Gerhard Tintner’s Review of The Lognormal Distribution (1958)

Gerhard Tintner’s journal book review assesses J. Aitchison and J. A. C. Brown’s The Lognormal Distribution, with Special Reference to Its Use in Economics. Its central judgment is strongly favorable: the monograph brings together statistical theory, estimation procedures, and economic applications in a form valuable to mathematical statisticians and especially to economic statisticians and econometricians. Tintner’s account moves from the distribution’s mathematical definition through the book’s methodological coverage to the economic applications he regards as its most valuable and original contributions.

The review first explains the family of distributions under discussion. If (Y=\log X) is normally distributed with mean (\mu) and variance (\sigma^2), then (X) follows the two-parameter lognormal distribution and ranges over positive values. A shift by (\tau) produces the three-parameter version, while the transformation ((X-\tau)/(\theta-X)) gives the four-parameter version bounded by (\tau) and (\theta). This opening establishes that the book treats lognormality as a flexible family rather than a single fixed specification. Tintner then identifies its generative principle: the derivation resembles that of the normal distribution, but small errors combine multiplicatively rather than additively. Historical explanation and general properties are followed by a thorough treatment of estimation, illustrated through artificial samples.

The breadth of the monograph matters to Tintner’s assessment. He notes chapters on probit analysis in biological applications, conventional statistical tests, and truncated and censored distributions. A survey of examples extending from astronomy to philology situates the distribution beyond economics. Yet the review’s emphasis shifts decisively when it reaches income and consumption:

Perhaps the most valuable part of the book is the applications of the lognormal distribution to economic problems.

Tintner values the discussion of personal income distribution because it links theoretical reasons for selecting a lognormal model with supporting empirical evidence. Its principal conceptual achievement is to connect a parameter of the distribution with measures of inequality:

Perhaps the most important idea is the application of the Lorenz diagram, which shows that concentration of income is related to the parameter σ² of the lognormal distribution, and the relation of the theory to Gini's coefficient of mean difference.

Here the distribution is useful not merely as a fitted statistical form but as a means of relating income dispersion to economically interpretable measures of concentration. Tintner’s qualified phrasing—“Perhaps” and “Some empirical evidence”—also keeps his praise from becoming a claim of universal applicability.

The treatment of consumer behavior receives the review’s strongest claim to novelty. Centering on the Engel curve, the book derives a lognormal relation between income and purchases first for indivisible goods and then for divisible goods. Inferior goods require the three-parameter form. Tintner also stresses aggregation: under assumptions he calls very reasonable, aggregation yields lognormal distributions again. This links the treatment of consumption at different levels without requiring abandonment of the distributional framework.

The parameters acquire economic interpretations as scales of consumption and income, associated respectively with saturation and cheapness. One consequence is stated succinctly:

The income elasticity is a decreasing function of income.

Tintner reports further extensions that accommodate prices through the saturation parameter, as well as changes in preferences and household composition. These details explain why he treats the consumption chapters as especially novel: the model connects distributional form with substantive features of demand rather than remaining only a descriptive device.

The review closes by returning to practical resources and scholarly coverage. The final chapter addresses computation, and electronically produced tables support estimation and analysis of artificial samples. Tintner praises the 217-item bibliography for including Dutch, French, Italian, and German scholarship alongside English-language work. His sole explicit criticism is the omission of H. T. Davis’s earlier work on income distribution. The resulting assessment is enthusiastic but specific: the monograph’s importance lies in joining a coherent statistical theory to estimation tools and economic interpretations, while its bibliography leaves one significant antecedent unacknowledged.