Knowledge that Transforms

alone. Problems may arise, however, if a firm employs more than one factor of production. This is because the firm's productivity derives from its ability to organize the collective behavior of its factors. This collective behavior often requires that the factors perform their tasks either simultaneously or consecutively and that their workdays bear some appropriate relationship to one another. Yet only by coincidence would the workdays preferred by each factor conform to this relationship. The firm would therefore hardly be content to offer each factor its going wage and allow it to choose for itself its hours of work. Yet this is precisely the way firms are assumed to behave in traditional theories of factor markets. We would expect instead that the firm will itself decide both the length of the workday and the rate of compensation for each factor. It must choose these wage-hours combinations not only to maximize its own productivity, but also to lure factors successfully away from competing opportunities elsewhere in the economy. These considerations complicate both the theory of the firm and the theory of the consumer, with results which we will see later in this paper. However, we should first note two alternative methods of reconciling the conflicting interests of firms and factors, methods which could conceivably justify the traditional theories of their behavior. The first method assumes heterogeneous preferences among consumers and heterogeneous technologies among firms. In that case, while the

[Any misspecification of the disturbance error process in a linear regression may lead to an inefficient estimator. Although spectral methods proposed by Hannan will always be asymptotically efficient, they are frequently used because they are computationally demanding and very large samples are presumably required. This paper presents Monte Carlo evidence from a variety of typical econometric situations which indicates that the estimators perform quite well for moderate-sized samples (100) when the error process is highly dependent, and even for small samples when the error process is simple. The results are used to estimate a second order term in the asymptotic expansion for the variance.]

An approximate solution, based on the method of dynamic programming, is provided for the optimal control of a system of nonlinear structural equations in econometrics with unknown parameters using a quadratic loss function. It generalizes the methods previously proposed by the author for the control of a nonlinear econometric model with constant parameters and of a linear econometric model with uncertain parameters. It is an improvement over the method of certainty equivalence which replaces the unknown parameters by their mathematical expectations and utilizes the solution for the resulting model. Since the solution is given in the form of feedback control equations, many of the useful concepts and techniques developed in the theory of optimal feedback control for linear systems are now applicable to the control of nonlinear systems using the method proposed, including the calculation of the expected loss of the system under control by analytical rather than Monte Carlo techniques. IN THIS PAPER, I present an approximate solution to the optimal control of a system of nonlinear structural equations using a quadratic welfare loss function when the parameters of the system are unknown. This is a generalization of ths solution given in Chapter 12 of Chow [2] for the control of nonlinear econometric systems with known parameters. It is also a generalization of the solution given in Chow [1] for the control of linear econometric systems with unknown parameters. The method of dynamic programming is applied to solve an optimal control problem involving a nonlinear econometric system with unknown parameters. As it turns out, the solution amounts to linearizing the nonlinear model about some nearly optimal control solution path and then applying a method for controlling the resulting linear model with uncertain parameters. This paper advances the state of the art in the control of nonlinear econometric systems as it improves upon the certainty-equivalence solution which is obtained by replacing the random parameters in a system by their mathematical expectations. It provides for a set of numerical feedback control equations based on a system of nonlinear structural equations in econometrics. It will show that many useful analytical concepts and tools developed in the theory of control of linear systems are indeed applicable to the control of nonlinear systems. Furthermore, in the derivation of an approximate solution using the method of dynamic programming, it will indicate precisely where the approximation takes place and why an exact solution is difficult to achieve. In Section 2, we set up the control problem and provide an exact solution to the optimal control problem for the last period. In Section 3, we give an approximate solution to the multiperiod control problem using dynamic programming. In Section 4, the mathematical expectations required in the solution of Section 3

[Previous work on non-tâtonnement processes has allowed only trading of titles to commodities or promises to produce to take place out of equilibrium. The present work allows production and consumption to take place. The basic model used is that of the Hahn Process, since the making of irreversible commitments in production and consumption seems especially suited to a model whose basic feature is the decline of target profits and utilities. The attempt to introduce production and consumption out of equilibrium brings to the fore a number of problems implicit in most stability analysis which must now be explicitly faced.]

[A certain set of weak rationality conditions is shown to be necessary and sufficient for a social decision function to be a cooperative game according to the formulation of von Neumann and Morgenstern. In exhibiting this broad connection between game theory and the theory of social choice, attention is focused on the critical role played by the blocking coalitions in such games.]

ITS DISAGREEABLE IMPLICATIONS about social choices have convinced many people that Arrow's impossibility theorem rests on unacceptably strong conditions. Dissatisfaction has centered on (but is not limited to) the conditions that the social ordering should be a weak ordering (WO) and that it should be independent of irrelevant alternatives (IIA). A long line of research, culminating in the results of Mas-Colell and Sonnenschein [18] and Fishburn [7], has shown, however, that the impossibility theorem is robust against reasonable relaxations of WO. (More accurately, if WO is weakened, say by waiving completeness or transitivity of the social ordering, and the remaining conditions are correspondingly strengthened to keep the problem interesting, the impossibility remains.) It seems that this line

[A specialization of the Edgeworth type formulae due to Chambers [4] to approximate the marginal distribution of an econometric estimator is presented, and its application to improvement of the use of asymptotic limits in significance testing is discussed. Appendices discuss the validity of Nagar approximations to estimator moments, the exact distribution of the instrumental variable estimator, the Edgeworth approximations for 3SLS and FIML estimators, and the use of Monte Carlo procedures for assessing the appropriate probability to attach to a given significance test.]