Adam Smith on lotteries: an interpretation and formal restatement

Abstract

The paper concerns a neglected aspect of the Wealth of Nations (with the notable exception of D. Levy 1999), dealing directly with decision under risk. In a few pages from book I, chapter 10, Adam Smith explicitly named “lotteries” various objects of choice (possible occupations, or investment opportunities, for instance) and provided an analysis which standard expected utility glasses would hardly fit. Taking this into account allows a better understanding of the part played by typical characters like the “projector” or the “sober man”, in such matters as Smith’s conception of entrepreneurship or of the credit market. The use of some modern concepts in decision analysis (inverse stochastic dominance, rank dependent utility, prudence toward risk), is a means to show the existence, in Smith’s work, of an original theory from decision under risk, where his analysis of lotteries in the Wealth of Nations is consistent with statements from his moral philosophy on asymmetric sensitivity to gains and losses and to the regulating part played by the impartial spectator.

Keywords:

• Adam Smith
• decision
• risk
• lotteries
• stochastic dominance
• rank-dependent utility
• asymmetric sensitivity
• prudence

JEL CLASSIFICATION::

• B12
• B31
• D01
• D81

Acknowledgements

Previous versions of this paper were discussed on the occasion of several academic events in 2012–2013: HES 2012 Conference in St Catharines (Ontario, Canada); Seminar of the Gredeg in Nice (France), 2012; Workshop on risk in the history of economic analysis in Paris (France), 2013; Seminar of Phare in Paris, 2013. We are particularly indebted for valuable comments to Alain Chateauneuf, Marc-Arthur Diaye, Louis Eeckhoudt, Gilbert Faccarello, Pierre Garrouste, Nathalie Sigot, Peter Wakker and Kazimierz Zaras. The referees of this journal also led us to render explicit our methodological viewpoint. Nonetheless, the usual caveat remains.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Adam Smith’s works are abbreviated as follows (see complete references in the bibliography): TMS = Theory of Moral Sentiments; WN = Wealth of Nations; LJ = Lectures on Jurisprudence (A and B respectively refer to the manuscript from 1762–63 and to the manuscript dated 1766). References are given according to the divisions of the Glasgow edition.

2 A more qualified point of view can be seen in contributions on risk which favor a historical perspective. P.-C. Pradier, for instance, is clearly aware of what is at stake in Smith’s approach which compares fair and unfair lotteries, but he considers that it is of no consequence at a macroscopic level, and that it is so linked to moral philosophy, that it will not be taken up among Smith’s followers (see Pradier 2006, 28–30).

3 Rigorously speaking, one might argue that there is a difference, at least from the potential insured point of view, since Smith adds that he also has to “pay the expence of management” and a profit at a normal rate (WN, I, 10, b, 27).

4 The use of first-degree stochastic dominance plays a crucial part in the appraisal of Smith’s comparison between an initial (La) and a hypothetical modified ($L{\prime }_a$) lottery, because FSD leads to preferences independent from the attitude toward risk. Such is clearly not the case, for instance, when L_a is a state lottery, and $L{\prime }_a$ a “lottery in which no prize exceeded twenty pounds”, though their probability is higher and $L{\prime }_a$ comes closer to a perfectly fair lottery (WN, p. 125). This time, the expected value of the lottery is modified not through an increase in the highest outcome, but through a decrease in the outcome and an increase in its probability. It would be easy to check that, in such a situation, $L{\prime }_a$ does not first-degree stochastically dominate L_a any more, since for all x belonging to $\left]x{\prime }_{a2},x_{a2}\right[,\hspace{0.17em}F{\prime }_a\left(x\right)-F_a\left(x\right)>0$. However, $L{\prime }_a$ second-degree stochastically dominates L_a (SSD) because for all x belonging to $\left[x_{a1},x{\prime }_{a2}\right]$, the expression $H_2\left(x\right)={\int }_{x_{a1}}^x\left[F{\prime }_a\left(t\right)-F_a\left(t\right)\right]dt\le 0$. And, Smith rightly concludes from his comparison that “there would not be the same demand for tickets” (WN, 125). Indeed, although all risk-averters would prefer the modified lottery $L{\prime }_a$ to the initial state lottery L_a, other people (a majority, according to Smith), among which risk-lovers, would prefer L_a to $L{\prime }_a$.

5 A systematic account of results concerning the links between stochastic dominance, direct and indirect up to degree 3, and the properties of the utility function, can be found in Zaras (1989) and (except for TISD1 and TISD2) in Levy (2006, chap. 3) who provides both necessary and sufficient conditions.

6 In standard rank-dependent utility models, U(L) is usually written:

$U\left(L\right)=u\left(x_1\right)+\cdots +\phi (\sum _{i=j+1}^n{p}_i)\left[u\left(x_{j+1}\right)-u\left(x_j\right)\right]+\cdots +\phi \left(p_n\right)\left[u\left(x_n\right)-u\left(x_{n-1}\right)\right]$

which is equivalent to (22) (when $\phi \prime \hspace{0.17em}>\hspace{0.17em}0$; $\phi \left(0\right)=0$; $\phi \left(1\right)=1$). In order to keep notations consistent with those of the previous section, we have kept up the use of ranking outcomes in an increasing order, from the lowest outcome x₁ to the highest x_n. Nonetheless, in recent literature, rank-dependent utility models usually favor notations in which outcomes are ranked in a decreasing order.

7 The same notion of “prudence” was previously identified as “downside risk-aversion” by C. Menezes, C. Geiss and J. Tressler (1980). A general framework has been described by Eeckhoudt and Schlesinger (2006).