Knowledge that Transforms

Equilibria in stochastic economic models are often time series which fluctuate in complex ways. But it is sometimes possible to summarize the long run, characteristics of these fluctuations. For example, if the law of motion determined by economic interactions is Markovian and if the equilibrium time series converges in a specific probabilistic sense then the long run behavior is completely determined by an invariant probability distribution. This paper develops and unifies a number of results found in the probability literature which enable one to prove, under very general conditions, the existence of an invariant distribution and the convergence of the corresponding Markov process. VIRTUALLY ALL OF ECONOMIC THEORY focuses upon the study of economic equilibrium. This concept has recently undergone several subtle elaborations. No longer must a system of markets in equilibrium be thought of as one at rest in a static steady state. Instead there is a growing body of literature (e.g., [4, 5, 12, 16, 20, 21]) which defines equilibrium as a stochastic process of market clearing prices and quantities which is consistent with the self-interested behavior of economic agents. Needless to say equilibrium stochastic processes can be very complex time series which fluctuate in irregular ways. For theoretical and econometric purposes it is useful to have a convenient way of summarizing the average behavior of such processes over time. This paper draws together and unifies a number of fundamental results from the probability literature which enable one to do this for discrete time, Markov processes on general state spaces. The starting point of the analysis is a set S of economic states (e.g., prices and/or quantities). The only technical restriction placed upon S is that it be a Borel subset of a complete, separable metric space. The second datum is a transition probability P(s, ) on S. The number P(s,A) records the probability that the economic system moves from the state s to some state in the Borel subset A of S during one unit of elapsed time. In economic applications the transition probability is usually derived from hypotheses about market clearing and the maximizing behavior of economic agents. The transition probability (together with an initial probability measure on S) defines a discrete time Markov process. One way of summarizing the dynamic behavior implied by P is to look for an invariant probability. A probability measure X on S is invariant for P if for all Borel subsets A of S one has the equality f P(s, A )X(ds) = X(A). An invariant probability is a kind of probabilistic steady state for the dynamics defined by P. Of course there may be no invariant probability for P at all; and even if one exists it may convey no information about the behavior of the process over time except under very special initial conditions. There is a second way of summarizing the behavior of Markov processes defined by the transition probability P. Let P (s,A) denote the n step transition

Both the class position of agents and their status as exploiters or exploited is endogenously determined as they optimize against asset constraints which limit their capacity to produce revenue. The Class Exploitation Correspondence Principle (CECP) asserts that class and exploitation status are related in a classical way. It is further shown that the class structure associated with a labor market can be generated isomorphically by a credit market, demonstrating the functional equivalence of these markets. Morever, these results hold in models of precapitalist, subsistence economy, showing that the phenomena of Marxian exploitation and class are applicable in economic mechanisms other than capitalist ones. The possibility for a general theory of exploitation is thereby suggested.

[We present a model of information acquisition in a competitive market in which traders can learn both from costly (and diverse) private enquiry and price, which costlessly (but partially) reveals the total amount of information known to all traders. Our major purpose is to show that an equilibrium exists in such a market: that is, there exists a rational expectations competitive equilibrium in which the amount of costly diverse information each trader acquires is endogenously determined. From this result we investigate the change in the informativeness of price relative to changes in the level of noise, the cost of acquiring information, and the distribution of traders' risk preferences.]

[Recent results have shown that, for bargaining over the distribution of commodities, or other riskless outcomes, Nash's solution predicts that risk aversion is a disadvantage in bargaining. Here we consider bargaining games which may concern risky outcomes as well as riskless outcomes, and we demonstrate that, in such games, risk aversion need not always be a disadvantage in bargaining. Intuitively, for bargaining games in which potential agreements involve lotteries which have a positive probability of leaving one of the players worse off than if a disagreement had occurred, the more risk averse a player, the better the terms of the agreement which had occurred, the more risk averse a player, the better the terms of agreeement which he must be offered in order to induce him to reach an agreement, and to compensate him for the risk involved. For bargaining games whose disagreement outcome involves no uncertainty, we characterize when risk aversion is advantageous, disadvantageous, or irrelevant from the point of view of Nash's solution.]

[This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.]

[Over half a century ago Frank Graham argued that decreasing costs could justify protection. Although this contention stimulated a huge literature, a correct analysis has never been made. The present paper attempts to fill this gap. It is shown that Graham's case applies to trade between approximately equally-sized economies and that a greater degree of increasing returns actually reduces its likelihood. Furthermore, increasing returns yield a positive analysis nearly completely symmetric to that of Ricardian constant costs. A new analytical tool, the allocation curve, is introduced, with which Marshallian stability is fully analogous to Walrasian instability with offer curves.]

statistics of a linear hypothesis in the standard linear model. These test statistics, which correspond to Wald, likelihood ratio, and Lagrange multiplier tests, are shown to have the same limiting chi-square behavior under mild regularity conditions on design and the distribution of errors. The asymptotic theory of the tests is derived for a large class of error distributions; thus in Huber's [10] terminology we investigate the behavior of the likelihood ratio test under non-standard conditions. The asymptotic efficiency of the 11 tests involves a modest sacrifice of power compared to classical tests in cases of strictly Gaussian errors but may yield large efficiency gains in non-Gaussian situations. The Lagrange multiplier test seems particularly attractive from a computational standpoint. We derive the asymptotic distribution of the three alternative 11 test statistics for a simple linear exclusion hypothesis. Extension of these results to hypotheses of the form R,8 = r is a straightforward exercise. When the density of the error distribution is strictly positive at the median, all three test statistics have the same limiting central x2 behavior at the null and noncentral x2 behavior for local alternatives to the null. When the variance of the error distribution is bounded, analogous results are well known for classical forms of the Wald, likelihood ratio, and Lagrange multipler tests based on least-squares methods. See, for example, Silvey [18] and the discussion in Section 4 below.

[The maximum likelihood approach is used to derive Hausman's specification test. Its asymptotic power is compared with the more conventional test procedures.]

Douglas W. Caves, Laurits R. Christensen, W. Erwin Diewert, The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity, Econometrica, Vol. 50, No. 6 (Nov., 1982), pp. 1393-1414