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    {\bf  The  Posner   Argument for   Transferring  Health Spending   from   
Old Women to Old Men}   \\
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 October 7, 1996 \\
                    \bigskip
                    Eric Rasmusen \\

Published: Economics Letters 53: 337- 339 (December 1996).\\
                    
                    {\it Abstract} 
                    \end{center}
    Richard Posner suggests several arguments for increasing health care 
spending on males and reducing it on females in his book {\it Aging and Old 
Age}.  I offer a new formalization of his verbal  argument.\\
   Keywords: Aging, Health Economics, Lifetime, Marriage.\\
 JEL Code: I18. 
   		   
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          \noindent 
\hspace*{20pt}	  	  Indiana University
School of Business,    1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
 Email: Erasmuse@Indiana.edu. 
  Web:  http://ezinfo.ucs.indiana.edu/$\sim$erasmuse. 

 FOR EDITING:  Office: (812) 855-9219.  Fax: 812-855-3354.

             \end{small}
 
\begin{small}
               \noindent 
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business Economics and 
Public Policy and Sanjay Subhedar Faculty Fellow,   Indiana University,
Kelley School of Business, BU 456,   
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.   
Php.indiana.edu/$\sim$erasmuse.

 \end{small}

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  On pages 273-278 of {\it Aging and Old Age}, Richard Posner  discusses a 
variety of arguments for reallocating government resources  from ailments of old 
women to ailments of old men.   This note represents a formalization of   two of 
the main arguments for this, arguments that rely on diminishing marginal utility 
and on the utility of marriage.   


   

Assume that  everyone in a population  is married. Let  $L_m$ and $L_w$  be real 
numbers that denote the  lifespan of   men and women, and assume that  $L_w > 
L_m$ under current policy.  Let the utility functions of men and women be
 \begin{equation}
  U_m = f(L_m)  
\end{equation}
  and  
  \begin{equation}
  U_w = f(L_m + \theta(L_w-L_m)), 
\end{equation}
 with $f'>0$, $f''<0$, and $\theta <1$.     Let us take society's total benefit  
to be $U_m + U_w$, which weights the utility of both sexes equally.   

  The man's utility function is simple, since his entire life is married. The 
woman's utility is a function of  $L_m + \theta(L_w-L_m)$ because she lives 
$L_m$ married years and $(L_w-L_m)$ widowed.  


   Note two important features of these assumptions:  (a)   $f''<0$, so 
marginal utility diminishes with lifespan, which can be interpreted as the 
ordinary human response to increased consumption or as a result of the  poor 
health humans experience in their later years.   (b)   $\theta <1$, so 
the woman's utility is higher if her husband is still alive.\footnote{One could   
deduce  $\theta <1$    rather than assuming it, since  anyone for whom it is 
false  could get divorced.  Anyone for whom death rather than divorce ends a 
marriage must prefer marriage to being single, anti-divorce scruples aside.  
(And, of course, assuming that the death is not a murder.)} 
 
 


\noindent
{\it PROPOSITION:  Society's marginal gain from increasing the male lifetime 
exceeds the marginal loss from reducing the female lifetime.} 

\noindent
 PROOF: 
  Reducing   the woman's lifetime has a marginal   cost  of
   \begin{equation}  
    f'_w (L_m + \theta(L_w-L_m)) \cdot \theta,  
\end{equation}
  which is the marginal cost  to her of  reducing her  widowed life, starting 
from her long  base lifetime of $L_w$.  

 Increasing the man's lifetime has a marginal benefit of 
   \begin{equation}
   f'_m (L_m) +     f'_w  (L_m + \theta(L_w-L_m))\cdot  (1-\theta)   ,   
\end{equation} 
 which is the  net marginal benefit to the woman of  increasing her married 
lifetime but  reducing her period of widowhood, plus the marginal benefit to the 
man of increasing his married lifetime.  

 Expression (4) exceeds expression (3) because of three reasons.  (i)  Because 
$L_w > L_m$,   $ L_m + \theta(L_w-L_m) > L_m$ and so   $ f'_w (L_m + \theta(L_w-
L_m))<  f'_m (L_m)$.  (ii) Because $\theta <1$,  and given the first reason,  $ 
f'_w (L_m + \theta(L_w-L_m)) \cdot \theta<    f'_m (L_m)$.   (iii)  $f'_w >0$, 
so  the second term in equation (4) is positive.   Therefore, the  marginal 
effect on social welfare of increasing the man's lifetime is greater than the 
effect of reducing the woman's, which was to be shown.  

\bigskip

  The three   reasons in the proof   each have intuitive explanations.  
 
 First,   the man  starts from a shorter lifetime, and hence values extra life 
more: $f'_m > f'_w$, because  $ L_m < L_m + \theta(L_w-L_m)$ and $f'' <0$.  Even 
if $\theta=1$, so living alone would be  no worse than living married, this 
would remain true.  Hence, assumption (a) alone would be sufficient for  the 
Proposition. 
 
Second, increasing the woman's lifespan does not benefit her as much because 
single life is not as happy as married life: $\theta <1$.  Even if $f``=0$,  so 
there were no diminishing returns to extra lifespan, this would remain true,  
because   $f'_w \cdot \theta <f'_m$.  Hence, assumption (b) alone would be 
sufficient for  the Proposition. 

Third,  increasing  the man's lifespan  benefits both himself and the woman, 
because it reduces her period of widowhood:   the  marginal benefit includes the  
extra term  $f'_w \cdot  (1-\theta)$.   Even if $f''=0$, so there were no 
diminishing returns to extra lifespan, this would remain true.  This is another 
reason why assumption (b) is sufficient to prove the Proposition. 

 As Posner  notes in his book,  implementation of  the policy suggested by these 
arguments   depends on  other variables such as the marginal cost of  increasing 
lifespans, which may be higher for men than for women.  He  himself  is quite 
diffident about applying the idea  to actual policymaking.   The argument is a 
powerful one, however, and   this note shows that it is rigorous enough to be   
consistently modelled.  


\noindent
 REFERENCES


  Posner, Richard (1995) {\it Aging and Old Age}, Chicago: University of Chicago 
Press, 1995. 

 
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