The Australia
A
an National Univ
versity
Centre fo
or Econ
nomic Policy
P
Researrch
DISCUS
D
USSION PAPER
R
The IInadequacy o
of Friedman an
nd Savaage’s Crittique o
of Dimin
nishingg Margiinal Utiility Wiilliam Colem
man
Research School of Economics
E
The
T Australlian Nationaal University
DISCUSSI
D
ION PAPE
ER NO. 67
76
Novvember, 2012
2
ISSN
N: 1442-8636
ISBN
N: 978-1-9216693-60-1
The Inadequacy of Friedman and Savage’s Critique of Diminishing Marginal Utility William Coleman Research School of Economics The Australian National University 1 Introduction In a well‐known paper Friedman and Savage (1948) advocated a utility function in which the marginal utility of wealth is increasing in wealth above some critical level of wealth. In order to win a reception for this novel conception of the utility function Friedman and Savage took some pains in the first part of their paper to try to shake the grip that the ‘law of diminishing marginal utility ‘ had on the mind of their fellow economists. To that end they criticised a classic ‘elemental argument’ in favour marginal utility being diminishing in wealth. It is a testimony to the strength of the belief in diminishing marginal utility that it has taken so long for the possibility of interpreting gambling and similar phenomena as a contradiction of universal diminishing marginal utility, rather than of utility maximization, to be recognized. The initial mistake must have been at least partly a product of a strong introspective belief in diminishing marginal utility: a dollar must mean less to a rich man than to a poor man; see how much more a man will spend when he is rich than when he is poor to avoid any given amount of pain or discomfort. This elemental argument seems so clearly to justify diminishing marginal utility that it may be desirable even now to state explicitly how this phenomenon can be rationalized equally well on the assumption of increasing marginal utility of money. It is only necessary to suppose that the avoidance of pain and the other goods that can be bought with money are related goods and that, while the margin‐
al utility of money increases as the amount of money increases, the marginal utility of avoiding pain increases even faster. Friedman and Savage, p282‐3 It is the purpose of the note to argue that Friedman and Savage’s attempt to reconcile the rich man’s avoidance of pain with increasing marginal utility is inadequate. More specifically, the note demonstrates that if the rich man spends more than the poor man on both incurring pleasure and avoiding pain then the marginal utility of wealth is diminishing in wealth. We conclude that the ‘elemental argument’ that concludes in favour of diminishing marginal utility ‐ on the basis of a comparison of rich and poor ‐ is valid. 2 The Model We consider someone who can spend their wealth, Y, on either ‘obtaining pleasure’ or ‘avoiding pain’. Spending wealth on obtaining pleasure is denoted C, and spending wealth on avoiding pain is consequently Y‐C. We assume that total utility is the sum of utility from spending to obtain pleasure , P, and the utility from spending to avoid pain, . We also assume that both the marginal utility of spending to obtain pleasure and the marginal utility of spending to avoid pain is positive. U  P (C )   (Y  C ) P '  0,  '  0
Maximisation implies, First Order Condition P ' (C )   ' (Y  C )  0
Second Order Condition P""  0
The second order condition brings out there is no necessity for either P” or ”to be negative. Thus under utility maximization marginal utility of incurring pleasure may be increasing in the quantity of pleasure incurred, and marginal utility of avoiding pain may be increasing in the quantity of pain avoided. However, it is a simple matter to infer from the FOC the ‘marginal propensity to spend on incurring pleasure’, and the ‘marginal propensity to spend on avoiding pain’, are as follows: C
"

Y " P" U"
[Y  C ]

Y
"U "
3 With the second order condition in hand, it is a trivial matter to infer from these two equalities that if both marginal propensities to spend are to be positive ‐if a rich man both obtains more pleasure and avoids more pain than a poor man – then it follows , P" 0
and " 0
That is, both the marginal utility of spending to obtain pleasure is diminishing in such spending, and the marginal utility of spending to avoid pleasure is diminishing in such spending. This implies that the marginal utility of wealth , Y, is diminishing in Y.  2U
0
Y 2
This last and critical inference may seem perfectly obvious implication of P”< 0 and U”< 0. But can we be sure P”< 0 and U”< 0 imples the marginal utility of wealth is diminishing in wealth? The easiest way to demonstrate this is to cast the original utility maximisation problem in a constrained optimisation format, and take advantage of the ‘Langrangian multiplier’ equaling the change in the maximand consequent upon a relaxation of the constraint. U  P (C )   (  )   (Y  C   )
 = spending on avoiding pain Assuming the satisfaction of these FOC, U
 P ' (C )    0
C
4 U
  ' ( )    0

then the theory of constrained optimization tells us, U

Y
It follows from the above  2U  C
P" (C )


Y 2 Y Y
But it was earlier shown that if both propensities to spend are positive then P”(C) is necessarily negative. Thus if both propensities to spend are positive then dC/dY P”(C) is negative, and so the marginal utility of wealth is, indeed, diminishing in wealth. To summarise: if the rich man spends more than the poor man on both obtaining pleasure and avoiding pain – as he surely does – then the marginal utility of wealth is diminishing. Comment Has the above effectively disposed of the Friedman and Savage criticism of the ‘elemental argument’ for the diminishing marginal utility of wealth? It is hard to be sure on account of certain passages in their case quoted at the beginning of this note. In particular Friedman and Savage seem to premise their critique on the avoidance of pain and the obtaining of pleasure being ‘related goods’, presumably meaning a non‐zero cross price elasticity between the two. How such an assumption would be relevant is not made clear by Friedman and Savage; and why it might be relevant is equally unclear. In any case, our modelling of the maxisation problem definitely allows non‐zero cross price elasticity avoiding pain and obtaining pleasure. So our modeling encompasses ‘relatedness’ and still refutes Friedman and Savage. What this paper does claim is that its modeling captures the intuition of the classic argument for diminishing margial utility of wealth on account of the ‘pain avoiding rich man’. That argument works by invoking some event that causes an equal reduction in utility regardless of the wealth of how rich or poor one is. So being caught outdoors in a freezing downpour – or in a suffocating heatwave – will reduce the utility the rich man by the same amount it reduces 5 utility of the poor man.1 But, goes the ‘elemental argument’, the rich man is more likely to hail a taxi to escape this painful weather. This can only mean that a bit of income at the margin (ie the taxi fare), contributes less utility to the rich man than the poor man. 2 This seems a good argument, and this paper has formally vindicated it.3 REFERENCE Friedman, Milton and L. J. Savage 1948, ‘The Utility Analysis of Choices Involving Risk’, Journal of Political Economy, Vol. 56, No. 4 , pp. 279‐304 1
We can multiply painful circumstances that are equally painful to the rich man and the poor; sleep deprivation, sea sickness etc. The rich man is more likely to spend on a comfortable berth, anti‐nausea medication etc. 2
We have not explicitly modeled the utility reduction of the painful circumstance (‘the downpour’), but it is implicit in our utility function. If we suppose that a sufficient expenditure on pain avoidance, X*, can reduce the utility reduction of the painful circumstance to zero then we may say Utility without downpour = P(C) + (X*) But Utility with downpour and no pain avoidance = P(C) + (0) Consequently Utility reduction of downpour with no pain avoidance = (X*) ‐(0) This reduction, notice, is the same for the rich man and the poor man alike, which is what the ‘elemental argument’ turns upon. 3
The argument does operate within the framework of cardinal utility. If that framework is refused in favour of ordinal utility , then any claim that marginal utility is diminishing is meaningless from the outset; just as meaningless any claim that marginal utility is increasing. 6