This paper provides deterministic approximation results for stochastic processes that arise when ¯nite populations recurrently play ¯nite games.The deterministic approximation is de¯ned in continuous time as a system of ordinary di®erential equations of the type studied in evolutionary game theory.We establish precise connections between the long-run behavior of the stochastic process, for large populations, and its deterministic approximation.In particular, we show that if the deterministic solution through the initial state of the stochastic process at some point in time enters a basin of attraction, then the stochastic process will enter any given neighborhood of that attractor within a ¯nite and deterministic time with a probability that exponentially approaches one as the population size goes to in¯nity.The process will remain in this neighborhood for a random time that almost surely exceeds an exponential function of the population size.During this time interval, the process spends almost all time at a certain subset of the attractor, its so-called Birkho® center.We sharpen this result in the special case of ergodic processes.

Using a large sample of limit orders on NYSE stocks, we find that investors submit more limit orders with even-eighth prices than odd-eight prices. However, even though there are a greater number of even-eighth limit orders, the proportion of executing limit orders submitted with even prices is greater than those submitted with odd prices for a large portion of our sample. We find that clustering on even prices is a positive function of stock price and various proxies for the dispersion in investors’ reservation prices. The preference for even prices affects stock quotes: Investors are more likely to submit a quote improving limit order if the limit price is even and quoted depth is higher for even quotes than for odd quotes.

We examine the provision of termination fee clauses in merger agreements between 1989 and 1998. Target-payable fees are observed more frequently when bidding is costly and the potential for information expropriation by third parties is significant. Fee provisions appear to benefit target shareholders through higher deal completion rates and greater negotiated takeover premiums. We conclude that target-payable fees serve as an efficient contracting device, rather than a means by which to deter competitive bidding. Bidder fee provisions appear to be used to secure target wealth gains in deals with higher costs associated with negotiation and bid failure.

This paper argues that long-run trends in geographic segregation are inconsistent with models where residential choice depends solely on local public goods (the Tiebout hypothesis). We develop an extension of the Tiebout model that predicts as mobility costs fall, the heterogeneity across communities of individual public good preferences and of public good provision must (weakly) increase. Given the secular decline in mobility costs, these predictions can be evaluated using historical data. We find decreasing heterogeneity in policies and proxies for preferences across (i) a sample of U.S. municipalities (1870–1990); (ii) all Boston-area municipalities (1870–1990); and (iii) all U.S. counties (1850–1990).

I agree with the point made by Edoh Y. Amiran and Daniel A. Hagen (2003) that there can be a substantial, or even infinite, divergence between the WTA and WTP for a public good even where there is a nonzero elasticity of substitution between market goods and the public good, provided that the indifference curves are asymptotically bounded with respect to market goods in the manner they describe. This is an important point. They are also correct to point out that the elasticity of substitution is a local concept, whereas their asymptotic boundedness condition applies also for discrete changes. My 1991 paper used a local analysis because it was following the structure of the analysis in Robert D. Willig (1976) and Alan Randall and John R. Stoll (1980); I wanted to show that, while Randall and Stoll appeared to extend Willig’s local result on WTA versus WTP from price changes to changes in the quantity of a public good, the relevant elasticity was in fact different and involved the substitution elasticity as well as the income elasticity. I view these points by Amiran and Hagen as not two separate results but essentially the same result: their asymptotic boundedness condition generalizes my zero elasticity of substitution condition to discrete changes. The asymptotic boundedness condition can be expressed as follows: assuming a bivariate utility function u( x, q), and given a reference point ( x*, q*) associated with a reference utility level u* u( x*, q*), there exists some q q* such that, for all q q , there exists no x such that u( x , q) u*. In other words, no amount of x can substitute for the reduction in public good from q* to q q . In the case of a zero elasticity of substitution, q q* but, as Amiran and Hagen show in their Theorem 1, this is unnecessarily restrictive when dealing with a discrete reduction in q. Furthermore, their Theorem 2 can be viewed as a special case of their Theorem 1 in which q 0, which makes q an essential commodity. It is well known in consumer theory that the WTA to avoid the loss of an essential market good is infinite; their Theorem 2 extends this result to the case of an essential nonmarket good. But, as their Theorem 1 shows, essentialness is not necessary for an infinite WTA. The boundedness condition is the key, and this implies a fundamental lack of substitutability between money (market goods) and the public good.

In many empirical studies of the effect of social programs researchers assume that, conditional on a set of observed covariates, assignment to the treatment is exogenous or unconfounded (aka selection on observables). Often this assumption is not realistic, and researchers are concerned about the robustness of their results to departures from it. One approach (e.g., Charles Manski, 1990) is to entirely drop the exogeneity assumption and investigate what can be learned about treatment effects without it. With unbounded outcomes, and in the absence of alternative identifying assumptions, there are no restrictions on the set of possible values for average treatment effects. This does not mean, however, that all evaluations are equally sensitive to departures from the exogeneity assumption. In this paper I explore an alternative approach, developed by Paul Rosenbaum and Donald Rubin (1983), where the assumption of exogeneity is explicitly relaxed by allowing for a limited amount of correlation between treatment and unobserved components of the outcomes. The starting point of the sensitivity analysis is the assumption that the exogeneity assumption is satisfied only conditional on an additional unobserved covariate. Making assumptions about the effect of the unobserved covariate on the outcome and its correlation with the treatment, I trace out the set of possible values for the treatment effect of interest. By considering a sufficiently large set of possible correlations with outcomes and treatment, one can recover the bounds on the treatment effect derived by Manski (1990). The approach here, in the spirit of Rosenbaum and Rubin (1983) and Rosenbaum (1995), is to allow only a limited amount of correlation and to judge the sensitivity of average treatment-effect estimates to such correlations. There are two novel features of the proposed analysis. First, rather than formulate the sensitivity in terms of coefficients on the unobserved covariate, the sensitivity results are presented in terms of partial R values, which may be easier to interpret. Second, the partial R values of the unobserved covariates are compared to those for the observed covariates in order to facilitate judgments regarding the plausibility of values necessary to substantially change results obtained under exogeneity. The proposed sensitivity analysis is conceptually related to the practice of assessing sensitivity of estimates by comparisons with results obtained by discarding one or more observed covariates (James Heckman and V. Joseph Hotz, 1989; Rajeev Dehejia and Sadek Wahba, 1999; Jeffrey Smith and Petra Todd, 2001). The attraction of the sensitivity analysis is that it is more directly relevant: one is not interested in what would have happened in the absence of covariates actually observed, but in biases that are the result from not observing all relevant covariates.

This paper extends the standard principal–agent model to allow for subjective evaluation. The optimal contract results in more compressed pay relative to the case with verifiable performance measures. Moreover, discrimination against an individual implies lower pay and performance, suggesting that the extent of discrimination as measured after controlling for performance may underestimate the level of true discrimination. Finally, the optimal contract entails the use of bonus pay rather than the threat of dismissal, hence neither “efficiency wages” nor the right to dismiss an employee are necessary ingredients for an optimal incentive contract.

By placing a ceiling on the amount of possible depositor loss, deposit insurance should result in a lower deposit risk premium. However, this effect may be modified if either the insurance promise has low credibility or the moral hazard incentives generated by deposit insurance result in a greater probability of bank default. Using financial and institutional panel data from thirteen countries, we find that the risk premium is over 40 basis points higher on average in uninsured countries than in countries that offer insurance up to some pre-specified maximum. However, the risk premium has a non-linear relationship with the level of maximum insurance coverage, suggesting that the market recognizes the moral hazard potential. Moreover, the effect of deposit insurance on the risk premium is weaker in countries with strong creditor rights, consistent with the view that investors view the latter as a substitute for explicit deposit insurance.

In sharp contrast to results in the United States, the average stock price response to an announcement of a seasoned equity issue in Japan is positive. Offer prices in Japan, unlike the United States, are announced several days before the beginning of the subscription period and incorporate a substantial discount. We suggest that the positive announcement effects in Japan are consistent with the underwriter's certification of the issuing firm's value. We characterize the underwriter's risk as a put option and find a positive association between the underwriter's risk and the announcement returns, as well as other results consistent with underwriter certification. Copyright 2003, Oxford University Press.

Although institutional investors have a preference for large capitalization stocks, over time they have shifted their preferences toward smaller, riskier securities. These changes in aggregate preferences have arisen primarily from changes in the preferences of each class of institution, rather than changes in the importance of different classes. Evidence also suggests that recent growth in institutional investment combined with this shift in preferences helps explain why markets in general, and smaller stocks in particular, have exhibited greater firm-specific risk and liquidity in recent years. Additional analyses suggest that institutional investors moved toward smaller securities because such securities offer “greener pastures.”