This paper explores a new approach to the Nash bargaining problem in which the axiom of symmetry is dropped and it is assumed that the final allocation depends on both the status quo and the threat point. The resulting final allocation, unlike that formalized by Nash, cannot be represented by a simple analytic expression; rather, it leads to a whole class of solutions. Properties of the final allocation are analyzed. It is shown that for every initial allocation there exists a Nash fiber, corresponding to the Nash allocation, that it is possible to determine the sign of the derivatives of the final allocation with respect to changes in the threat point, and that a Slutsky-like equation relates these derivatives to the derivatives with respect to the initial allocation. It is also shown that, under certaiia conditions, as play is repeated the final allocation asymptotically converges to the Nash allocation.

This paper considers tests for structural instability of short duration, such as at the end of the sample. The key feature of the testing problem is that the number, m, of observations in the period of potential change is relatively small—possibly as small as one. The well-known F test of Chow (1960) for this problem only applies in a linear regression model with normally distributed iid errors and strictly exogenous regressors, even when the total number of observations, n+m, is large. We generalize the F test to cover regression models with much more general error processes, regressors that are not strictly exogenous, and estimation by instrumental variables as well as least squares. In addition, we extend the F test to nonlinear models estimated by generalized method of moments and maximum likelihood. Asymptotic critical values that are valid as n→∞ with m fixed are provided using a subsampling-like method. The results apply quite generally to processes that are strictly stationary and ergodic under the null hypothesis of no structural instability.

The description for this book, Adaptive Control Processes: A Guided Tour, will be forthcoming.

WHEN TO REPLACE individual units of durable equipment by similar or improved units is one of the main problems, upon which the success of industrial enterprise depends. Nevertheless, no unified presentation of its many aspects appears to have been published up to the present. The principal writers refer to replacement merely incidentally, when discussing the subject of depreciation. From the theoretical point of view, such an approach really amounts to putting the cart before the horse.' Replacement is the basic problem, because it actually affects the composition and productivity of a plant. Calculations of depreciation are mere figures entered into books, the significance of which depends entirely on the use to which they are put. The concept of depreciation does not enter into the theory of capital value at all. In practice, on the other hand, differences in depreciation methods do to some extent influence the judgment of traders in the negotiable symbols of composite capital goods. This anomaly is due partly to defective accounting methods. A study of the replacement problem by itself must precede attempts to correct the situation. The value aspect of replacement or arises from the familiar phenomenon that many types of machines outlive their usefulness. The income stream derived from their operation gradually declines, until a more attractive alternative becomes available. The theory that the economic life of a machine is a period which makes the unit cost (plus interest) of the product a minimum, appears to have been originated by Professor J. S. Taylor.2 His algebraic presentation was simplified and refined by Professor Harold Hotelling,3 who employs continuous functions for the purpose. The basic formula given by the latter writer is:4

IN A PREVIOUS article1 I made a brief and incomplete survey of the theory of depreciation. In the present paper I discuss its practice. One obstacle to practical progress in this field is that mathematically trained minds are seldom well informed on what accountants actually do. The latter are therefore more often criticized for methods which they are not using than for those which they are. Even otherwise valuable contributions thus elicit opposition quite unnecessarily. The inappropriate antithesis tends to discredit the rest of the argument and prompts general retorts, for instance that is a of . . . determined by the practices of men.-Where accounting treatment diverges from economic theory, a similar divergence is likely to be found between economic theory and practice.2 Such an attitude, in turn, is not very helpful or progressive, even if the dangerous phrase tool of business is interpreted only in its best possible sense. In the article cited, I probably added to the already existing confusion by calling sample methods by certain names without proper qualification, although the same names are commonly applied to substantially different methods. The truth is that the familiar singlemachine formulae permit of different interpretations. To clarify the situation, the present paper identifies a greater number of methods unequivocally by developing their basic many-machine equations and comparing the results. References to practice and to individual writers' ideas are made wherever possible, before choosing a method which appears best suited to the practical needs of large enterprises and the investing public.