Knowledge that Transforms

We consider discriminatory and uniform price auctions for multiple identical units of a good. Players have private values, possibly asymmetrically distributed and for multiple units. Our setting allows for aggregate uncertainty about demand and supply. In this setting, equilibria generally will be inefficient. Despite this, we show that such auctions become arbitrarily close to efficient if they are "large, " and use this to derive an asymptotic characterization of revenue and bidding behavior.

UNIT ROOT TESTING has been developed through numerous papers since the work of Ž. Dickey and Fuller 1979 . The idea is to test the hypothesis that the differences of an observed time series do not depend on its levels, or in other words, the levels of the time series have a unit root that can be removed by differencing. While it is in general possible to have multiple unit roots, only the hypothesis of exactly one unit root is considered Ž. here. The available tests therefore hinge on two assumptions: i the levels of the time Ž. series have exactly one unit root which can be removed by differencing, and ii the remaining characteristic roots of the time series are stationary roots. In this paper it is proved that for the likelihood ratio test and a number of other likelihood based statistics Ž. Ž . the assumption ii is redundant whereas i is necessary. It is also shown that for some tests that are not likelihood based it is indeed necessary to assume that the differences have stationary roots. The consequences of the result are perhaps best understood from the implications of Ž. condition i . For autoregressive models of order two or higher, that condition is not satisfied in the entire parameter space and the asymptotic distribution of the likelihood ratio test for a unit root depends on unknown nuisance parameters. In this situation the test statistic is not pivotal; hence the test is not similar, and this complicates the testing. Ž. For non-likelihood based tests the necessity of condition ii implies an additional similarity problem. The practitioner is therefore faced with a trade off between likelihood based tests with fewer similarity problems and other tests that may have other advantageous properties. There are thus two empirical implications of the result. First, when analyzing time series with stationary roots that have modulus close to one so that Ž. condition ii is nearly violated, then the likelihood based tests are preferable and other tests should be used cautiously. Secondly, if explosive roots are found in an application, most of the statistical analysis is actually valid and should not necessarily be disregarded because of the presence of explosive roots. Section 2 presents a Gaussian autoregressive model along with its statistical analysis Ž. and the result showing that condition ii is redundant for likelihood based tests. Robustness with respect to innovations that are martingale difference is also discussed. The results of Section 2 are given for a model without deterministic trends. In Section 3 these are generalized to models with deterministic terms. The mathematical proofs Ž. following in two Appendices are based on the work of Lai and Wei 1983 and Chan and Ž. Wei 1988 .

This paper considers testing problems where several of the standard regularity conditions fail to hold. We consider the case where (i) parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and (ii) there may be a nuisance parameter that appears under the alternative hypothesis, but not under the null. The paper establishes the asymptotic null and local alternative distributions of quasi-likelihood ratio, rescaled quasi-likelihood ratio, Wald, and score tests in this case. The results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models. Examples treated in the paper are: (i) tests of the null hypothesis of no conditional heteroskedasticity in a GARCH(1, 1) regression model and (ii) tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with (possibly) correlated random coefficients.

In expected utility theory, risk attitudes are modeled entirely in terms of utility. In the rank-dependent theories, a new dimension is added: chance attitude, modeled in terms of nonadditive measures or nonlinear probability transformations that are independent of utility. Most empirical studies of chance attitude assume probabilities given and adopt parametric fitting for estimating the probability transformation. Only a few qualitative conditions have been proposed or tested as yet, usually quasi-concavity or quasi-convexity in the case of given probabilities. This paper presents a general method of studying qualitative properties of chance attitude such as optimism, pessimism, and the "inverse-S shape" pattern, both for risk and for uncertainty. These qualitative properties can be characterized by permitting appropriate, relatively simple, violations of the sure-thing principle. In particular, this paper solves a hitherto open problem: the preference axiomatization of convex ("pessimistic" or "uncertainty averse") nonadditive measures under uncertainty. The axioms of this paper preserve the central feature of rank-dependent theories, i.e. the separation of chance attitude and utility.

The logic of the linkage principle of Milgrom and Weber (1982) extends to price discrimination. A non-linear pricing monopolist who sells to a single buyer always prefers to commit to publicly reveal information affiliated to the valuation of the buyer. This is also valid when the value of the buyer affects the opportunity cost of the seller.

Evidence such as the Ellsberg Paradox shows that decision-makers do not assign probabilities to all events. It is intuitive that they may differ not only in the probabilities assigned to given events but also in the identity of the events to which they assign probabilities. This paper describes a theory of probability that is fully subjective in the sense that both the domain and the values of the probability measure are derived from preference. The key is a formal definition of `subjectively unambiguous event.'

We experimentally investigate the sensitivity of bidders demanding multiple units of a homogeneous commodity to the demand reduction incentives inherent in uniform price auctions. There is substantial demand reduction in both sealed bid and ascending price clock auctions with feedback regarding rivals’ drop-out prices. Although both auctions have the same normal form representation, bidding is much closer to equilibrium in the ascending price auctions. We explore the behavioral process underlying these differences along with dynamic Vickrey auctions designed to eliminate the inefficiencies resulting from demand reduction in the uniform price auctions. Key words: multi-unit demand auctions, uniform price auction, dynamic Vickrey auction, demand reduction, experiment.

This paper presents a full characterization of the equilibrium value set of a Ramsey tax model. More generally, it develops a dynamic programming method for a class of policy games between the government and a continuum of households. By selectively incorporating Euler conditions into a strategic dynamic programming framework, we wed two technologies that are usually considered competing alternatives, resulting in a substantial simplification of the problem.

The fiscal theory says that the price level is determined by the ratio of nominal debt to the present value of real primary surpluses. I analyze long-term debt and optimal policy in the fiscal theory. I find that the maturity structure of the debt matters. For example, it determines whether news of future deficits implies current inflation or future inflation. When long-term debt is present, the government can trade current inflation for future inflation by debt operations; this tradeoff is not present if the government rolls over short-term debt. The maturity structure of outstanding debt acts as a ‘‘budget constraint’’ determining which periods’ price levels the government can affect by debt variation alone. In addition, debt policythe expected pattern of future state-contingent debt sales, repurchases and redemptionsmatters crucially for the effects of a debt operation. I solve for optimal debt policies to minimize the variance of inflation. I find cases in which long-term debt helps to stabilize inflation. I also find that the optimal policy produces time series that are similar to U.S. surplus and debt time series. To understand the data, I must assume that debt policy offsets the inflationary impact of cyclical surplus shocks, rather than causing price level disturbances by policy-induced shocks. Shifting the objective from price level variance to inflation variance, the optimal policy produces much less volatile inflation at the cost of a unit root in the price level; this is consistent with the stabilization of U.S. inflation after the gold standard was abandoned.