of avoiding obesity-related deaths, specifically deaths resulting from four diseases for which obesity is listed as a risk factor, equals 22 billion dollars. Finally, we demonstrate that gains resulting from decreased mortality associated with Acute Lymphoblastic Leukemia (all) in children exceed the amount that sp spends to treat the disease.
This paper is organized as follows: in section I we describe the model and explain how it was calibrated; in section II we apply the model to calculate the social value of gains in longevity from 1930 until 2010, of avoiding obesity-related deaths, as well as to establish thresholds for the sp program in terms of cost-benefit analysis for the specific case of ALL; section III concludes.
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The Framework
I.1. Valuing Changes in Longevity
Murphy and Topel's model measures the value of health and longevity by the amount a person is willing to pay to avoid death or to maintain their health. Under this model, a rational agent will maximize their expected remaining utility function subject to a budget constraint. The utility and constraint of an individual age a are shown in equations (1) and (2), respectively. Utility is a function of consumption, work effort, health status and a discount factor, with the last two variables being exogenous to the representative individual. H(t) measures health status, c(t) measures consumption in goods other than healthcare, l(t) measures leisure, and S(t, a) is the probability that an individual age a survives from age a to age t. The budget constraint equation shows that the expected value of the difference between income and consumption plus an exogenous wealth value equals zero. (2) The value of utility after death is normalized to zero; A(a) is the value of assets at age a, and y(t) measures income.
[mathematical expression not reproducible] (1)
Subject to
[mathematical expression not reproducible] (2)
Equation (1) assumes that health improves the quality of life through a multiplicative effect in the utility function (the health status effect), and that utility can also be improved by an increase in the probability of survival (the mortality effect). Income in each period, y(t), is endogenously determined by the choice of leisure since y(t) = w(t) [1 - l(t)] + b(t), where w(t) is the wage rate and b(t) is life-contingent non-wage income, such as defined benefit pension receipts, which is exogenous to the individual. The maximum amount of non-market time is normalized at unity. Optimality conditions are shown in equation (3), where [mu] is the multiplier associated with the constraint.
[H(t)[u'.sub.c](c(t), l(t)) = [[micron]e.sup.-(r- p)(t - a)]
[H(t)[u'.sub.l](c(t), l(t)) = w(t)[[micron]e.sup.-(r- p)(t - a)] (3)
The optimal values that result from this system can be used to write the indirect function in equation (4), and the VRL at age a can be calculated as the marginal rate of substitution between changes in factors that affect survival probabilities and assets. It is important to note that wages equal the marginal rate of substitution between leisure and non-health consumption (w = [u.sub.l]/[u.sub.c]).
[mathematical expression not reproducible](4)
Where:
[mathematical expression not reproducible] (5)
The VRL is the present value of a life year (v(t)) in future periods adjusted for the survival probability and the discount factor. The indirect function v(t) measures the monetary value of the utility plus net savings, which enter the value of a life year because they are used to finance future consumption.
Using this framework we can analyze the impact of an exogenous medical improvement, [alpha], that increases the probability of surviving from age a onward. In this exercise we exclude improvements in health through H(t). Equation (6) shows the willingness to pay (wtp) for this medical improvement at age a.
[mathematical expression not reproducible] (6)
where S(t, a)[[GAMMA].sub.[alpha]] (t, a) = [partial derivative]S(t,a)/[partial derivative][alpha]. The authors restrict the utility function to be homothetic. Thus, u(c, l) = u(z(c, l)), and z is a composite commodity, homogenous of degree 1, that aggregates consumption and non-market time (z = [z.sub.c]c + [z.sub.l]l). With these assumptions, indirect utility equals full income ([y.sub.F]) plus consumer surplus multiplied by full consumption ([c.sup.F]), as seen in equation (7).
[mathematical expression not reproducible] (7)
Full consumption and full income include the value of non-market time: [c.sup.F]= c + [lz.sub.l]/[z.sub.c] = [z.sup.-1.sub.z] and [y.sup.F]= y + [lz.sub.l]/[z.sub.c] = [z.sup.-1.sub.z], where [z.sub.l]/[z.sub.c] is the relative price of non-market time and consumption (w for the labor force participants). The term in brackets measures the consumer surplus per unit of consumption of z. Based on the following equation [PHI](z) = [u(z)/zu'(z)-1], the value of a life year is given by equation (8).
v = [y.sup.F] + [c.sup.F] [PHI](z) (8)
The next section uses equations (6) and (8) to estimate the value of longevity, which enables us to analyze the value of variations in survival probabilities.
I.2. Calibrating the Model for the Case of Mexico The empirical analogue of equation (6) can be written as
[mathematical expression not reproducible] (9)
where [S.sub.t] measures the probability of surviving from birth to age t, [S.sub.a,t] = [S.sub.t]/[S.sub.a].
[DELTA][S.sub.a,t] = [S.sup.2.sub.t]/[S.sup.1.sub.a] - [S.sup.1.sub.t]/[S.sup.1.sub.a], [lambda] = [S.sup.2.sub.t]/[S.sup.1.sub.t] - [S.sup.1.sub.t]/[S.sup.1.sub.t] and R = 1/ 1 + r
Thus, [lambda] measures small changes in the probability of death that affects homogenously through all periods, and [W.sub.a] is interpreted as the value of a statistical life (VSL, Kniesner...
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