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Wealth from optimal health
OF
Journal of Health Economics 14 (1995) 65-79
ELSEVIER
Wealth from optimal health Per-Olov Johansson
a,*
Karl-Gustaf I_/Sfgren b
a Centre for Health Economics, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden b Department of Economics, University of Umegz, S-901 87 Umed, Sweden
Received April 1993; revised December 1993
Abstract Recently, much research has been devoted to the question of how the conventional net national product measure should be augmented so as to cover changes in the stocks of natural resources. This paper investigates the treatment of health (capital) and the risk of 'doomsday' caused by pollution in such welfare measures. Our problem is not a standard optimal control problem because the survival probability depends on state variables. We show how to handle this complication. The resulting welfare measure is contrasted with the conventional net national product measure. Finally, we address the matter of how to design a subsidy on health investment such that a market economy provides the optimal level of health. Keywords: National product; Welfare measures; Health capital; Health investment JEL classification: D61; D62; H21; H51; I10; I18
1. Introduction
T h e net national product ( N N P ) m e a s u r e is the prevalent indicator of w e l f a r e in a country. In an elegant paper, W e i t z m a n (1976) s h o w e d that under perfect foresight N N P is a static e q u i v a l e n t o f wealth, and that it m e a s u r e s the m a x i m u m
* Corresponding author. Fax: +46 8 302115. 0167-6296/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 2 9 6 ( 9 4 ) 0 0 0 3 7 - 9
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sustainable consumption level. Since Weitzman's paper, there has been much research on the relationship between the standard net national product measure and economic welfare in a country; see, for example, Hartwick (1990, Hartwick 1991), Dasgupta and M~iler (1990), Aronsson and L6fgren (1993), and M~iler (1991). Much of this research has been focused on the fact that the NNP-measure does not cover changes in future consumption resulting from current changes in the resource stocks. Various authors have shown that the NNP-measure should be augmented with (positive or negative) net investment in natural resources in order to be a reasonable measure of welfare. Typically, a Ramsey growth model is used to derive these welfare measures. To the best of our knowledge, no one has used a Ramsey model to examine how to incorporate investment in human health in an augmented NNP-measure; see, however, Mushkin's (Mushkin, 1962) discussion which, though informal, is in similar terms. One possible interpretation of the model presented in Section 2 of this paper is in terms of such an augmented NNP-measure. The interpretation of our Theorem, stated in Section 2, is then as follows: many authors have objected on moral grounds to the common practice of discounting the interests of future generations. Dasgupta and Heal (1979), who provide a survey of the discounting debate, claim, just like Solow (1986), that it still may be reasonable to discount the welfare of future generations if the probability of doomsday is strictly positive; the probability of doomsday is then used as a kind of a discount rate. Within such a context, our basic model, developed in Section 2, can be interpreted as follows. Suppose there is a strictly positive extinction probability for mankind due to pollution of the air and the water. The deterioriation of environmental quality also has a direct and adverse impact on human welfare through its impact on health. Investment in an improved environmental quality can be viewed as investment in health capital. In turn, such investment both improves utility and reduces the doomsday probability, though, of course, directly reducing emissions causes the same effects. In any case, one of the principal results derived in this paper is that a positive and endogeneous death risk invalidates the conventional wisdom as stated in Weitzman (1976) and M~iler (1991), for example (saying that the Hamiltonian along an optimal trajectory is a static equivalent of wealth). We also discuss the properties of a market economy when health can be viewed as a factor of production influencing the productivity of the economy. As modelled in this paper, health is taken as a datum by firms and hence causes a kind of positive external effect. In other words, the market economy provides too little health investment. This raises the question how to design a subsidy on health investment, this is addressed in the final section of the paper. The paper is structured as follows. In Section 2, we present the basic model employed in this paper, and derive our main result, which is stated in Theorem 1. In Section 3, we relate our results to the working of a market economy, and discuss how to design a subsidy on health investment such that the market economy produces an optimal solution.
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2. The basic model: Health and national accounting Our way of viewing health capital in this paper is much inspired by Grossman's (Grossman, 1972) model, as is further explained in, Aronsson, Johansson and L6fgren (Aronsson et al., 1994) though we interpret the model in an entirely different way. We interpret it, not as Grossman does in terms of a finitely-lived individual, but in terms of a society consisting of a succession of identical and finitely-lived individuals. This interpretation is used in order to relate our results to those of Weitzman (1976) and others on the welfare significance of the net national product measure. The instantaneous utility function of the single individual (generation) living at time t is written as follows: u =u(c(t),
h(t)),
(1)
where u(.) is the twice continuously differentiable instantaneous cardinal utility function, which is strictly concave and increasing in its arguments, c(t) is consumption, and h(t) stands for the volume of the health capital at time t. Health capital is produced by refraining from consuming goods and investing the resources in the health sector. Goods are produced by capital, labor, and emissions (through the use of energy inputs). Moreover, the health status of the labor force is assumed to affect its productivity, implying that the stock of health capital is included as a separate argument in the production function. The labor endowment is assumed to be fixed, and is normalized to unity. Assuming that the production function is homogeneous of degree 1, the production per capita can be written as follows: q = f ( k , ei, h),
(2)
where q denotes net output, so depreciation has been accounted for, f(.) is the production function, k = K / L is the capital-labor ratio, K is capital, L ( = 1) is labor, e i is energy used per unit of labor, h is health capital per unit of labor. The production function is assumed to be strictly concave, twice continuously differentiable, increasing in all arguments, all inputs are necessary, i.e. f(0, e i, h ) = f ( k, O, h) = f ( k, e i, 0) = 0, the sign of the cross derivatives, i.e. fkh(') -= O2f(') /Okoh etc., are strictly positive, and production cannot be increased without bound by increasing the use of a single input. These are basicly the assumptions employed by Tahvonen and Kuuluvainen (1993). The accumulation of capital follows the accumulation equation: k = f ( k , ei, h) - c - x ,
(3)
where k = d k / d t , and x denotes goods per unit of labor that are used as an input in the health sector to augment the health capital. The accumulation of health capital follows the equation: /~ = g* ( x , z ) - "yh = g( x, z, h, ~/ ),
(4)
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where z is the stock of pollution built up through emissions caused by the use of energy inputs in the production of ordinary goods, 3t is a depreciation factor, and g *(x, z) is a strictly concave production function, g * ~ C 2, which is increasing (decreasing) with its first (second) argument, x is a necessary input, and gxz(X, z) < 0 for x > 0. The last assumption means that the marginal productivity of investments in health capital decreases with the stock of pollution. Eq. (4) introduces a quite conventional health production function; see, for example, Grossman (1972) and Wagstaff (1986). Thus, health is treated as any other durable. At the individual level, ageing can be introduced by allowing for a time dependency of the health production function and by letting time (age) appear as an argument in the utility function; see, for example, Jones-Lee (1976, p. 81) for an 'age-dependent' utility function. Our model is however an aggregate one though we have ignored any signs of a birth and death process spanning generations. We have ignored this process and hence also ageing in order to be able to concentrate on some other issues. Finally, we assume that emissions are accumulated in nature, though the environment has an assimilative capacity. The stock of pollution follows the following equation: = e -
c~z,
(5)
where e is the flow of emissions from the production of ordinary goods, and a is a parameter reflecting the environment's assimilative capacity (0 < a < 1). The treatment of emissions means that we have exploited an assumed relationship between a production factor, energy, and the ' production' of emissions. In order to simplify the notation, without any loss of generality, we will in the remainder of this paper assume that e ( t ) = el(t) for all t, and suppress the production of energy. The optimization problem of society is to maximize the expected utility of discounted future utility. More precisely, society is assumed to maximize:
u(c(t),h(t))e-°tdt
MaxE[ut] = M a x E r c,x,e
c,x,e
,
(6)
I 0
subject to:
k(t)=f[k(t),e(t),h(t)]-c(t)-x(t); h(t)=g[x(t),z(t),h(t),~/]; 2(t) =e(t)--otz(t);
k(O)=ko, h(O) = ho,
z(0)=z0,
(6a) (6b) (6c)
where T is the stochastic time to doomsday (or length of human life), and 0 is the non-negative rate of time preference or subjective discount rate. The distribution of T is given by the hazard function (the probability that mankind becomes extinct in the short interval (T, T + d T ) conditional on having survived to T): [ f (T)/(1 -F(T))]dT=
6[h(T), z(T),T]dT,
(7)
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69
where F(.) is the cumulative distribution function for doomsday, i.e. T, and F (.) is the density function for T. The hazard is, hence, a function of both the stocks of health capital and pollution at time T, and the 'age' of society. We assume that the intensity parameter 6(.) is non-negative, bounded away from zero, and decreasing in h and increasing in z and T. Integration of (7) yields:
1 - F(T)
= e- f0~th(S)'Z(s)'slos,
(8)
where 1 - F ( T ) is the survival probability for mankind at time ('age') T. Eq. (8) means that we assume that health capital not only increases the quality of life through its presence in the utility function, but that it also has an impact on the expected life-length of humanity. On the other hand, the stock of pollution has a direct impact on the human race's life expectancy, but only an indirect influence on instantaneous utility through its impact on health capital. These assumptions seem realistic. In contrast to the 'value of life' literature, see, for example, Jones-Lee (1976) and Rosen (1988), in our model the death probability depends on (two) state variables, h and z. As is shown below, this fact means that our maximization problem is not a conventional optimal control problem. Using Eq. (8) and the results on the equivalency between areas of integration presented in the appendix at the end of the paper, the optimization problem can be formulated in the following manner: ~c
MaxE[ut] = M a x f c,x,e
e-
fo~lh(s)'z(~)'~ld~u[c(t),h(t)]e-°tdt,
(9)
c,x,e~O
subject to:
Jc(t)=f[k(t),e(t),h(t)]
=-c(t)-x(/);
k ( 0 ) = k 0'
h(t) =g[x(t),z(t),h(t),y]; h(0) = h o, ~(t) = e ( t ) - a z ( t ) ; z(0) = z 0. Unfortunately, this is not a standard optimal control
(9a) (9b) (9c)
problem with an infinite planning horizon since the discount factor contains the state variable h. However, using a standard trick, (9) can be written as follows: oc
MaxE[ut] = M a x f c,x,e
u[c(t), h(t)]e-t°t+a(t)ldt,
(9')
c,x,e~O
with A(t) defined as: a(t) =
fo6[h(s), z ( s ) , s]ds,
(10)
i.e. A(t) is given by the differential equation: A(t) = 6 [ h ( t ) , z ( t ) , t ] ;
A(0) = 0 .
(9d)
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Thus, by maximizing (9') subject to (9a)-(9d) we turn our problem into a standard control problem with an infinite planning horizon. This approach should be useful for health economists working with intertemporal problems where the survival probability depends on the health capital or other state variables. We can write the current value Hamiltonian as follows:
H( t) = u[ c( t ) , h ( t)] + A ( t ) k ( t ) + tx( t)h( t) + @( t)~( t) + A( t) /i( t), (11) where A(t) = A(t)e °t+ a(t), Iz(t) =/2(t)e °t+a(O, ~ ( t ) = ~(t)e °t+ a(O, and A(t) = y~(t)e0t+ a(o are costate variables. The necessary conditions for an optimal control are stated in the appendix. Combining (i) and (ii) of Eq. (A.3) in the appendix, we have that:
Uc(.) = / z g x ( . ) = A.
(12)
Along an optimal path, the marginal utility of consumption equals the value of the marginal product of the input in health investment, which in turn equals the value (in utility terms) of a unit of capital A. Assuming for the moment that society faces a constant death risk, 8 = 8 0, the following conditions hold in a steady state: 1 fk(-) = O+ 8 o, +/h(.)]
(12') = 0+
+
According to the first-line in (12'), the stock of capital should be such that the marginal product of capital (net of depreciation) is equal to the sum of the marginal rate of time preference and the death probability. This is a kind of Modified Golden Rule result, with the rate of population growth replaced by the 'death rate' ~; see, for example, Blanchard and Fischer (1989). The second-line expression yields a similar result for the optimal stock of health capital, once it is recognized that depreciation is explicitly accounted for (through T) in the second, but not in the first, line of (12'). Note, however, that improved health not only increases human utility but also has a positive impact on firms' productivity through the term fh(.). This is a kind of positive external effect caused by health investment, which a market economy is unable to correctly account for, as is further explained in Section 3. In other words, ceteris paribus underinvestment in health will result unless some policy instrument is introduced. There is a striking similarity between this result and those obtained in endogenous growth theory, where human capital accumulated within a firm through, for example, experience
1 For a version of the model in Section 3 one can show that there exists exactly one steady state, and that this steady state is a saddle point towards which all bounded solutions will converge. The proofs, which are available from the authors, are similar to the corresponding proofs in Tahvonen and Kuuluvainen (1993).
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and education, causes an external effect at the aggregate level: firms underinvest in human capital since the benefits in part accrue or spill over to other firms through such things as labor mobility; see, for example, Lucas (1988), Romer (1986), and Saint-Paul (1992). Our model identifies an additional possible reason, namely human health, for a slower than optimal economic growth in a market economy. We are now ready to derive the main result of this section. The Hamiltonian is a function of control, state, and costate variables. We can write the Hamiltonian as follows: 2
H( t) = H( c( t), x( t), e( t), k( t), h( t), z( t), A( t ), A ( t ) , / z ( t ) , 0 ( t ) , A(t)).
(13)
Using Eqs. (A.4) and (A.5) in the Appendix, the total derivative of the Hamiltonian with respect to time can be shown to be equal to:
O H / a t = [ 0 + 6 [ h ( t ) , z ( t ) , t]] [ H ( t ) - u ( t ) ] + A ( t ) 6 t [ h ( t ), z ( t ) , t], (14) where 6t is the partial derivative of 6 with respect to time; 6, is assumed to be strictly positive. Integrating Eq. (14) forwards we get: 30
S
H( t) = oft U( S)e- f,[~(h('~)'z(~)"~)d~+O(s-t)lds oc
s
+ ft 3( h ( s ) , z ( s ) , s) u( s)e [f, 8(h(~),z,~),~)d~+O(s-,)lOs
-f, A(s)Sjh(s), (s),s]e
s
(15)
After rewriting Eq. (15) we have the following theorem:
Theorem. Interest on the expected future (lifetime) utility at time t is adequately measured by the Hamiltonian at time t minus the marginal loss in expected future utility from 'getting one year older' plus the (negative) present value of the future marginal utility (A < O) lost through an age dependent (6 t > O) conditional death risk. More formally: ~c
s
oft u[ c ( s ) , h( s)] e-If, ~(h(~),z(~,,~)d~+o(~-t,] d S
= u[c(t), h ( t ) ] + A ( t ) ( d k / d t ) + I ~ ( t ) ( d h / d t ) + O ( t ) ( d z / d t ) + A(t)(da/dt) 2The state variable A(t) does not show up explicitlyin the current value Hamiltonian H(t) but in the present value Hamiltonian HP(t) since H(t)= HP(t)e °t+art)
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- ft ~(~( h ( s ) , z ( s ) , s) u[ c ( s ) , h( s)] e-If, sS(h(~)'z(7)'~)d~+0(s-t)lds oo
s
+ ft A(s) 6s( h ( s ) , z ( s ) , s)e-if, ~(h(~),z(7),~)d~+0(s-t)] d S. Thus, the Hamiltonian at time t overestimates the maximum sustainable expected utility of society at time t, at least if the death risk increases over time, i.e. (St > 0. This means that the results derived in, for example, M~iler (1991, p. 11) on the definition of a societal net welfare measure, based on the Hamiltonian, are no longer valid in the presence of an endogeneous death risk. This finding is similar to a result in Aronsson and L6fgren (1993) where it is shown that under endogeneous externalities the correct measure of welfare today is the value of the current value Hamiltonian plus the present value of the marginal (positive or negative) external effect. However, a death risk which depends on state variables add further complications, as our Theorem illustrates. In the special case where the death risk 6 is constant and equal to (5o, we have the following result: (0 + 60)E[u~* ] = u [ c ( t ) , h(t)] + I x ( t ) ( d h / d t ) + qJ(t)(dz/dt).
+ A(t)(dk/dt)
(16)
Now the value of the Hamiltonian at time t measures, up to a factor of proportionality, (0 + 60), the expected future lifetime utility of society. In other words, the utility from consumption and the stock of the health capital at time t plus the value in utility units from the investments in health and ordinary capital at time t, are proportional to future expected lifetime utility; the factor of proportionality being the rate of time preference plus the death risk. In the constant death risk case, a net welfare change measure of the kind derived by Hartwick (1990) and M~iler (1991) can be defined by performing a linear approximation of u(0,0) around c(t), h(t) in (16) using the first-order conditions (A.3). After straightforward calculations, one obtains: NWM = A. (c + x ) + A - ( d k / d t ) + u h • h + Ix" ( d h / d t ) + ~b. ( d z / d t )
- A .x,
(16')
where A = Uc[C(t), h(t)] is interpreted as the market price of private goods times the marginal utility of income, see Eq. (A.3i) in the Appendix, u h is the marginal utility of health, ( d k / d t ) is net investment in capital, ( d h / d t ) is net investment in health capital, ( d z / d t ) is the net accumulation of pollution, Ix is the shadow price of or the marginal willingness to pay for health capital times the marginal utility of income, and qt is the negative shadow price of pollution. The first line in Eq. (16') yields the conventional net national product measure (but converted to units of utility since the shadow price of goods A is in units of utility). This measure does not cover the 'unpriced' health and pollution terms in the NWM-measure, but
P.-O. Johansson, K.-G. L6fgren /Journal of Health Economics 14 (1995) 65-79
73
includes purchases of market goods for health purposes. From (16') it can be seen that purchases of health goods net out from the welfare measure, since health goods are inputs in the production of health. On the other hand, health changes are covered by the two first terms in the second line of (16'). If the health capital is eroded over time, i.e. ( d h / d t ) < 0, the conventional NNP measure overestimates welfare, and vice versa. This is analogous to the treatment of unpriced renewable and nonrenewable natural resources in a net welfare change measure; see, for example, Hartwick (Hartwick, 1990; Hartwick 1991) and M~iler (1991). Similarly, the conventional NNP-measure does not cover utility derived from the flow of health (through the unpriced term Uh). Also note that we do not adjust the measure for the impact of improved health on the economy's productivity, though the second line of Eq. (12') may suggest such an adjustment. The reason is that productivity changes are reflected in firms' profits and hence implicitly show up in the first line of (16') through the value of the costate variable )t of the investment equation. This is one of many useful insights our approach produces. Using the optimum conditions (12), (12') and (A.3), it is possible to see how one, at least in principle, can use market data to calculate the shadow price of health capital /x as well as a monetary counterpart to u h. However, such an approach requires an assumption on the properties of the health production function g * ( x ) in Eq. (4), since we must know the magnitude of ~ g * ( . ) / ~ x . Alternatively, the health shadow prices can be estimated by using the contingent valuation method, i.e. by asking samples of the population about their willingness to pay for changes in health. This is also true for changes in pollution, though in the model used in this paper, one can alternatively use market data to estimate Eq. (A.3iii).
3. The market economy and optimal health In this section, we will show that the market economy is unable to correctly account for the positive production externality caused by health capital. The question of how to design a subsidy so as to obtain the socially optimal level of health investment is, therefore, addressed. In order to concentrate on these issues without having to devote too much space to mathematical hair-splitting, we will use a simplified version of the model developed in Section 2: pollution is ignored and the survival probability of mankind is set equal to unity. The reader should note that the NWM-measure for this simplified version looks almost identical to the one in Eq. (16'), the difference being that tp, the shadow price of pollution, now is equal to zero. To reach our goal we start by solving for the command optimum, and we compare this solution to the decentralized solution. This comparison will give us an idea how to internalize the health externality in a market economy. The current value Hamiltonian of the command problem is: n(s) = u[c(s),h(s)] + A(s)[f[k(s),h(s)] - c(s) -x(s)] + tz(s)g[x(s),h(s),y].
(17)
74
p.-o. Johansson, K.-G. Lbfgren/Journal of Health Economics14 (1995) 65-79 The necessary conditions for an optimal path are: uc(. ) - A = 0, - A = 0,
- 0h = - Af , ( . ) ,
(lSa) (18b)
(18c)
/k - 0/x = - Afh (.) -- tzgh(.),
(lSd)
]c = f ( . ) - c - x ,
(18e)
h = g(.),
(18f)
plus transversality conditions, as is shown in Seierstad and Sydsaeter (1987, Theorem 3.17). The maximization problem of the representative individual in our market economy version of the model is assumed to be as follows: MaxE[ ut]c,x
= ft~u[c(s), h(s)]e-°
(19)
subject to the budget constraint 3 and the health accumulation equation:
k( s) = w( s) + r( s)k( s) + 7r( s) - c( s) - x( s),
(19a)
h( s) = g( x( s), h( s), y ).
(19b)
The left hand side of the budget constraint is the net accumulation of real capital (assets) at time s. The right hand side of the budget constraint consists of labor income, w(s), plus capital income at time s less the sum of consumption and resources set aside for health investments. Capital income consists of two components: compensation for capital services provided to firms, r(s)k(s), where r(s) is the market interest at time s, and pure profits, 7r(s). Because of our assumption that the production function is homogeneous of degree 1, pure profits will be equal to zero in equilibrium. See e.g. Blanchard and Fischer (1989) for details. Also note that since the firm is owned by the individual and the labor endowment is put equal to one, labor income is indistinguishable from profit income. The necessary conditions for the optimization problem of the representative individual in the decentralized case are as follows: u c ( . ) - Am = 0 ,
(20a)
I~mgx(.) --A m = 0 ,
(20b)
~ra -- Ol~rn = --t~mF,
(20c)
3 To avoid that debt will be unbounded a No-Ponzi game condition lim e-°tk(t)=O has to be t-~ added.
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75
/-Zm -- Ol~ m = -- i X m g h ( . ) ,
(20d)
k = w + rk + I t - c - x ,
(20e)
h = g( x, h, "/),
(20 0
where Am and tzm are the costate variables corresponding to real capital and health capital, respectively. The economy's representative finn is assumed to maximize its profits at each point of time: max'n'(t) = f [ k ( t ) , h ( t ) ] - w ( t ) - r ( t ) k ( t ) ,
(21)
k(t)
where f[. ] is the production function, and w ( t ) is the equilibrium wage rate, i.e. such that the fixed supply of labor is demanded by the firm. From Eq. (21) it is obvious that the firm will choose its stock of capital so that: (22)
fk[ "] = r ( t )
at each point of time (net of depreciation). It also holds that the value of the marginal product of labor is equal to the wage rate in optimum. Profits plus wages are given by: 7r + w : f ( k ,
(23)
h ) --fk k,
where, in a general equilibrium, relative prices are such that the firm earns a pure profit or quasi-rent, i.e. zr > 0, since workers' provide a factor of production, their health capital, free of charge. Still, the firm has no incentive to further expand production since the workers' health status is known but exogenous to the firm; this is similar to fixed real capital in the short-run. Substituting (22) and (23) into (20e) and comparing the resulting equations with the corresponding equations for the command optimum shows that the only equation which 'fails' is (20d). The individual is not compensated for the fact that his accumulation of health capital contributes to production. Hence the individual is not given the correct incentives to accumulate health capital. The correct way to internalize this externality is to give the individual a unit subsidy Ph = fh('), where fh(') is the marginal productivity of health capital along the optimal path. The individual's flow budget constraint now becomes: k = 7r+ w + rk + p h h -- C --X.
(19a')
Maximization of (19) subject to (19a') and (19b) produces the following version of Eq. (20d): /-£m -- O]£rn = - - I ~ m g h ( " )
-- A m f h ( ' ) "
(20d')
The subsidy is financed by taxing the firm according to the tax function T ( s ) = Phh. Since the firm cannot adjust its use of health capital, T ( s ) is a lump sum tax. The first-order condition for profit maximization is once again (22), and the residual, i.e. profits plus wages, is now: 7r + w = f ( k, h ) - f ~ k - f h h ,
(23')
76
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where the relative prices are different from those in (23) since the general equilibrium behind (23') is different from the one behind (23). By using (22) and (23') in the necessary conditions for the individual's optimum it follows that the market solution will coincide with the command optimum. We have thus shown:
Proposition. If health investment is subsidized at a rate Ph = fh( k *,h *), where an asterisk refers to a socially optimal level, the market economy will attain the socially optimal solution of the decision problem captured by Eq. (17). It is worth noting that although the firm cannot control h it would find, if it was possible to buy health capital at the unit price Ph, that health capital is already correctly provided since Ph = f h along the optimal path. It should also be noted that even if a firm was able to support health investment by its employees, there may be an externality at the aggregate level. This is so because once an employer has invested in his workers health, these workers become attractive to other firms; these firms can pay higher wages since they do not have to pay for the initial investment. It is not easy to construct a subsidy on the rather abstract concept of health capital. An obvious alternative, therefore, seems to be to subsidize purchases of health goods x(t). Reinterpreting the health subsidy in (19a') as being equal to Px x, where Px denotes an ad valorem subsidy on health goods, and repeating the maximization experiment, one obtains the following condition for the steady state case:
[ u h ( ' ) / A x P x ] gx(') = 0 + y,
(24)
where Ax is a costate variable corresponding to A and Am. The problem with this design of the subsidy is that Px cannot be constructed in such a way that the market economy becomes efficient. The reason is that Eq. (20b) is replaced by t Z x g x ( . ) - Axp x = 0 and that the new version of (20d') no longer contains fh(.), i.e. we cannot construct Px so as to induce the market economy to 'replicate' conditions (18). This means that the market economy will not provide the optimal level of health investment. In closing, we have shown that the market economy is unable to provide the optimal level of health capital, unless a subsidy on the stock of health capital is introduced. One can view this as if the firm pays the household for the use of its health capital. The seemingly simpler way of subsidizing purchases of health goods is unable to induce the market economy to fully account for the positive externality caused by improved health. In any case, the enormous amount of information needed in designing optimal health subsidies should be emphasized. In principle, we must solve the social planner's maximization problem in order to be able to design an optimal health subsidy for the market economy.
P.-O. Johansson, K.-G. Lffgren /Journal of Health Economics 14 (1995) 65-79
77
Acknowledgements We are grateful for detailed comments and suggestions from two anonymous referees. Extremely useful comments from a person very familiar with the 1987 book on optimal control theory by Seierstad and Sydsaeter are also acknowledged.
Appendix 1 Using Eq. (9) in the main text, the objective function to be maximized can be written as follows: E[ u,] = fo~6[ h(T), z ( T ) , T ] e - ffath(s,,~(~),~la~
× [forU(C(t ), h( t) )e-°'dtldT.
(A.1)
The following equivalency between areas of integration is true: 0 < T < o o ~ ¢~ [ t < T < o o O-<_t<_T) ~O
E[u,] = ~o f ~ e - [fc'~th(', )'~Cs)"ld" 0
u c(t),h(t)le
°'d,,
(A.2)
where e -/~()ds= 1 - F ( t ) is the 'survival probability' at age t. Hence the expected lifetime utility of society is obtained by summing, from zero to infinity, the product at each point of time of the survival probability and utility. The necessary conditions for an (interior) optimal control is that for each t it holds that:
OH/Oc = uc(c, h) - A = O,
(A.3i)
OH/Ox = tXgx( x, z, h) - A = O,
(ii)
OH/Oe = hfe ( k, e, h) + ~b= O,
(iii)
-- (0 + a ( t ) ) A = -OH/Ok,
(iv)
[z - (0 + A( t))l.t = -OH~Oh,
(v)
¢,- (o+
-OH/Oz,
(vi)
-OH/OA,
(vii)
A-(O+
A(t))A=
k=f(k,e,h)
-c-x,
(viii)
78
P.-O. Johansson, K.-G. Lffgren /Journal of Health Economics 14 (1995) 65-79 h = g ( x, z, h, y ) ,
(ix)
2 = e - otz,
(x)
lim hke -(°t+ a(t)) = 0,
(xi)
t - - ~ oo
lim tzhe -(°t+ a(t)) = 0,
(xii)
l ---~ oo
lira ~0he -(°t+ a(t)) = 0,
(xiii)
where in addition OH/Ok = Ark(.), OH/Oh = uh(.) + hfh(.) + ixgh(.) + A6n(.), and OH/Oz = lzg~(.) - Oa + AO~(.). The transversality conditions written in the manner in Eqs. (xi)-(xiii) presuppose a certain growth condition on the state variables. For details, the reader is referred to Theorem 3.17 in Seierstad and Sydsaeter (1987). At a steady state, it holds that ]c = h --- 2 = ,~ = / ~ = ~, = 0. These assumptions are used in arriving at equations (12'). The total derivative of the Hamiltonian (13) in the main text with respect to time is equal to:
dH/dt = (OH/Oc)(dc/dt) + (OH/Ox)(dx/dt) + (OH/Oe)(de/dt) + (OH/Ok)(dk/dt) + (OH/Oh)(dh/dt) + (OH/Oz)(dz/dt)
+ (OH/Oa)(da/dt) + ( o n / o h ) ( d h / d t ) + (OH/O~)(d~/dt) + (OH/O~b)(d~b/dt) + ( b H / O A ) ( d A i d 0 + (0H/Ot~) 6, where 6 t denotes the partial derivative of 6 with respect to t. Now since
OH/Oc = OH/Ox = OH/Oe = O, OH/Oh = d k / d t , OH/Otz = d h / d t , d z / d t , and OH/OA = d A / d t we can write (A.4) as follows:
OH/O~O=
d H / d t = ( OH/Ok )( d k / d t ) + ( OH/Oh )( dh / d t ) + ( OH/Oz )( d z / d t )
+ (oi-i/oa)(da/dt) + (Ok/at)(dh/dt) + (Oh/Ot)(d~,/dt) + (Oz/Ot)(dO/dt) + ( O a / O t ) ( d a / d t ) +
(OH/O6)6,
= [o+ a(t)] [ h(dk/dt) + ~(dh/dt) + ~0(dz/d/) + a(da/dt] + A 6 t. The final-line expression in (A.5) yields Eq. (14) in the main text.
References Aronsson, T., P.-O. Johansson and K.-G., L6fgren, 1994, Welfare measurement and the health environment, Annals of Operations Research 54, 203-215. Aronsson T. and K.-G. L6fgren, 1993, Welfare consequences of technological and environmental externalities in the Ramsey Growth Model, Natural Resource Modeling. Blanchard, O.J. and S. Fischer, 1989, Lectures on macroeconomics(MIT Press, Cambridge, MA).
P.-O. Johansson, K.-G. Li~fgren /Journal of Health Economics 14 (1995) 65-79
79
Dasgupta, P.S and M., Heal, 1979, Economic theory and exhaustible resources (Cambridge University Press, Oxford). Dasgupta, P.S. and K.G., M~iler, 1990, The environment and emerging developing issues (Wider, Helsinki). Beijer Institute Reprint Series No. 1 (1991), The Royal Swedish Academy of Sciences. Grossman, M., 1972, The demand for health: A theoretical and empirical investigation, NBER Occational Paper 119 (National Bureau of Economic Research, New York). Hartwick, J.M., 1990, Natural resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291-304. Hartwick, J.M., 1991, Degradation of environmental capital and national accounting procedures, European Economic Review 35, 642-649. Jones-Lee, M.W., 1976, The value of life: An economic analysis (Martin Robertson, London). Lucas, R., 1988, On the mechanics of industrial development, Journal of Monetary Economics 22, 3-48. M~iler, K.G., 1991, National accounts and environmental resources, Environmental and Resource Economics 1, 1-15. Mushkin, S.J., 1962, Health as investment, Journal of Political Economy 70, 129-157. Romer, P., 1986, Increasing returns and long-run growth, Journal of Political Economy 94, 1002-1037. Rosen, S., 1988, The value of changes in life expextancy, Journal of Risk and Uncertainty 1,285-304. Saint-Paul, G., 1992, Fiscal policy in an endogenous growth model, Quarterly Journal of Economics 107, 1243-1259. Seierstad, A. and Sydsaeter, K., 1987, Optimal control theory with economic applications (Elsevier, New York). Solow, R.M., 1986, On the intergenerational allocation of natural resources, Scandinavian Journal of Economics 88, 141-149. Tahvonen, O. and J. Kuuluvainen, 1993, Economic growth, pollution, and renewable resources, Journal of Environmental Economics and Management 24. Wagstaff, A., 1986, The demand for health: Theory and applications, Journal of Epidemiology and Community Health 40, 1-11. Weitzman, M., 1976, On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156-162.
Journal of Health Economics 14 (1995) 65-79
ELSEVIER
Wealth from optimal health Per-Olov Johansson
a,*
Karl-Gustaf I_/Sfgren b
a Centre for Health Economics, Stockholm School of Economics, Box 6501, S-113 83 Stockholm, Sweden b Department of Economics, University of Umegz, S-901 87 Umed, Sweden
Received April 1993; revised December 1993
Abstract Recently, much research has been devoted to the question of how the conventional net national product measure should be augmented so as to cover changes in the stocks of natural resources. This paper investigates the treatment of health (capital) and the risk of 'doomsday' caused by pollution in such welfare measures. Our problem is not a standard optimal control problem because the survival probability depends on state variables. We show how to handle this complication. The resulting welfare measure is contrasted with the conventional net national product measure. Finally, we address the matter of how to design a subsidy on health investment such that a market economy provides the optimal level of health. Keywords: National product; Welfare measures; Health capital; Health investment JEL classification: D61; D62; H21; H51; I10; I18
1. Introduction
T h e net national product ( N N P ) m e a s u r e is the prevalent indicator of w e l f a r e in a country. In an elegant paper, W e i t z m a n (1976) s h o w e d that under perfect foresight N N P is a static e q u i v a l e n t o f wealth, and that it m e a s u r e s the m a x i m u m
* Corresponding author. Fax: +46 8 302115. 0167-6296/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 2 9 6 ( 9 4 ) 0 0 0 3 7 - 9
66
P.-O. Johansson, K.-G. L6fgren /Journal of Health Economics 14 (1995) 65-79
sustainable consumption level. Since Weitzman's paper, there has been much research on the relationship between the standard net national product measure and economic welfare in a country; see, for example, Hartwick (1990, Hartwick 1991), Dasgupta and M~iler (1990), Aronsson and L6fgren (1993), and M~iler (1991). Much of this research has been focused on the fact that the NNP-measure does not cover changes in future consumption resulting from current changes in the resource stocks. Various authors have shown that the NNP-measure should be augmented with (positive or negative) net investment in natural resources in order to be a reasonable measure of welfare. Typically, a Ramsey growth model is used to derive these welfare measures. To the best of our knowledge, no one has used a Ramsey model to examine how to incorporate investment in human health in an augmented NNP-measure; see, however, Mushkin's (Mushkin, 1962) discussion which, though informal, is in similar terms. One possible interpretation of the model presented in Section 2 of this paper is in terms of such an augmented NNP-measure. The interpretation of our Theorem, stated in Section 2, is then as follows: many authors have objected on moral grounds to the common practice of discounting the interests of future generations. Dasgupta and Heal (1979), who provide a survey of the discounting debate, claim, just like Solow (1986), that it still may be reasonable to discount the welfare of future generations if the probability of doomsday is strictly positive; the probability of doomsday is then used as a kind of a discount rate. Within such a context, our basic model, developed in Section 2, can be interpreted as follows. Suppose there is a strictly positive extinction probability for mankind due to pollution of the air and the water. The deterioriation of environmental quality also has a direct and adverse impact on human welfare through its impact on health. Investment in an improved environmental quality can be viewed as investment in health capital. In turn, such investment both improves utility and reduces the doomsday probability, though, of course, directly reducing emissions causes the same effects. In any case, one of the principal results derived in this paper is that a positive and endogeneous death risk invalidates the conventional wisdom as stated in Weitzman (1976) and M~iler (1991), for example (saying that the Hamiltonian along an optimal trajectory is a static equivalent of wealth). We also discuss the properties of a market economy when health can be viewed as a factor of production influencing the productivity of the economy. As modelled in this paper, health is taken as a datum by firms and hence causes a kind of positive external effect. In other words, the market economy provides too little health investment. This raises the question how to design a subsidy on health investment, this is addressed in the final section of the paper. The paper is structured as follows. In Section 2, we present the basic model employed in this paper, and derive our main result, which is stated in Theorem 1. In Section 3, we relate our results to the working of a market economy, and discuss how to design a subsidy on health investment such that the market economy produces an optimal solution.
P.-O. Johansson, K.-G. LOfgren/Journal of Health Economics 14 (1995) 65-79
67
2. The basic model: Health and national accounting Our way of viewing health capital in this paper is much inspired by Grossman's (Grossman, 1972) model, as is further explained in, Aronsson, Johansson and L6fgren (Aronsson et al., 1994) though we interpret the model in an entirely different way. We interpret it, not as Grossman does in terms of a finitely-lived individual, but in terms of a society consisting of a succession of identical and finitely-lived individuals. This interpretation is used in order to relate our results to those of Weitzman (1976) and others on the welfare significance of the net national product measure. The instantaneous utility function of the single individual (generation) living at time t is written as follows: u =u(c(t),
h(t)),
(1)
where u(.) is the twice continuously differentiable instantaneous cardinal utility function, which is strictly concave and increasing in its arguments, c(t) is consumption, and h(t) stands for the volume of the health capital at time t. Health capital is produced by refraining from consuming goods and investing the resources in the health sector. Goods are produced by capital, labor, and emissions (through the use of energy inputs). Moreover, the health status of the labor force is assumed to affect its productivity, implying that the stock of health capital is included as a separate argument in the production function. The labor endowment is assumed to be fixed, and is normalized to unity. Assuming that the production function is homogeneous of degree 1, the production per capita can be written as follows: q = f ( k , ei, h),
(2)
where q denotes net output, so depreciation has been accounted for, f(.) is the production function, k = K / L is the capital-labor ratio, K is capital, L ( = 1) is labor, e i is energy used per unit of labor, h is health capital per unit of labor. The production function is assumed to be strictly concave, twice continuously differentiable, increasing in all arguments, all inputs are necessary, i.e. f(0, e i, h ) = f ( k, O, h) = f ( k, e i, 0) = 0, the sign of the cross derivatives, i.e. fkh(') -= O2f(') /Okoh etc., are strictly positive, and production cannot be increased without bound by increasing the use of a single input. These are basicly the assumptions employed by Tahvonen and Kuuluvainen (1993). The accumulation of capital follows the accumulation equation: k = f ( k , ei, h) - c - x ,
(3)
where k = d k / d t , and x denotes goods per unit of labor that are used as an input in the health sector to augment the health capital. The accumulation of health capital follows the equation: /~ = g* ( x , z ) - "yh = g( x, z, h, ~/ ),
(4)
68
P.-O. Johansson, K.-G. Li~fgren/Journal of Health Economics 14 (1995)65-79
where z is the stock of pollution built up through emissions caused by the use of energy inputs in the production of ordinary goods, 3t is a depreciation factor, and g *(x, z) is a strictly concave production function, g * ~ C 2, which is increasing (decreasing) with its first (second) argument, x is a necessary input, and gxz(X, z) < 0 for x > 0. The last assumption means that the marginal productivity of investments in health capital decreases with the stock of pollution. Eq. (4) introduces a quite conventional health production function; see, for example, Grossman (1972) and Wagstaff (1986). Thus, health is treated as any other durable. At the individual level, ageing can be introduced by allowing for a time dependency of the health production function and by letting time (age) appear as an argument in the utility function; see, for example, Jones-Lee (1976, p. 81) for an 'age-dependent' utility function. Our model is however an aggregate one though we have ignored any signs of a birth and death process spanning generations. We have ignored this process and hence also ageing in order to be able to concentrate on some other issues. Finally, we assume that emissions are accumulated in nature, though the environment has an assimilative capacity. The stock of pollution follows the following equation: = e -
c~z,
(5)
where e is the flow of emissions from the production of ordinary goods, and a is a parameter reflecting the environment's assimilative capacity (0 < a < 1). The treatment of emissions means that we have exploited an assumed relationship between a production factor, energy, and the ' production' of emissions. In order to simplify the notation, without any loss of generality, we will in the remainder of this paper assume that e ( t ) = el(t) for all t, and suppress the production of energy. The optimization problem of society is to maximize the expected utility of discounted future utility. More precisely, society is assumed to maximize:
u(c(t),h(t))e-°tdt
MaxE[ut] = M a x E r c,x,e
c,x,e
,
(6)
I 0
subject to:
k(t)=f[k(t),e(t),h(t)]-c(t)-x(t); h(t)=g[x(t),z(t),h(t),~/]; 2(t) =e(t)--otz(t);
k(O)=ko, h(O) = ho,
z(0)=z0,
(6a) (6b) (6c)
where T is the stochastic time to doomsday (or length of human life), and 0 is the non-negative rate of time preference or subjective discount rate. The distribution of T is given by the hazard function (the probability that mankind becomes extinct in the short interval (T, T + d T ) conditional on having survived to T): [ f (T)/(1 -F(T))]dT=
6[h(T), z(T),T]dT,
(7)
P.-O. Johansson, K.-G. L6fgren /Journal of Health Economics 14 (1995) 65-79
69
where F(.) is the cumulative distribution function for doomsday, i.e. T, and F (.) is the density function for T. The hazard is, hence, a function of both the stocks of health capital and pollution at time T, and the 'age' of society. We assume that the intensity parameter 6(.) is non-negative, bounded away from zero, and decreasing in h and increasing in z and T. Integration of (7) yields:
1 - F(T)
= e- f0~th(S)'Z(s)'slos,
(8)
where 1 - F ( T ) is the survival probability for mankind at time ('age') T. Eq. (8) means that we assume that health capital not only increases the quality of life through its presence in the utility function, but that it also has an impact on the expected life-length of humanity. On the other hand, the stock of pollution has a direct impact on the human race's life expectancy, but only an indirect influence on instantaneous utility through its impact on health capital. These assumptions seem realistic. In contrast to the 'value of life' literature, see, for example, Jones-Lee (1976) and Rosen (1988), in our model the death probability depends on (two) state variables, h and z. As is shown below, this fact means that our maximization problem is not a conventional optimal control problem. Using Eq. (8) and the results on the equivalency between areas of integration presented in the appendix at the end of the paper, the optimization problem can be formulated in the following manner: ~c
MaxE[ut] = M a x f c,x,e
e-
fo~lh(s)'z(~)'~ld~u[c(t),h(t)]e-°tdt,
(9)
c,x,e~O
subject to:
Jc(t)=f[k(t),e(t),h(t)]
=-c(t)-x(/);
k ( 0 ) = k 0'
h(t) =g[x(t),z(t),h(t),y]; h(0) = h o, ~(t) = e ( t ) - a z ( t ) ; z(0) = z 0. Unfortunately, this is not a standard optimal control
(9a) (9b) (9c)
problem with an infinite planning horizon since the discount factor contains the state variable h. However, using a standard trick, (9) can be written as follows: oc
MaxE[ut] = M a x f c,x,e
u[c(t), h(t)]e-t°t+a(t)ldt,
(9')
c,x,e~O
with A(t) defined as: a(t) =
fo6[h(s), z ( s ) , s]ds,
(10)
i.e. A(t) is given by the differential equation: A(t) = 6 [ h ( t ) , z ( t ) , t ] ;
A(0) = 0 .
(9d)
70
P.-O. Johansson, K.-G. Li~fgren/Journal of Health Economics14 (1995) 65-79
Thus, by maximizing (9') subject to (9a)-(9d) we turn our problem into a standard control problem with an infinite planning horizon. This approach should be useful for health economists working with intertemporal problems where the survival probability depends on the health capital or other state variables. We can write the current value Hamiltonian as follows:
H( t) = u[ c( t ) , h ( t)] + A ( t ) k ( t ) + tx( t)h( t) + @( t)~( t) + A( t) /i( t), (11) where A(t) = A(t)e °t+ a(t), Iz(t) =/2(t)e °t+a(O, ~ ( t ) = ~(t)e °t+ a(O, and A(t) = y~(t)e0t+ a(o are costate variables. The necessary conditions for an optimal control are stated in the appendix. Combining (i) and (ii) of Eq. (A.3) in the appendix, we have that:
Uc(.) = / z g x ( . ) = A.
(12)
Along an optimal path, the marginal utility of consumption equals the value of the marginal product of the input in health investment, which in turn equals the value (in utility terms) of a unit of capital A. Assuming for the moment that society faces a constant death risk, 8 = 8 0, the following conditions hold in a steady state: 1 fk(-) = O+ 8 o, +/h(.)]
(12') = 0+
+
According to the first-line in (12'), the stock of capital should be such that the marginal product of capital (net of depreciation) is equal to the sum of the marginal rate of time preference and the death probability. This is a kind of Modified Golden Rule result, with the rate of population growth replaced by the 'death rate' ~; see, for example, Blanchard and Fischer (1989). The second-line expression yields a similar result for the optimal stock of health capital, once it is recognized that depreciation is explicitly accounted for (through T) in the second, but not in the first, line of (12'). Note, however, that improved health not only increases human utility but also has a positive impact on firms' productivity through the term fh(.). This is a kind of positive external effect caused by health investment, which a market economy is unable to correctly account for, as is further explained in Section 3. In other words, ceteris paribus underinvestment in health will result unless some policy instrument is introduced. There is a striking similarity between this result and those obtained in endogenous growth theory, where human capital accumulated within a firm through, for example, experience
1 For a version of the model in Section 3 one can show that there exists exactly one steady state, and that this steady state is a saddle point towards which all bounded solutions will converge. The proofs, which are available from the authors, are similar to the corresponding proofs in Tahvonen and Kuuluvainen (1993).
P.-O. Johansson, K.-G. LOfgren/Journal of Health Economics 14 (1995) 65-79
71
and education, causes an external effect at the aggregate level: firms underinvest in human capital since the benefits in part accrue or spill over to other firms through such things as labor mobility; see, for example, Lucas (1988), Romer (1986), and Saint-Paul (1992). Our model identifies an additional possible reason, namely human health, for a slower than optimal economic growth in a market economy. We are now ready to derive the main result of this section. The Hamiltonian is a function of control, state, and costate variables. We can write the Hamiltonian as follows: 2
H( t) = H( c( t), x( t), e( t), k( t), h( t), z( t), A( t ), A ( t ) , / z ( t ) , 0 ( t ) , A(t)).
(13)
Using Eqs. (A.4) and (A.5) in the Appendix, the total derivative of the Hamiltonian with respect to time can be shown to be equal to:
O H / a t = [ 0 + 6 [ h ( t ) , z ( t ) , t]] [ H ( t ) - u ( t ) ] + A ( t ) 6 t [ h ( t ), z ( t ) , t], (14) where 6t is the partial derivative of 6 with respect to time; 6, is assumed to be strictly positive. Integrating Eq. (14) forwards we get: 30
S
H( t) = oft U( S)e- f,[~(h('~)'z(~)"~)d~+O(s-t)lds oc
s
+ ft 3( h ( s ) , z ( s ) , s) u( s)e [f, 8(h(~),z,~),~)d~+O(s-,)lOs
-f, A(s)Sjh(s), (s),s]e
s
(15)
After rewriting Eq. (15) we have the following theorem:
Theorem. Interest on the expected future (lifetime) utility at time t is adequately measured by the Hamiltonian at time t minus the marginal loss in expected future utility from 'getting one year older' plus the (negative) present value of the future marginal utility (A < O) lost through an age dependent (6 t > O) conditional death risk. More formally: ~c
s
oft u[ c ( s ) , h( s)] e-If, ~(h(~),z(~,,~)d~+o(~-t,] d S
= u[c(t), h ( t ) ] + A ( t ) ( d k / d t ) + I ~ ( t ) ( d h / d t ) + O ( t ) ( d z / d t ) + A(t)(da/dt) 2The state variable A(t) does not show up explicitlyin the current value Hamiltonian H(t) but in the present value Hamiltonian HP(t) since H(t)= HP(t)e °t+art)
72
P.-O. Johansson, K.-G. Li~fgren/Journal of Health Economics 14 (1995) 65-79
- ft ~(~( h ( s ) , z ( s ) , s) u[ c ( s ) , h( s)] e-If, sS(h(~)'z(7)'~)d~+0(s-t)lds oo
s
+ ft A(s) 6s( h ( s ) , z ( s ) , s)e-if, ~(h(~),z(7),~)d~+0(s-t)] d S. Thus, the Hamiltonian at time t overestimates the maximum sustainable expected utility of society at time t, at least if the death risk increases over time, i.e. (St > 0. This means that the results derived in, for example, M~iler (1991, p. 11) on the definition of a societal net welfare measure, based on the Hamiltonian, are no longer valid in the presence of an endogeneous death risk. This finding is similar to a result in Aronsson and L6fgren (1993) where it is shown that under endogeneous externalities the correct measure of welfare today is the value of the current value Hamiltonian plus the present value of the marginal (positive or negative) external effect. However, a death risk which depends on state variables add further complications, as our Theorem illustrates. In the special case where the death risk 6 is constant and equal to (5o, we have the following result: (0 + 60)E[u~* ] = u [ c ( t ) , h(t)] + I x ( t ) ( d h / d t ) + qJ(t)(dz/dt).
+ A(t)(dk/dt)
(16)
Now the value of the Hamiltonian at time t measures, up to a factor of proportionality, (0 + 60), the expected future lifetime utility of society. In other words, the utility from consumption and the stock of the health capital at time t plus the value in utility units from the investments in health and ordinary capital at time t, are proportional to future expected lifetime utility; the factor of proportionality being the rate of time preference plus the death risk. In the constant death risk case, a net welfare change measure of the kind derived by Hartwick (1990) and M~iler (1991) can be defined by performing a linear approximation of u(0,0) around c(t), h(t) in (16) using the first-order conditions (A.3). After straightforward calculations, one obtains: NWM = A. (c + x ) + A - ( d k / d t ) + u h • h + Ix" ( d h / d t ) + ~b. ( d z / d t )
- A .x,
(16')
where A = Uc[C(t), h(t)] is interpreted as the market price of private goods times the marginal utility of income, see Eq. (A.3i) in the Appendix, u h is the marginal utility of health, ( d k / d t ) is net investment in capital, ( d h / d t ) is net investment in health capital, ( d z / d t ) is the net accumulation of pollution, Ix is the shadow price of or the marginal willingness to pay for health capital times the marginal utility of income, and qt is the negative shadow price of pollution. The first line in Eq. (16') yields the conventional net national product measure (but converted to units of utility since the shadow price of goods A is in units of utility). This measure does not cover the 'unpriced' health and pollution terms in the NWM-measure, but
P.-O. Johansson, K.-G. L6fgren /Journal of Health Economics 14 (1995) 65-79
73
includes purchases of market goods for health purposes. From (16') it can be seen that purchases of health goods net out from the welfare measure, since health goods are inputs in the production of health. On the other hand, health changes are covered by the two first terms in the second line of (16'). If the health capital is eroded over time, i.e. ( d h / d t ) < 0, the conventional NNP measure overestimates welfare, and vice versa. This is analogous to the treatment of unpriced renewable and nonrenewable natural resources in a net welfare change measure; see, for example, Hartwick (Hartwick, 1990; Hartwick 1991) and M~iler (1991). Similarly, the conventional NNP-measure does not cover utility derived from the flow of health (through the unpriced term Uh). Also note that we do not adjust the measure for the impact of improved health on the economy's productivity, though the second line of Eq. (12') may suggest such an adjustment. The reason is that productivity changes are reflected in firms' profits and hence implicitly show up in the first line of (16') through the value of the costate variable )t of the investment equation. This is one of many useful insights our approach produces. Using the optimum conditions (12), (12') and (A.3), it is possible to see how one, at least in principle, can use market data to calculate the shadow price of health capital /x as well as a monetary counterpart to u h. However, such an approach requires an assumption on the properties of the health production function g * ( x ) in Eq. (4), since we must know the magnitude of ~ g * ( . ) / ~ x . Alternatively, the health shadow prices can be estimated by using the contingent valuation method, i.e. by asking samples of the population about their willingness to pay for changes in health. This is also true for changes in pollution, though in the model used in this paper, one can alternatively use market data to estimate Eq. (A.3iii).
3. The market economy and optimal health In this section, we will show that the market economy is unable to correctly account for the positive production externality caused by health capital. The question of how to design a subsidy so as to obtain the socially optimal level of health investment is, therefore, addressed. In order to concentrate on these issues without having to devote too much space to mathematical hair-splitting, we will use a simplified version of the model developed in Section 2: pollution is ignored and the survival probability of mankind is set equal to unity. The reader should note that the NWM-measure for this simplified version looks almost identical to the one in Eq. (16'), the difference being that tp, the shadow price of pollution, now is equal to zero. To reach our goal we start by solving for the command optimum, and we compare this solution to the decentralized solution. This comparison will give us an idea how to internalize the health externality in a market economy. The current value Hamiltonian of the command problem is: n(s) = u[c(s),h(s)] + A(s)[f[k(s),h(s)] - c(s) -x(s)] + tz(s)g[x(s),h(s),y].
(17)
74
p.-o. Johansson, K.-G. Lbfgren/Journal of Health Economics14 (1995) 65-79 The necessary conditions for an optimal path are: uc(. ) - A = 0, - A = 0,
- 0h = - Af , ( . ) ,
(lSa) (18b)
(18c)
/k - 0/x = - Afh (.) -- tzgh(.),
(lSd)
]c = f ( . ) - c - x ,
(18e)
h = g(.),
(18f)
plus transversality conditions, as is shown in Seierstad and Sydsaeter (1987, Theorem 3.17). The maximization problem of the representative individual in our market economy version of the model is assumed to be as follows: MaxE[ ut]c,x
= ft~u[c(s), h(s)]e-°
(19)
subject to the budget constraint 3 and the health accumulation equation:
k( s) = w( s) + r( s)k( s) + 7r( s) - c( s) - x( s),
(19a)
h( s) = g( x( s), h( s), y ).
(19b)
The left hand side of the budget constraint is the net accumulation of real capital (assets) at time s. The right hand side of the budget constraint consists of labor income, w(s), plus capital income at time s less the sum of consumption and resources set aside for health investments. Capital income consists of two components: compensation for capital services provided to firms, r(s)k(s), where r(s) is the market interest at time s, and pure profits, 7r(s). Because of our assumption that the production function is homogeneous of degree 1, pure profits will be equal to zero in equilibrium. See e.g. Blanchard and Fischer (1989) for details. Also note that since the firm is owned by the individual and the labor endowment is put equal to one, labor income is indistinguishable from profit income. The necessary conditions for the optimization problem of the representative individual in the decentralized case are as follows: u c ( . ) - Am = 0 ,
(20a)
I~mgx(.) --A m = 0 ,
(20b)
~ra -- Ol~rn = --t~mF,
(20c)
3 To avoid that debt will be unbounded a No-Ponzi game condition lim e-°tk(t)=O has to be t-~ added.
P.-O. Johansson, K.-G. Lffgren /Journal of Health Economics 14 (1995) 65-79
75
/-Zm -- Ol~ m = -- i X m g h ( . ) ,
(20d)
k = w + rk + I t - c - x ,
(20e)
h = g( x, h, "/),
(20 0
where Am and tzm are the costate variables corresponding to real capital and health capital, respectively. The economy's representative finn is assumed to maximize its profits at each point of time: max'n'(t) = f [ k ( t ) , h ( t ) ] - w ( t ) - r ( t ) k ( t ) ,
(21)
k(t)
where f[. ] is the production function, and w ( t ) is the equilibrium wage rate, i.e. such that the fixed supply of labor is demanded by the firm. From Eq. (21) it is obvious that the firm will choose its stock of capital so that: (22)
fk[ "] = r ( t )
at each point of time (net of depreciation). It also holds that the value of the marginal product of labor is equal to the wage rate in optimum. Profits plus wages are given by: 7r + w : f ( k ,
(23)
h ) --fk k,
where, in a general equilibrium, relative prices are such that the firm earns a pure profit or quasi-rent, i.e. zr > 0, since workers' provide a factor of production, their health capital, free of charge. Still, the firm has no incentive to further expand production since the workers' health status is known but exogenous to the firm; this is similar to fixed real capital in the short-run. Substituting (22) and (23) into (20e) and comparing the resulting equations with the corresponding equations for the command optimum shows that the only equation which 'fails' is (20d). The individual is not compensated for the fact that his accumulation of health capital contributes to production. Hence the individual is not given the correct incentives to accumulate health capital. The correct way to internalize this externality is to give the individual a unit subsidy Ph = fh('), where fh(') is the marginal productivity of health capital along the optimal path. The individual's flow budget constraint now becomes: k = 7r+ w + rk + p h h -- C --X.
(19a')
Maximization of (19) subject to (19a') and (19b) produces the following version of Eq. (20d): /-£m -- O]£rn = - - I ~ m g h ( " )
-- A m f h ( ' ) "
(20d')
The subsidy is financed by taxing the firm according to the tax function T ( s ) = Phh. Since the firm cannot adjust its use of health capital, T ( s ) is a lump sum tax. The first-order condition for profit maximization is once again (22), and the residual, i.e. profits plus wages, is now: 7r + w = f ( k, h ) - f ~ k - f h h ,
(23')
76
P.-O. Johansson, K.-G. Lffgren /Journal of Health Economics 14 (1995) 65-79
where the relative prices are different from those in (23) since the general equilibrium behind (23') is different from the one behind (23). By using (22) and (23') in the necessary conditions for the individual's optimum it follows that the market solution will coincide with the command optimum. We have thus shown:
Proposition. If health investment is subsidized at a rate Ph = fh( k *,h *), where an asterisk refers to a socially optimal level, the market economy will attain the socially optimal solution of the decision problem captured by Eq. (17). It is worth noting that although the firm cannot control h it would find, if it was possible to buy health capital at the unit price Ph, that health capital is already correctly provided since Ph = f h along the optimal path. It should also be noted that even if a firm was able to support health investment by its employees, there may be an externality at the aggregate level. This is so because once an employer has invested in his workers health, these workers become attractive to other firms; these firms can pay higher wages since they do not have to pay for the initial investment. It is not easy to construct a subsidy on the rather abstract concept of health capital. An obvious alternative, therefore, seems to be to subsidize purchases of health goods x(t). Reinterpreting the health subsidy in (19a') as being equal to Px x, where Px denotes an ad valorem subsidy on health goods, and repeating the maximization experiment, one obtains the following condition for the steady state case:
[ u h ( ' ) / A x P x ] gx(') = 0 + y,
(24)
where Ax is a costate variable corresponding to A and Am. The problem with this design of the subsidy is that Px cannot be constructed in such a way that the market economy becomes efficient. The reason is that Eq. (20b) is replaced by t Z x g x ( . ) - Axp x = 0 and that the new version of (20d') no longer contains fh(.), i.e. we cannot construct Px so as to induce the market economy to 'replicate' conditions (18). This means that the market economy will not provide the optimal level of health investment. In closing, we have shown that the market economy is unable to provide the optimal level of health capital, unless a subsidy on the stock of health capital is introduced. One can view this as if the firm pays the household for the use of its health capital. The seemingly simpler way of subsidizing purchases of health goods is unable to induce the market economy to fully account for the positive externality caused by improved health. In any case, the enormous amount of information needed in designing optimal health subsidies should be emphasized. In principle, we must solve the social planner's maximization problem in order to be able to design an optimal health subsidy for the market economy.
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77
Acknowledgements We are grateful for detailed comments and suggestions from two anonymous referees. Extremely useful comments from a person very familiar with the 1987 book on optimal control theory by Seierstad and Sydsaeter are also acknowledged.
Appendix 1 Using Eq. (9) in the main text, the objective function to be maximized can be written as follows: E[ u,] = fo~6[ h(T), z ( T ) , T ] e - ffath(s,,~(~),~la~
× [forU(C(t ), h( t) )e-°'dtldT.
(A.1)
The following equivalency between areas of integration is true: 0 < T < o o ~ ¢~ [ t < T < o o O-<_t<_T) ~O
E[u,] = ~o f ~ e - [fc'~th(', )'~Cs)"ld" 0
u c(t),h(t)le
°'d,,
(A.2)
where e -/~()ds= 1 - F ( t ) is the 'survival probability' at age t. Hence the expected lifetime utility of society is obtained by summing, from zero to infinity, the product at each point of time of the survival probability and utility. The necessary conditions for an (interior) optimal control is that for each t it holds that:
OH/Oc = uc(c, h) - A = O,
(A.3i)
OH/Ox = tXgx( x, z, h) - A = O,
(ii)
OH/Oe = hfe ( k, e, h) + ~b= O,
(iii)
-- (0 + a ( t ) ) A = -OH/Ok,
(iv)
[z - (0 + A( t))l.t = -OH~Oh,
(v)
¢,- (o+
-OH/Oz,
(vi)
-OH/OA,
(vii)
A-(O+
A(t))A=
k=f(k,e,h)
-c-x,
(viii)
78
P.-O. Johansson, K.-G. Lffgren /Journal of Health Economics 14 (1995) 65-79 h = g ( x, z, h, y ) ,
(ix)
2 = e - otz,
(x)
lim hke -(°t+ a(t)) = 0,
(xi)
t - - ~ oo
lim tzhe -(°t+ a(t)) = 0,
(xii)
l ---~ oo
lira ~0he -(°t+ a(t)) = 0,
(xiii)
where in addition OH/Ok = Ark(.), OH/Oh = uh(.) + hfh(.) + ixgh(.) + A6n(.), and OH/Oz = lzg~(.) - Oa + AO~(.). The transversality conditions written in the manner in Eqs. (xi)-(xiii) presuppose a certain growth condition on the state variables. For details, the reader is referred to Theorem 3.17 in Seierstad and Sydsaeter (1987). At a steady state, it holds that ]c = h --- 2 = ,~ = / ~ = ~, = 0. These assumptions are used in arriving at equations (12'). The total derivative of the Hamiltonian (13) in the main text with respect to time is equal to:
dH/dt = (OH/Oc)(dc/dt) + (OH/Ox)(dx/dt) + (OH/Oe)(de/dt) + (OH/Ok)(dk/dt) + (OH/Oh)(dh/dt) + (OH/Oz)(dz/dt)
+ (OH/Oa)(da/dt) + ( o n / o h ) ( d h / d t ) + (OH/O~)(d~/dt) + (OH/O~b)(d~b/dt) + ( b H / O A ) ( d A i d 0 + (0H/Ot~) 6, where 6 t denotes the partial derivative of 6 with respect to t. Now since
OH/Oc = OH/Ox = OH/Oe = O, OH/Oh = d k / d t , OH/Otz = d h / d t , d z / d t , and OH/OA = d A / d t we can write (A.4) as follows:
OH/O~O=
d H / d t = ( OH/Ok )( d k / d t ) + ( OH/Oh )( dh / d t ) + ( OH/Oz )( d z / d t )
+ (oi-i/oa)(da/dt) + (Ok/at)(dh/dt) + (Oh/Ot)(d~,/dt) + (Oz/Ot)(dO/dt) + ( O a / O t ) ( d a / d t ) +
(OH/O6)6,
= [o+ a(t)] [ h(dk/dt) + ~(dh/dt) + ~0(dz/d/) + a(da/dt] + A 6 t. The final-line expression in (A.5) yields Eq. (14) in the main text.
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