Knowledge that Transforms

Pontryagin's Principle can be and has been used in inventory theory, production theory, capital theory and growth theory. The idea presented in this paper shows how the principle can be used also when a firm operates in a market economy or a country in a world market. The resulting jumps in the state variables-amount of capital, amount sold of a commodity, etc.-will disappear through a reinterpretation of the system. The idea is that time is one of the variables and the speed of time can be controlled. By stopping time and letting the other variables change, one can get jumps with respect to time.

A new method is proposed for deriving skew distributions of business finn sizes from the assumption of Gibrat's Law. The growth of the firm is decomposed into an industry-wide component and an individual component, the latter governed by a one-period Markov process. The model is fitted to data on the recent growth of large American firms. A NUMBER of stochastic models, embodying various forms of Gibrat's law of proportionate effect, have been shown to generate skew distribution functions resembling the actual size distributions of business firms. (See [2] and references cited there.) In a previous paper [1] we presented some results of the simultation of such a model permitting serial correlations over time in the size changes of individual firms. The aim of the present paper is to carry further the analysis of autocorrelated growth, by proposing an economically meaningful scheme for its analysis, and applying the scheme to some data on large American firms. In studying business firm growth, we often encounter cases where a firm suddenly acquires an impetus for growth. Perhaps by innovating in production or marketing processes, or perhaps as an effect of new management staffs or techniques, the firm grows much more rapidly than the other firms in the industry, as measured, say, by the ratio of the current firm size to its size in the previous time period. Thus, we may observe that, while most of the firms in the industry are growing at, say, 5% a year, some firms grow 10%. Furthermore, a firm that grew 10% last year is likely to grow more rapidly than average again this year as a result of the carry-over effects of an innovation that occurred in a previous year on operations in subsequent periods. This carry-over becomes more and more likely as we shorten the length of the time period we are considering from a year to a month, week, or day. Moreover, on the average, a firm which grew rapidly in one year subsequently retains a greater share of the industry assets (or market share if sales are used as a measure of firm size) from that time on than do firms that have enjoyed only the average industry growth. Therefore, not only the growth rate over and above the average growth rate, but also the period when the extra growth took place are important factors in the individual firm's growth relative to the industry growth. In this paper, we develop a model to represent such characteristics of firms' growth, so that the process may be analysed further. In the final section we estimate the key parameter of the model for the recent growth of large American business

This paper surveys the results of mostly recent research on optimal aggregate economic growth models, and comments on the difficulties encountered and on desirable directions of further research.

THE PROBLEMS of distribution in an economic system may be analysed either by means of the behavioral assumptions of a competitive model or by the more flexible techniques of n person game theory. In the competitive model, consumers are assumed to respond to a set of prices by maximizing utility subject to a budget constraint and producers by maximizing profit. Consistent production decisions and an allocation of commodities are obtained by the determination of a set of prices at which all markets are in equilibrium. The analysis of these problems by means of n person game theory requires us to specify the production and distribution activities that are available to an arbitrary coalition of economic agents. It is frequently sufficient to summarize the detailed strategic possibilities open to a coalition by the set of possible utility vectors that can be achieved by the coalition. For example, in a pure exchange economy each coalition will have associated with it the collection of all utility vectors that can be obtained by arbitrary redistributions of the resources of that coalition. The core of an n person game is a generalization of Edgeworth's contract curve. A vector of utility levels is suggested which is feasible for all of the players acting collectively, and an arbitrary coalition is examined to see whether it can provide higher utility levels for all of its members. If this is possible, the utility vector which was originally suggested is said to be blocked by the coalition. The core of the n person game consists of those utility vectors which are feasible for the entire group of players and which can be blocked by no coalition. As we have seen during the last several years, there is an intimate connection between these two methods of analysis. If the conventional assumptions of the competitive model are made, such as convexity of preferences and convexity and constant returns to scale for the production set, then there will be a price system at which all markets are in equilibrium and a resulting assignment of commodity bundles to consumers. The utility vector associated with this competitive equilibrium may be shown to be in the core. Even further, if the number of consumers tends

Near the end of World War II, a number of economists advanced the famous balanced-budget multiplier theorem: an equal increase in tax collections and government expenditures will increase the national income by a like amount. The balanced-budget multiplier is equal to one. This important theorem pointed out the existence of a third road to full employment, in addition to the two already well-known (though not yet heavily traveled) deficit-increasing roads, an expenditure increase or a tax cut. Since then, the literature on the subject has concentrated on introducing qualifications into the simple balanced-budget multiplier model. The purpose of this note is to show the relationships among a number of the qualifications. These apparently unrelated qualifications will be seen to fall into place as special cases of a more general formulation. They may all be fruitfully viewed as considering redistribution among sectors of the economy having different marginal propensities to consume. A simple diagrammatic approach that emphasizes this basic similarity will be used. The same diagrammatic approach will finally be used to construct a balanced-budget multiplier model that embraces all of the redistributional qualifications already discussed plus another one of some importance.