Uday Rajan

Equilibrium in a Dynamic Limit Order Market

We provide an algorithm for solving for equilibrium in a dynamic limit order market. Our model relaxes many of the restrictive assumptions in the prior literature, leading to a more realistic framework for policy experiments on market design. We formulate a limit order market as a stochastic sequential game and use a simulation technique based on Pakes and McGuire (2001) to find a stationary equilibrium. Given the stationary equilibrium, we generate artificial time series and perform comparative dynamics. We explicitly determine investor welfare in our numerical solution. We find that the effective spread is {\it negatively} correlated with transactions costs and uncorrelated with welfare. As one policy experiment, we evaluate the effect of changing tick size.

The Failure of Models that Predict Failure: Distance, Incentives and Defaults

Statistical default models, widely used to assess default risk, fail to account for a change in the relations between different variables resulting from an underlying change in agent behavior. We demonstrate this phenomenon using data on securitized subprime mortgages issued in the period 1997–2006. As the level of securitization increases, lenders have an incentive to originate loans that rate high based on characteristics that are reported to investors, even if other unreported variables imply a lower borrower quality. Consistent with this behavior, we find that over time lenders set interest rates only on the basis of variables that are reported to investors, ignoring other credit-relevant information. As a result, among borrowers with similar reported characteristics, over time the set that receives loans becomes worse along the unreported information dimension. This change in lender behavior alters the data generating process by transforming the mapping from observables to loan defaults. To illustrate this effect, we show that the interest rate on a loan becomes a worse predictor of default as securitization increases. Moreover, a statistical default model estimated in a low securitization period breaks down in a high securitization period in a systematic manner: it underpredicts defaults among borrowers for whom soft information is more valuable. Regulations that rely on such models to assess default risk could, therefore, be undermined by the actions of market participants.

Competition in Loan Contracts

We present a model of an unsecured loan market. Many lenders simultaneously offer loan contracts (a debt level and an interest rate) to a borrower. The borrower may accept more than one contract. Her payoff if she defaults increases in the total amount borrowed. If this payoff is high enough, deterministic zero-profit equilibria cannot be sustained. Lenders earn a positive profit, and may even charge the monopoly price. The positive-profit equilibria are robust to increases in the number of lenders. Despite the absence of asymmetric information, the competitive outcome does not obtain in the limit.

Cause Marketing: Spillover Effects of Cause-Related Products in a Product Portfolio

The number of firms carrying a cause-related product has significantly increased in recent years. We consider a duopoly model of competition between firms in two products to determine which products a firm will link to a cause. We first test the behavioral underpinnings of our model in two laboratory experiments to demonstrate the existence of both a direct utility benefit to consumers from cause marketing (CM) and a spillover benefit onto other products in the portfolio. Linking one product in a product portfolio to a cause can therefore increase sales both of that product and, via a spillover effect, of other products in the firms portfolio. We construct a CM game in which each firm chooses which products, if any, to place on CM. In the absence of a spillover benefit, a firm places a product on CM if and only if it can increase its price by enough to compensate for the cost of CM. Thus, in equilibrium, firms either have both products or neither product on CM. However, with the introduction of a spillover benefit to the second product, this result changes. We show that if a single firm in the market links only one product to a cause, it can raise prices on both products and earn a higher profit. We assume each firm has an advantage in one product and show that there is an equilibrium in which each firm links only its disadvantaged product to a cause. If the spillover effect is strong, there is a second equilibrium in which each firm links only its advantaged product to a cause. In each case, firms raise their prices on both products and earn higher profits than when neither firm engages in CM. We also show that a firm will never place its entire portfolio on CM. Overall, our work implies that, by carrying cause-related products, companies can not only improve their image in the public eye but also increase profits.

Mechanism design for coalition formation and cost sharing in group-buying markets

Abstract We study the mechanism design problem of coalition formation and cost sharing in a group-buying electronic marketplace, where buyers can form coalitions to take advantage of volume based discounts. The desirable mechanism properties include stability (in the core), and incentive compatibility with good efficiency. We show the impossibility to simultaneously satisfy efficiency, budget balance and individual rationality at a Bayesian–Nash equilibrium, and propose a mechanism in the core of the game. We then present and evaluate a group of reasonable mechanisms. Empirical results show positive correlation between stability and incentive compatibility (which is in turn related to efficiency).

Signaling Quality via Queues

We consider an M/M/1 queueing system with impatient consumers who observe the length of the queue before deciding whether to buy the product. The product may have high or low quality, and consumers are heterogeneously informed. The firm chooses a slow or (at a cost) a fast service rate. In equilibrium, informed consumers join the queue if it is below a threshold. The threshold varies with the quality of the good, so an uninformed consumer updates her belief about quality on observing the length of the queue. The strategy of an uninformed consumer has a “hole”: she joins the queue at lengths both below and above the hole, but not at the hole itself. We show that if the prior probability the product has high quality and the proportion of informed consumers are both low, a high-quality firm may select a slower service rate than a low-quality firm. The queue can therefore be a valuable signaling device for a high-quality firm. Strikingly, in some scenarios, the high-quality firm may choose the slow service rate even if the technological cost of speeding up is zero. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Rationing in IPOs

We provide a model of bookbuilding in IPOs, in which the issuer can choose to ration shares. Before informed investors submit their bids, they know that, in the aggregate, winning bidders will receive only a fraction of their demand. We demonstrate that this mitigates the winner’s curse, that is, the incentive of bidders to shade their bids. It leads to more aggressive bidding, to the extent that rationing can be revenue-enhancing. In a parametric example, we characterize bid and revenue functions, and the optimal degree of rationing. We show that, when investors’ information is diffuse, maximal rationing is optimal. Conversely, when their information is concentrated, the seller should not ration shares. We provide testable predictions on bid dispersion and the degree of rationing. Our model reconciles the documented anomaly that higher bidders in IPOs do not necessarily receive higher allocations.

Min-Max payoffs in a two-player location game

We consider a two-player, sequential location game with arbitrarily distributed consumer demand. Players alternately select locations from a feasible set so as to maximize the consumer mass in their vicinity. Our main result is a complete characterization of feasible market shares, when locations form a finite set in R^d.

Implementation in Principal-Agent Models of Adverse Selection

This paper studies implementation in a principal-agent model of adverse selection. We explore ways in which the additional structure of principal agent models (compared to general implementaion models) simplifies the implementation problem. We develop a connection between the single crossing property and monotonicity conditions which are necessary for Nash and Bayesian Nash implementation.

Corporate Hedging, Investment and Value

We consider the effect of hedging with foreign currency derivatives on Brazilian firms in the period 1997 through 2004, a period that includes the Brazilian currency crisis of 1999. We find that, derivative users have valuations that are 6.7-7.8% higher than non-user firms. Hedging with currency derivatives allows firms to sustain larger capital investments, and also removes the sensitivity of investment to internally generated funds. Thus, it mitigates the underinvestment friction of Froot, Scharfstein, and Stein (1993), at a time when capital in the economy as a whole is scarce. We further show that hedging increases the foreign currency debt capacity of a firm, and that foreign debt is a cheaper source of capital than domestic debt during our period of study.