Knowledge that Transforms

SUMMARY WHEN LISTING addresses within sample blocks for surveys the field interviewer hastily assigns economic ratings of L, M, and H (for low, medium, and high) to dwellings. The means and the standard deviations differ greatly among these strata with regard to socioeconomic characteristics; hence they may be used for allocating different sampling rates to decrease the variance of sample means and totals. Disproportionate sampling rates bring gains in precision for the means of skewed financial items and for estimates based on higher economic subclasses, but they bring losses in estimating most proportions. The many diverse purposes of economic surveys lead to conflicting allocations. To facilitate rational decision among them we developed condensed ways of analyzing and presenting data. The tables display for many variables the relative precision of several allocation schemes, including the optimum one. Standard statistics are extended to provide estimators for subclasses from stratified samples. Then these are used to investigate optimal and other allocations for the subclasses.

This paper considers the problem of criteria for the identifiability of a structural equation which is one of a set of equations linear in the parameters but not in the variables. The criteria are developed in terms of parameter restrictions of the rank condition type. By expansion in Taylor's series and combination with the results of Fisher [2], these results can be easily extended to far more general nonlinear systems. THE USUAL treatment of identifiability criteria is one of linear structures and homogeneous linear restrictions on the coefficients of a single structural equation.2 Recently, I generalized this to the case where restrictions on such coefficients merely have continuous first derivatives; however, it was still the case that only linear structural equations were considered.3 While Anderson and Rubin showed how to obtain limited information, maximum likelihood estimates for the parameters of a linear equation which is part of a nonlinear system and have shown that such estimates have the usual consistency properties,4 they assumed that the equation in question was identifiable under coefficient restrictions of the usual type and did not consider the prior question of the application of the rank and order conditions for identifiability to nonlinear systems.5 This paper considers that question explicitly for a restricted (but highly important) class of nonlinear systems, namely, for systems linear in the unknown coefficients and in the residuals, but not necessarily linear in the variables. Extension to far more general nonlinear cases can be readily accomplished by expansion in Taylor series in the parameters and combina

Edited by L.H. Klaassen, L.M. Koyck and H.J. Witteveen, Presented on the occasion of his 25 years jubilee as a professor at the Netherlands School of Economics at Rotterdam. (With a bibliography of professor Tinbergen's work).

a new family of stable non-gaussian random functions, U(t), and it is intimately connected with random walks of log U(t). Its principal feature is, however, that no use is made of the principle of proportionate effect, the model being rather based upon the fact that there exist certain limits for sums of random functions, upon which the effect of chance in time is multiplicative. (Actually, the result is more general.) This feature provides a new type of motivation for the widespread, convenient, and frequently fruitful use of the logarithm of income, considered as a moral wealth. Looking at the results from another viewpoint, one may say that the approach is based upon a new kind of diachronic factor analysis, the use of which will be justified in detail. (2) I wish to point out the wider role which I believe that these new stochastic processes will eventually play in linear economics, for example in certain problems of aggregation. The reader will also find that the results are translatable with little effort into terms of theories of variation of various economic quantities similar to income. It may even turn out that this approach will be more reasonable for some other such quantities or that the empirical fit will be better in other cases. As a result, the tools to be introduced may be as important as the immediate results which are hopefully to be achieved. In particular, the problem of the distribution and variation of city sizes is very similar to the problem raised by income. A much less obvious generalization

A FUNCTION representing the distribution of income in a society can serve a number of purposes. It may be used to smooth out irregularities in the observed income distribution caused by the misreporting of income. In this role it is similar to the graduation formulae used in demographic work to correct an age distribution distorted by mistatements of age. An income function may also form the basis of a model explaining how an income distribution is generated. Interest here lies in the success with which the model generates a distribution close to that of the observed values and in the meaning that can be ascribed to the parameters in the model. In addition, an income function can assist in the analysis of income distributions by highlighting the more important characteristics of such distributions and providing measures, which can be compared spatially or temporally, of those characteristics. Other uses can no doubt be suggested. For a function to serve these purposes adequately, it is desirable that it should approximate observed distributions of income closely when particular values, usually estimated from the observed data, are given to the parameters. This criterion is the least satisfied by the formulae that have been suggested to date except, perhaps, over limited segments of the income range. The Pareto curve fits income distributions at the extremities of the income range but provides a poor fit over the whole income range. The log normal (or Gibrat) distribution fits reasonably well over a large part of the income range but diverges markedly at the extremities. A function suggested

one is concerned with the interplay between economies of scale and an anticipated persistent growth in demand for capacity. The generalizations discussed here are of two types: (a) the use of probabilities in place of a constant rate of growth in demand; and (b) a study of the economies and the penalties involved in accumulating backlogs of unsatisfied demand. The possibility of accumulating such backlogs raises considerable doubt with respect to Chenery's excess capacity hypothesis. Surprisingly enough, generalization (b) leads to greater difficulties in analysis than (a). The use of probabilities to describe the growth process does little-if anything-to complicate matters. A probabilistic version of Chenery's model turns out to be closely related to the classical problem of gambler's ruin, and a powerful tool can be borrowed from that area-the Laplace transform for the duration of the game. Thanks to this transform, the zero-backlog probabilistic model becomes no more difficult to study than the corresponding deterministic one. A direct implication is that a probabilistic growth course makes it necessary to incur higher expected costs, and also makes it desirable to install plant capacity of a somewhat larger size than would be optimal if demand were growing at a steady rate equal to the expected value of the probabilistic increments. Uncertainty, in this sense, has a stimulating effect upon the magnitude of individual investments.