- Email: [email protected]
Uncertainty and investment in health
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Journal of Health Economics 15 (1996) 369-376
Uncertainty and investment in health Fwu-Ranq Chang * Department of Economics, Indiana Unit, ersi~_ , Ballantine Hall 806, Bloomington, IN 47405, USA Received 15 May 1995; revised 15 October 1995
JEL classification: I 11 Keywords: Health investment; Portfolio selection; Absolute risk aversion
1. Introduction
Great strides have been made in the theory of the demand for health ~ since Grossman (1972) introduced the concept of health capital as a durable good. However, his pure investment model has the unfortunate implication that the optimal investment in health is independent of the initial wealth. This is because, as pointed out by Dardanoni and Wagstaff (1987), the optimal investment in health maximizes the net present value of health care services. Consequently, the health investment decision is separated from consumption decision and the income elasticity of the demand for health is zero. They also pointed out that the introduction of uncertainty into the model could break this separation and showed that investment in health is a normal good if the utility function exhibits decreasing absolute risk aversion. The rich do invest more than the poor in health care.
* Corresponding author. Tel.: + I 812 855-6070; Fax: ~- I 812 855-3736" E-mail: changf@ indiana.edu. i For example, Muurinen (1982) generalized Grossman's model, Cropper (1977) derived the age profile of health investment and occupational choice, Wolfe (1985) analyzed health status and retirement decision, and Ehrlich and Chuma (t9901 extended Grossman's model and removed the indeterminacy problem by allowing the production function to exhibit decreasing returns to scale. 0167-6296/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved. PII SO 1 6 7 - 6 2 9 6 ( 9 6 ) 0 0 0 0 6 - 9
370
F. -R. Chang /" Journal of Health Economics 15 (1996) 369-376
Selden (1993), on the other hand, correctly argued that the sign of the income elasticity of health investment depends on the way uncertainty enters the model. In contrast to Dardanoni-Wagstaff's multiplicative shocks, he presented an example of additive shocks to health capital and showed that the income elasticity of health investment is negative if the utility function exhibits decreasing absolute risk aversion. Investment in health care thus becomes an inferior good. It appears that "relatively minor reformulations of uncertainty" (p. 113) lead to diametrically opposite results. There must be economic reasons, besides the algebraic ones, for these results. The purpose of this note is to provide such a theory. The problem of health investment is approached from a different perspective. The consumer has two choices: current consumption and health investment. Since the source of uncertainty in the model is the fluctuation in one's health capital, the return to health investment is stochastic. The investment in health can thus be regarded as a risky venture. By contrast, the income from savings is risk-free because the market interest rate is assumed constant. In other words, this consumer is an investor who is making a portfolio selection between a risky asset and a risk-flee asset. As such, the problem is a variant of Arrow's celebrated portfolio choice model (Arrow, 1965). This theory of health investment differs from Arrow's model in that there are two random variables in the model. The first is an exogenous source of uncertainty, which is usually assumed to be the fluctuation in health capital. The second is the endogenous rate of return to health investment, which is a transformation of the first. This rate of return is endogenous because it depends on the amount of investment. Moreover, the rate of return assumes both positive and negative values. It raises an interesting question: Is the rate of return necessarily negative when the realization comes from the lower tail of the source of uncertainty? Put it differently, is the rate of return negative when one falls ill? The answer to this question is central to the analysis. It is shown that an increase in the exogenous random variable has two effects on the rate of return. The first is a scale effect, whose sign depends on the way uncertainty transforms the existing health stock into post-shock health. The second is a diminishing-marginal-return effect, which is always non-positive. If the rate of return is an increasing (or a decreasing) function of the random variable, then health investment is a normal (or an inferior) good. Dardanoni-Wagstaff's and Selden's exampIes are special cases of the theory because there is only one non-trivial effect in their models. Specifically, in Dardanoni-Wagstaff's model, there is a positive scale effect and, therefore, health investment is a normal good. By contrast, in Selden's model, there is a negative diminishing-marginal-return effect and, therefore, health investment is an inferior good. When these two effects are opposite in sign (specifically, a positive scale effect against a negative diminishing-marginal-return effect), the net effect depends on their relative strength. Then health investment could have a positive, zero or negative income effect. To shed some light on the problem, consider Dardanoni-
F.-R. Chang / Journal of Health Economics 15 (1996) 369-376
371
Wagstaff's case of multiplicative shocks. It is shown that if the marginal incomegenerating function is elastic, then the negative diminishing-marginal-return effect dominates the positive scale effect. As a result, health investment is an inferior good. One cannot ascertain the sign of the income effect of health investment simply by identifying the type of shocks.
2. Health investment and portfolio selection The model of demand for health follows that of Dardanoni and Wagstaff (1987), which is a two-period version of Grossman's pure investment model (Grossman, 1972). Denote by H~ the stock of health capital in period i, i = 1, 2. Let M be the amount of health care purchased in period 1 and let I ( M ) be the stock of health capital produced by M. Assume I ( M ) is non-negative, twice continuously differentiable, strictly increasing and strictly concave, i.e., I(0) = 0, l ' ( - ) > 0 and 1"(.) < 0. The stock of health capital in period 2 before random shocks is
m = ( 1 - 8)/4, + I ( M ) ,
(,)
where g is the depreciation rate. The income in each period ~, i = 1, 2, is a function of the health stock. In certainty models, Y2 = q~(H2), where the function @(-) is twice continuously differentiable, strictly increasing and concave, i.e., + ' ( . ) > 0 and @"(-) < 0. Call @(.) the income-generating function. The source of uncertainty in the model are shocks to the income-generating function @(.). Notice that the income-generating function is in essence a production function. According to Brock and Mirman (1972), the most general way to introduce uncertainty into a production function is to make uncertainty an argument of the production function. The income-generating function under uncertainty is thus formulated as ~ ( H 2 , e ) , where • is a random variable satisfying ~12'1(H2,•) > 0, xlS'll(H2,e)~ 0 and ~IJ2(H2,•)> 0 for all values of e. Both Dardanoni and Wagstaff (1987) and Selden (1993) are special cases of this formulation. Specifically, Dardanoni and Wagstaff assumed xlr(H, ,e) -- • H 2 with e > 0, while Selden assumed ~ ( H 2 , e ) = q,(H 2 + e). Let A be the initial wealth (non-labor earnings), r be the interest rate, P,,, be the price of health care and C i be the consumption in period i, i = 1, 2. Given A, Y~, H~, r, ~ and P,,, the consumer in period 1 chooses the amount of consumption C~ and the amount of health care M so that in period 2 the consumer has income from savings and income from investment in health, i.e., C 2 = (I + r)(A + Y~ C I - P,,,M) + g'(H2,e). Assuming the utility function is additively separable over time, the consumer solves the following problem: MaxE{U(C,)+V[(I+r)(A+Y,-C,-P,,M)+Rr(H2,e)]}. C h,M
(2)
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F.-R. Chang~Journal of Health Economics 15 (1996) 369-376
As usual, the objective function of problem (2) is strictly concave in C~ and M. Assuming interior solutions, the first-order conditions of problem (2) are
E(U'(C,) - (1 + r)V'(C2) } = 0 , - ( 1 + r) P,.]} = 0.
(3) (4)
To have a better understanding of how uncertainty affects investment decisions, this note approaches problem (2) from a different perspective. This consumer has two choices. The first choice is to invest in the capital market; the rate of return to this financial asset is the market interest rate. Since the market interest rate is assumed constant, ~ the investment in the financial market is risk-free. The second choice is to invest in health capital, which is subject to fluctuations. Thus, the consumer becomes an investor who is making a portfolio choice between the risk-free financial asset and the risky asset, the investment in health. Such a model is a variant of Arrow's model of portfolio selection (Arrow, 1965). Arrow's model can be summarized as follows. An investor has the choice between a risky asset and a risk-free asset. The return to the risk-free asset is zero. The return to the risky asset is a random variable X, which assumes both positive and negative values. Let A and a be respectively the initial wealth and the amount of wealth invested in the risky asset. The investor maximizes the expected utility E[U(Y)] of stochastic income Y = A + a X by choosing an optimal portfolio. The investment in the risky asset is positive if and only if E[X] > 0. The first-order condition for an interior solution is E[U'(Y)X] = 0 and the income elasticity of the risky asset has the same sign as E[U"(Y)X]. The celebrated result is that the sign of E[U"(Y)X] is positive and, therefore, the demand for the risky asset is a normal good, if the utility function U(Y) exhibits decreasing absolute risk aversion. Eq. (4) bears a close resemblance to Arrow's first-order condition. To see this, let
G(~) = * , ( H z , e ) I ' ( M ) - ( 1
+ r)e,,.
(5)
By definition, the partial derivative of the income-generating function ~ ( H 2 , e ) with respect to M, qq(H2,~)I'(M), is the marginal product of health investment. Since the production process is lagged by one period, the cost of per unit health investment is (1 + r)P,,. Thus, the function G(¢) represents the rate of return to health investment. Rewrite Eq. (4) as
E[V'(C2)G(~)] = 0 .
(6)
The resemblance is apparent.
2 Uncertainty concerning future (physical) capital income is called capital risk, according to Sandmo (1969, 1970), The reader is referred to his classic papers for interesting implications.
F.-R. Chang~Journal of Health Economics 15 (1996) 369-376
373
There is, however, a fundamental difference between this model and that of Arrow. In Arrow's model the distribution of the rate of return is independent of the amount invested. He called such an assumption stochastic constant returns to scale. In this model there are two random variables: e, the source of uncertainty, and G({), the rate of return to the risky asset. The assumption of stochastic constant returns to scale does not apply to the rate of return G(e). More importantly, the relationship between e and G(e) needs to be specified. Is the rate of return to health investment necessarily negative when the realization comes from the lower tail of the distribution of e? To be precise, let e * be the particular level of e defined by G ( e * ) = 0 . That is, e* is the realization of e that corresponds to zero rate of return to health investment. What should be the sign of G(e) f o r e < e * and f o r e > e * ' ~ Since W2(H2,e) > 0, C 2 -- (t + r ) ( a + YI - C~ - Po, M ) + XP(H2,e) rises with e. The basic structure of Arrow's model and the normal good property of the demand for the risky asset will be retained if one assumes that G ( e ) < 0 for < e* and G ( e ) > 0 for e > e*. This raises a question. Suppose the source of uncertainty is the fluctuation in health capital. A realization from the lower tail of the distribution of e, which represents a health hazard, would no doubt satisfy < e*. Then the condition G(e) < 0 if e < e* says that the rate of return to health investment is negative when one falls ill. Similarly, G(e) > 0 if e > e* says that the rate of return to health investment is positit, e when one feels great. Must the rate of return to health investment be greater in health than in sickness'? The answer to this question depends on the function G(e). Suppose G(e) is increasing in e. Then G ( e ) < G ( e * ) = 0 for e < e * and G ( e ) > G ( e * ) = 0 for > e*. One thus concludes that health investment is a normal good if the utility function exhibits decreasing absolute risk aversion. Similarly, the income elasticity of health investment becomes negative if G(e) is decreasing in e. From G'(e) = xIrLz(H2,e)I'(M), G'(e) has the same sign as ~12(H2,e). What, then. determines the sign of xPl2(H2,e)?
3. Health-capital risk To bring out the economic intuition on the sign of 'ltr2( He ,e), consider a subclass of uncertainty called health-capital risk in which random shock changes H 2, but leaves the income-generating function q~(-) intact. Such a fluctuation in health capital represents a major illness, not minor ones like colds or flu. 3 Mathematically, xIr(H2,e) - + ( g ( H 2 , ~ ) ) for some function g, with ~ > 0
3 According to Cropper (1977, footnote no. 4), if one considers minor illnesses such as colds, viruses and influenza, then the stock of health capital is not affected and, consequently, preventive health care is planned in the initial time period for the entire lifetime. On the other hand, if one considers major illnesses, then the stock of health capital is reduced when one falls ill. The random shocks in Dardanoni and Wagstaff (1987) and Selden (1993) are all major illnesses, not minor ones,
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and g2 > 0. This class of uncertainty includes Dardanoni-Wagstaff' s and Selden's examples as special cases. The former is characterized by g(H2,e)= eH 2 and ~ ( g ) = g, while the latter is characterized by g(H2,e) = H 2 + ~ and t~"(g) < 0. It is shown in the appendix 4 that, for this class of uncertainty,
~M/~A--- - ( 1 + r ) U " ( C . ) E [ V ' ( C 2 ) G ( e ) ] / D ,
(7)
where D is the determinant of the Hessian matrix of problem (2) and D > 0 by the concavity assumption. Clearly, OM/OA has the same sign as E[V"(C2)G(e)]. This result is reminiscent of Arrow's famous result. Let R(C 2) = - V"(C2 ) / V ' ( C 2 ) be the absolute risk aversion of utility function V(C 2). Following Arrow et al., assume the utility function V(C) exhibits decreasing absolute risk aversion, i.e., R ' ( . ) < 0. It is shown in the appendix that health investment is a normal good (OM/OA > 0) if G'(e) > 0. It is also shown that health investment is an inferior good (~M/OA < 0) if G'(e) < 0. The borderline case G'(e) = 0 implies that income elasticity is zero (OM/OA = 0) and, hence there is a separation of consumption and investment. Under health capital risk, ~](H2,e) = t~'(g(H2,e))g~(H~,e), and
~]z( H2,e) =t~'( g)g,2 + ,"( g)g]g2=tO'( g)[ g , 2 - ( g, gz/g)'q],
(8)
where xI = -gt)"(g)/t~'(g)>_ 0 is the elasticity of the marginal income-generating function. The first term on the right-hand-side of Eq. (8) represents the scale effect of increased • on G(•) = d/(g(H2,e))g~(Hz,e)l'(M) - (1 + r)P m through the scale factor g~(H2,e). This effect has the same sign as g~2. The second term on the right-hand-side of Eq. (8) represents the diminishing-marginal-return effect of increased ~ on G(e) through ~'(g(Hz,e)). Since gj > 0, g2 > 0 and t~"(g) _< 0 (or xI > 0), the latter effect is non-positive.
Case 1. Assume
i.e., the scale effect is negative. Since the diminishingmarginal-return effect is non-positive, G(~) is decreasing in c and therefore investment in health is always an inferior good. By definition, random shocks transform the existing health stock H 2 into post-shock health g(H2,E). Then g]2 < 0 says that falling ill (a reduction in ~) would improve the marginal transformation of H z into post-shock health (an increase in g~). Presumably this is not the most common case. gl2 <~ 0,
Case 2. Assume g~2 = 0, i.e., the scale effect is zero. If + " ( g ) < 0 (or ~q > 0), then G(e) is decreasing in e and health investment is an inferior good. A celebrated example for this case is Selden's additive shocks: g(Hz,~) = 1-[2 + ~. Random shocks, which must have the same physical units as H 2, simply add or 4 The appendix is available upon request from the author.
F.-R. Chang/Journal of Health Economics 15 (1996) 369-376
375
subtract some units of health capital. If ¢ " ( g ) - - 0 (or 11 = 0), then G(e) is a constant and there is a separation of consumption and investment. An obvious example is ~ ( H ~ , ¢ ) = H~ + ~. Introducing uncertainty does not necessarily break this separation result. Case 3. Assume glz > 0, i.e., the scale effect is positive. If + " ( g ) = 0 (or "q = 0), then the scale effect is the only non-trivial effect. Therefore, G(¢) is increasing in and health investment is a normal good. A celebrated example for this case is Dardanoni-Wagstaff's multiplicative shocks: g ( H 2 , ¢ ) : ~H2, ¢ > 0 . Random shocks, which are scalars, simply expand or shrink in size the existing health capital.
On the other hand, if t~"(g) < I) (or "q > 0), then the diminishing-marginal-return effect is negative, which is opposite of the scale effect. The net effect clearly depends on their relative strength. Investment in health can be a normal good, an inferior good or independent of income. To gain some insight, let post-shock health capital be the p r o d u c t of H~ and ~ as in the case of Dardanoni-Wagstaff, i.e., g ( H 2 , ~ ) = ~ H 2. Then Eq. (8) becomes 'v,2(
= +'(g(
-
Clearly, the outcome is dictated by the elasticity "q. If 0 < "q < 1, then health investment is indeed a normal good. This class of functions includes the iso-elastic income-generating functions + ( g ) = g~ o. One may regard Dardanoni-Wagstaff's example as the limiting case when "q = 0. If ~q = 1, then the income effect is zero and the associated income-generating function is of logarithmic form. Again, there is a separation of consumption and investment under uncertainty. However, if -q > !, then health investment is an inferior good. It highlights the point that the income effect of health investment could be negative even if random shocks were introduced multiplicatively. The proposed approach clearly provides more insight than the simple comparison of additive vs. multiplicative shocks.
4. Conclusion Earlier works on the demand for health have had very different conclusions. Dardanoni and Wagstaff (1987) showed that health investment is a normal good if the utility function exhibits decreasing absolute risk aversion. Selden (1993) showed that the opposite is true if uncertainty enters the model additively instead of multiplicatively. This note presents an economic theory that integrates both results. Specifically, the problem of health investment is studied in the context of portfolio selection. Health investment is treated as a risky venture in contrast with
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F.-R. Chang/Journal of Health Economics 15 (1996) 369-376
risk-free financial savings. As a variant of Arrow's celebrated model, it is not surprising that the income elasticity of the demand for health depends on the stochastic rate of return to health investment (Arrow, 1965). As with any demand analysis, the result depends on the tastes and opportunities of the consumer. It is standard in the literature to assume that the utility function exhibits decreasing absolute risk aversion. Then the conclusion should be governed by the description of the opportunity set, in this case the specification of the income-generating function and the way uncertainty transforms the existing health stock into postshock health capital. The proposed theory shows that uncertainty can change the rate of return through a scale effect and a decreasing-marginal-return effect and that the elasticity of the marginal income-generating function plays a role in ascertaining the sign of the income effect of health investment. Dardanoni-Wagstaff's and Selden's examples are special case because one of the effects is trivial in their models. The note concludes with an example that multiplicative shocks to health capital could, under rather standard assumptions, imply an inferior good of health investment.
Acknowledgements I am indebted to two anonymous referees for their valuable comments and suggestions. Naturally, all remaining errors are mine.
References Arrow, K., 1965, The theory of risk aversion, in: Aspects of the theory of risk-bearing (Yrj6 Jahnssonin s~i~itiiS, Heisinki); also in: Collected papers of Kenneth J. Arrow, Individual choice under certainty and uncertainty, Vol. 3 (The Belknap Press of Harvard University Press, Cambridge, 1984). Brock, W. and L. Mirman, 1972, Optimal economic growth and uncertainty: The discounted case, Journal of Economic Theory 4, 479-513. Cropper, M., 1977, Health, investment in health, and occupational choice, Journal of Political Economy 85, 1273-1294. Dardanoni, V. and A. Wagstaff, 1987, Uncertainty, inequalities in health and the demand for health, Journal of Health Economics 6, 283-290. Ehrlich, I. and H. Chuma, 1990, A model of the demand for longevity and the value of life extension, Journal of Political Economy 98, 761-782. Grossman, M., 1972, On the concept of health capital and the demand for health, Journal of Political Economy 80, 223-255. Muurinen, J-M., 1982, Demand for health: A generalized Grossman model, Journal of Health Economics 1, 5-28. Sandmo, A., 1969, Capita] risk, consumption, and portfolio choice, Econometrica 37, 586-599. Sandmo, A., 1970, The effect of uncertainty on saving decisions, Review of Economic Studies 37, 353-360. Selden, T., 1993, Uncertainty and health care spending by the poor: The health capital model revisited, Journal of Health Economics 12, 109-115. Wolfe, J., 1985, A model of declining health and retirement, Journal of Political Economy 93, 1258-1267.
'"
,,
~it'
',,
,'I,
J ~
OF
EC01
'Zl
i"
ELSEVIER
Journal of Health Economics 15 (1996) 369-376
Uncertainty and investment in health Fwu-Ranq Chang * Department of Economics, Indiana Unit, ersi~_ , Ballantine Hall 806, Bloomington, IN 47405, USA Received 15 May 1995; revised 15 October 1995
JEL classification: I 11 Keywords: Health investment; Portfolio selection; Absolute risk aversion
1. Introduction
Great strides have been made in the theory of the demand for health ~ since Grossman (1972) introduced the concept of health capital as a durable good. However, his pure investment model has the unfortunate implication that the optimal investment in health is independent of the initial wealth. This is because, as pointed out by Dardanoni and Wagstaff (1987), the optimal investment in health maximizes the net present value of health care services. Consequently, the health investment decision is separated from consumption decision and the income elasticity of the demand for health is zero. They also pointed out that the introduction of uncertainty into the model could break this separation and showed that investment in health is a normal good if the utility function exhibits decreasing absolute risk aversion. The rich do invest more than the poor in health care.
* Corresponding author. Tel.: + I 812 855-6070; Fax: ~- I 812 855-3736" E-mail: changf@ indiana.edu. i For example, Muurinen (1982) generalized Grossman's model, Cropper (1977) derived the age profile of health investment and occupational choice, Wolfe (1985) analyzed health status and retirement decision, and Ehrlich and Chuma (t9901 extended Grossman's model and removed the indeterminacy problem by allowing the production function to exhibit decreasing returns to scale. 0167-6296/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved. PII SO 1 6 7 - 6 2 9 6 ( 9 6 ) 0 0 0 0 6 - 9
370
F. -R. Chang /" Journal of Health Economics 15 (1996) 369-376
Selden (1993), on the other hand, correctly argued that the sign of the income elasticity of health investment depends on the way uncertainty enters the model. In contrast to Dardanoni-Wagstaff's multiplicative shocks, he presented an example of additive shocks to health capital and showed that the income elasticity of health investment is negative if the utility function exhibits decreasing absolute risk aversion. Investment in health care thus becomes an inferior good. It appears that "relatively minor reformulations of uncertainty" (p. 113) lead to diametrically opposite results. There must be economic reasons, besides the algebraic ones, for these results. The purpose of this note is to provide such a theory. The problem of health investment is approached from a different perspective. The consumer has two choices: current consumption and health investment. Since the source of uncertainty in the model is the fluctuation in one's health capital, the return to health investment is stochastic. The investment in health can thus be regarded as a risky venture. By contrast, the income from savings is risk-free because the market interest rate is assumed constant. In other words, this consumer is an investor who is making a portfolio selection between a risky asset and a risk-flee asset. As such, the problem is a variant of Arrow's celebrated portfolio choice model (Arrow, 1965). This theory of health investment differs from Arrow's model in that there are two random variables in the model. The first is an exogenous source of uncertainty, which is usually assumed to be the fluctuation in health capital. The second is the endogenous rate of return to health investment, which is a transformation of the first. This rate of return is endogenous because it depends on the amount of investment. Moreover, the rate of return assumes both positive and negative values. It raises an interesting question: Is the rate of return necessarily negative when the realization comes from the lower tail of the source of uncertainty? Put it differently, is the rate of return negative when one falls ill? The answer to this question is central to the analysis. It is shown that an increase in the exogenous random variable has two effects on the rate of return. The first is a scale effect, whose sign depends on the way uncertainty transforms the existing health stock into post-shock health. The second is a diminishing-marginal-return effect, which is always non-positive. If the rate of return is an increasing (or a decreasing) function of the random variable, then health investment is a normal (or an inferior) good. Dardanoni-Wagstaff's and Selden's exampIes are special cases of the theory because there is only one non-trivial effect in their models. Specifically, in Dardanoni-Wagstaff's model, there is a positive scale effect and, therefore, health investment is a normal good. By contrast, in Selden's model, there is a negative diminishing-marginal-return effect and, therefore, health investment is an inferior good. When these two effects are opposite in sign (specifically, a positive scale effect against a negative diminishing-marginal-return effect), the net effect depends on their relative strength. Then health investment could have a positive, zero or negative income effect. To shed some light on the problem, consider Dardanoni-
F.-R. Chang / Journal of Health Economics 15 (1996) 369-376
371
Wagstaff's case of multiplicative shocks. It is shown that if the marginal incomegenerating function is elastic, then the negative diminishing-marginal-return effect dominates the positive scale effect. As a result, health investment is an inferior good. One cannot ascertain the sign of the income effect of health investment simply by identifying the type of shocks.
2. Health investment and portfolio selection The model of demand for health follows that of Dardanoni and Wagstaff (1987), which is a two-period version of Grossman's pure investment model (Grossman, 1972). Denote by H~ the stock of health capital in period i, i = 1, 2. Let M be the amount of health care purchased in period 1 and let I ( M ) be the stock of health capital produced by M. Assume I ( M ) is non-negative, twice continuously differentiable, strictly increasing and strictly concave, i.e., I(0) = 0, l ' ( - ) > 0 and 1"(.) < 0. The stock of health capital in period 2 before random shocks is
m = ( 1 - 8)/4, + I ( M ) ,
(,)
where g is the depreciation rate. The income in each period ~, i = 1, 2, is a function of the health stock. In certainty models, Y2 = q~(H2), where the function @(-) is twice continuously differentiable, strictly increasing and concave, i.e., + ' ( . ) > 0 and @"(-) < 0. Call @(.) the income-generating function. The source of uncertainty in the model are shocks to the income-generating function @(.). Notice that the income-generating function is in essence a production function. According to Brock and Mirman (1972), the most general way to introduce uncertainty into a production function is to make uncertainty an argument of the production function. The income-generating function under uncertainty is thus formulated as ~ ( H 2 , e ) , where • is a random variable satisfying ~12'1(H2,•) > 0, xlS'll(H2,e)~ 0 and ~IJ2(H2,•)> 0 for all values of e. Both Dardanoni and Wagstaff (1987) and Selden (1993) are special cases of this formulation. Specifically, Dardanoni and Wagstaff assumed xlr(H, ,e) -- • H 2 with e > 0, while Selden assumed ~ ( H 2 , e ) = q,(H 2 + e). Let A be the initial wealth (non-labor earnings), r be the interest rate, P,,, be the price of health care and C i be the consumption in period i, i = 1, 2. Given A, Y~, H~, r, ~ and P,,, the consumer in period 1 chooses the amount of consumption C~ and the amount of health care M so that in period 2 the consumer has income from savings and income from investment in health, i.e., C 2 = (I + r)(A + Y~ C I - P,,,M) + g'(H2,e). Assuming the utility function is additively separable over time, the consumer solves the following problem: MaxE{U(C,)+V[(I+r)(A+Y,-C,-P,,M)+Rr(H2,e)]}. C h,M
(2)
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F.-R. Chang~Journal of Health Economics 15 (1996) 369-376
As usual, the objective function of problem (2) is strictly concave in C~ and M. Assuming interior solutions, the first-order conditions of problem (2) are
E(U'(C,) - (1 + r)V'(C2) } = 0 , - ( 1 + r) P,.]} = 0.
(3) (4)
To have a better understanding of how uncertainty affects investment decisions, this note approaches problem (2) from a different perspective. This consumer has two choices. The first choice is to invest in the capital market; the rate of return to this financial asset is the market interest rate. Since the market interest rate is assumed constant, ~ the investment in the financial market is risk-free. The second choice is to invest in health capital, which is subject to fluctuations. Thus, the consumer becomes an investor who is making a portfolio choice between the risk-free financial asset and the risky asset, the investment in health. Such a model is a variant of Arrow's model of portfolio selection (Arrow, 1965). Arrow's model can be summarized as follows. An investor has the choice between a risky asset and a risk-free asset. The return to the risk-free asset is zero. The return to the risky asset is a random variable X, which assumes both positive and negative values. Let A and a be respectively the initial wealth and the amount of wealth invested in the risky asset. The investor maximizes the expected utility E[U(Y)] of stochastic income Y = A + a X by choosing an optimal portfolio. The investment in the risky asset is positive if and only if E[X] > 0. The first-order condition for an interior solution is E[U'(Y)X] = 0 and the income elasticity of the risky asset has the same sign as E[U"(Y)X]. The celebrated result is that the sign of E[U"(Y)X] is positive and, therefore, the demand for the risky asset is a normal good, if the utility function U(Y) exhibits decreasing absolute risk aversion. Eq. (4) bears a close resemblance to Arrow's first-order condition. To see this, let
G(~) = * , ( H z , e ) I ' ( M ) - ( 1
+ r)e,,.
(5)
By definition, the partial derivative of the income-generating function ~ ( H 2 , e ) with respect to M, qq(H2,~)I'(M), is the marginal product of health investment. Since the production process is lagged by one period, the cost of per unit health investment is (1 + r)P,,. Thus, the function G(¢) represents the rate of return to health investment. Rewrite Eq. (4) as
E[V'(C2)G(~)] = 0 .
(6)
The resemblance is apparent.
2 Uncertainty concerning future (physical) capital income is called capital risk, according to Sandmo (1969, 1970), The reader is referred to his classic papers for interesting implications.
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There is, however, a fundamental difference between this model and that of Arrow. In Arrow's model the distribution of the rate of return is independent of the amount invested. He called such an assumption stochastic constant returns to scale. In this model there are two random variables: e, the source of uncertainty, and G({), the rate of return to the risky asset. The assumption of stochastic constant returns to scale does not apply to the rate of return G(e). More importantly, the relationship between e and G(e) needs to be specified. Is the rate of return to health investment necessarily negative when the realization comes from the lower tail of the distribution of e? To be precise, let e * be the particular level of e defined by G ( e * ) = 0 . That is, e* is the realization of e that corresponds to zero rate of return to health investment. What should be the sign of G(e) f o r e < e * and f o r e > e * ' ~ Since W2(H2,e) > 0, C 2 -- (t + r ) ( a + YI - C~ - Po, M ) + XP(H2,e) rises with e. The basic structure of Arrow's model and the normal good property of the demand for the risky asset will be retained if one assumes that G ( e ) < 0 for < e* and G ( e ) > 0 for e > e*. This raises a question. Suppose the source of uncertainty is the fluctuation in health capital. A realization from the lower tail of the distribution of e, which represents a health hazard, would no doubt satisfy < e*. Then the condition G(e) < 0 if e < e* says that the rate of return to health investment is negative when one falls ill. Similarly, G(e) > 0 if e > e* says that the rate of return to health investment is positit, e when one feels great. Must the rate of return to health investment be greater in health than in sickness'? The answer to this question depends on the function G(e). Suppose G(e) is increasing in e. Then G ( e ) < G ( e * ) = 0 for e < e * and G ( e ) > G ( e * ) = 0 for > e*. One thus concludes that health investment is a normal good if the utility function exhibits decreasing absolute risk aversion. Similarly, the income elasticity of health investment becomes negative if G(e) is decreasing in e. From G'(e) = xIrLz(H2,e)I'(M), G'(e) has the same sign as ~12(H2,e). What, then. determines the sign of xPl2(H2,e)?
3. Health-capital risk To bring out the economic intuition on the sign of 'ltr2( He ,e), consider a subclass of uncertainty called health-capital risk in which random shock changes H 2, but leaves the income-generating function q~(-) intact. Such a fluctuation in health capital represents a major illness, not minor ones like colds or flu. 3 Mathematically, xIr(H2,e) - + ( g ( H 2 , ~ ) ) for some function g, with ~ > 0
3 According to Cropper (1977, footnote no. 4), if one considers minor illnesses such as colds, viruses and influenza, then the stock of health capital is not affected and, consequently, preventive health care is planned in the initial time period for the entire lifetime. On the other hand, if one considers major illnesses, then the stock of health capital is reduced when one falls ill. The random shocks in Dardanoni and Wagstaff (1987) and Selden (1993) are all major illnesses, not minor ones,
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and g2 > 0. This class of uncertainty includes Dardanoni-Wagstaff' s and Selden's examples as special cases. The former is characterized by g(H2,e)= eH 2 and ~ ( g ) = g, while the latter is characterized by g(H2,e) = H 2 + ~ and t~"(g) < 0. It is shown in the appendix 4 that, for this class of uncertainty,
~M/~A--- - ( 1 + r ) U " ( C . ) E [ V ' ( C 2 ) G ( e ) ] / D ,
(7)
where D is the determinant of the Hessian matrix of problem (2) and D > 0 by the concavity assumption. Clearly, OM/OA has the same sign as E[V"(C2)G(e)]. This result is reminiscent of Arrow's famous result. Let R(C 2) = - V"(C2 ) / V ' ( C 2 ) be the absolute risk aversion of utility function V(C 2). Following Arrow et al., assume the utility function V(C) exhibits decreasing absolute risk aversion, i.e., R ' ( . ) < 0. It is shown in the appendix that health investment is a normal good (OM/OA > 0) if G'(e) > 0. It is also shown that health investment is an inferior good (~M/OA < 0) if G'(e) < 0. The borderline case G'(e) = 0 implies that income elasticity is zero (OM/OA = 0) and, hence there is a separation of consumption and investment. Under health capital risk, ~](H2,e) = t~'(g(H2,e))g~(H~,e), and
~]z( H2,e) =t~'( g)g,2 + ,"( g)g]g2=tO'( g)[ g , 2 - ( g, gz/g)'q],
(8)
where xI = -gt)"(g)/t~'(g)>_ 0 is the elasticity of the marginal income-generating function. The first term on the right-hand-side of Eq. (8) represents the scale effect of increased • on G(•) = d/(g(H2,e))g~(Hz,e)l'(M) - (1 + r)P m through the scale factor g~(H2,e). This effect has the same sign as g~2. The second term on the right-hand-side of Eq. (8) represents the diminishing-marginal-return effect of increased ~ on G(e) through ~'(g(Hz,e)). Since gj > 0, g2 > 0 and t~"(g) _< 0 (or xI > 0), the latter effect is non-positive.
Case 1. Assume
i.e., the scale effect is negative. Since the diminishingmarginal-return effect is non-positive, G(~) is decreasing in c and therefore investment in health is always an inferior good. By definition, random shocks transform the existing health stock H 2 into post-shock health g(H2,E). Then g]2 < 0 says that falling ill (a reduction in ~) would improve the marginal transformation of H z into post-shock health (an increase in g~). Presumably this is not the most common case. gl2 <~ 0,
Case 2. Assume g~2 = 0, i.e., the scale effect is zero. If + " ( g ) < 0 (or ~q > 0), then G(e) is decreasing in e and health investment is an inferior good. A celebrated example for this case is Selden's additive shocks: g(Hz,~) = 1-[2 + ~. Random shocks, which must have the same physical units as H 2, simply add or 4 The appendix is available upon request from the author.
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subtract some units of health capital. If ¢ " ( g ) - - 0 (or 11 = 0), then G(e) is a constant and there is a separation of consumption and investment. An obvious example is ~ ( H ~ , ¢ ) = H~ + ~. Introducing uncertainty does not necessarily break this separation result. Case 3. Assume glz > 0, i.e., the scale effect is positive. If + " ( g ) = 0 (or "q = 0), then the scale effect is the only non-trivial effect. Therefore, G(¢) is increasing in and health investment is a normal good. A celebrated example for this case is Dardanoni-Wagstaff's multiplicative shocks: g ( H 2 , ¢ ) : ~H2, ¢ > 0 . Random shocks, which are scalars, simply expand or shrink in size the existing health capital.
On the other hand, if t~"(g) < I) (or "q > 0), then the diminishing-marginal-return effect is negative, which is opposite of the scale effect. The net effect clearly depends on their relative strength. Investment in health can be a normal good, an inferior good or independent of income. To gain some insight, let post-shock health capital be the p r o d u c t of H~ and ~ as in the case of Dardanoni-Wagstaff, i.e., g ( H 2 , ~ ) = ~ H 2. Then Eq. (8) becomes 'v,2(
= +'(g(
-
Clearly, the outcome is dictated by the elasticity "q. If 0 < "q < 1, then health investment is indeed a normal good. This class of functions includes the iso-elastic income-generating functions + ( g ) = g~ o. One may regard Dardanoni-Wagstaff's example as the limiting case when "q = 0. If ~q = 1, then the income effect is zero and the associated income-generating function is of logarithmic form. Again, there is a separation of consumption and investment under uncertainty. However, if -q > !, then health investment is an inferior good. It highlights the point that the income effect of health investment could be negative even if random shocks were introduced multiplicatively. The proposed approach clearly provides more insight than the simple comparison of additive vs. multiplicative shocks.
4. Conclusion Earlier works on the demand for health have had very different conclusions. Dardanoni and Wagstaff (1987) showed that health investment is a normal good if the utility function exhibits decreasing absolute risk aversion. Selden (1993) showed that the opposite is true if uncertainty enters the model additively instead of multiplicatively. This note presents an economic theory that integrates both results. Specifically, the problem of health investment is studied in the context of portfolio selection. Health investment is treated as a risky venture in contrast with
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risk-free financial savings. As a variant of Arrow's celebrated model, it is not surprising that the income elasticity of the demand for health depends on the stochastic rate of return to health investment (Arrow, 1965). As with any demand analysis, the result depends on the tastes and opportunities of the consumer. It is standard in the literature to assume that the utility function exhibits decreasing absolute risk aversion. Then the conclusion should be governed by the description of the opportunity set, in this case the specification of the income-generating function and the way uncertainty transforms the existing health stock into postshock health capital. The proposed theory shows that uncertainty can change the rate of return through a scale effect and a decreasing-marginal-return effect and that the elasticity of the marginal income-generating function plays a role in ascertaining the sign of the income effect of health investment. Dardanoni-Wagstaff's and Selden's examples are special case because one of the effects is trivial in their models. The note concludes with an example that multiplicative shocks to health capital could, under rather standard assumptions, imply an inferior good of health investment.
Acknowledgements I am indebted to two anonymous referees for their valuable comments and suggestions. Naturally, all remaining errors are mine.
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