SOME DISCUSSION has arisen recently as to whether the imposition of an entrance fee on arriving customers who wish to be serviced by a station and hence join a waiting line is a rational measure. Not much of this discussion has appeared in print; indeed this author is aware of only three short communications, representing an exchange of arguments between Leeman [1, 2] and Saaty [3]. The ideas advanced there were of qualitative character and no attempt was made to quantify the arguments. The problem under consideration is obviously analogous to one that arises in connection with the control of vehicular traffic congestion on a road network. It has been argued2 by traffic economists that the individual car driver on making an optimal routing choice for himself-does not optimize the system at large. The purpose of this communication is to demonstrate that, indeed, analogous conclusions can be drawn for queueing models if two basic conditions are satisfied:

In this paper, a 14-equation model of the United States potato industry is presented. Four of the equations contain endogenous variables lagged one time period. The solution to this system of first-order difference equations is presented to determine the system's stability. The stochastic stability is then investigated by obtaining estimates of the limiting variance-covariance matrix of endogenous variables. This matrix shows the cumulated effect of historical random shocks. This is followed by a similar study of the effect of erratic variation in exogenous variables. Next is a comparative static analysis, comparing actual values of variables with their stationary state values. The impact of the price support program on the industry is analyzed. Impact and stationary state multipliers are computed and short and long run effects of structural changes are evaluated. RELATIVELY LARGE fluctuations of prices and quantities have characterized the United States potato industry during the last three decades. These fluctuations have had a profound effect on growers' income, regional allocation of production, and the economic efficiency of the industry in general.2 The quantitative analysis of the stability properties of the United States potato industry and its stationary states under changing environmental conditions constitute the main objectives of the present study. In addition, the analysis is so designed as to focus on certain questions pertaining to the particular position maintained by California potato growers in the United States market. The method of analysis consists in formulating an econometric model of the United States potato market. Then, having estimated the parameters of the economic structure, a detailed analysis of the static and dynamic properties of the system is undertaken. The comparative static analysis seeks to evaluate equilibrium values of the endogenous variables both in the short and in the long run and to determine quantitatively the effects of conceivable variation in exogenous variables and certain parameters of the structural relations on these values.

Since network data commonly consists of observations from a single large network, researchers often partition the network into clusters in order to apply cluster‐robust inference methods. Existing such methods require clusters to be asymptotically independent. Under mild conditions, we prove that, for this requirement to hold for network‐dependent data, it is necessary and sufficient that clusters have low conductance, the ratio of edge boundary size to volume. This yields a simple measure of cluster quality. We find in simulations that when clusters have low conductance, cluster‐robust methods control size better than HAC estimators. However, for important classes of networks lacking low‐conductance clusters, the former can exhibit substantial size distortion. To determine the number of low‐conductance clusters and construct them, we draw on results in spectral graph theory that connect conductance to the spectrum of the graph Laplacian. Based on these results, we propose to use the spectrum to determine the number of low‐conductance clusters and spectral clustering to construct them.

This paper studies causal inference in randomized experiments under network interference. Commonly used models of interference posit that treatments assigned to alters beyond a certain network distance from the ego have no effect on the ego's response. However, this assumption is violated in common models of social interactions. We propose a substantially weaker model of “approximate neighborhood interference” (ANI) under which treatments assigned to alters further from the ego have a smaller, but potentially nonzero, effect on the ego's response. We formally verify that ANI holds for well‐known models of social interactions. Under ANI, restrictions on the network topology, and asymptotics under which the network size increases, we prove that standard inverse‐probability weighting estimators consistently estimate useful exposure effects and are approximately normal. For inference, we consider a network HAC variance estimator. Under a finite population model, we show that the estimator is biased but that the bias can be interpreted as the variance of unit‐level exposure effects. This generalizes Neyman's well‐known result on conservative variance estimation to settings with interference.