Hidden effort, learning by doing, and wage dynamics

Many occupations are subject to learning by doing: Effort at the workplace early in the career of a worker results in higher productivity later on. In such occupations, if effort at work is unobservable, a moral hazard problem also arises. We study a particular specification of learning by doing in which the conditional distribution of output depends on the sum of undepreciated efforts. With this specification, we can overcome the technical difficulties for solving for the optimal contract that arise because of the persistent effects of effort in time. Our numerical example shows that effort is frontloaded over the contractual relationship, and follows a steeper decreasing pattern than in the case without learning by doing. On the other hand, the properties of wage dynamics remain unchanged with respect to those of the optimal contract without learning by doing.Labor market

Optimal CEO compensation and stock options

We study the incentive problem between the owners of a firm and its CEO's due to the unobservability of the manager's actions. Our model departs from the literature in two ways. First, we acknowledge that, in contrast with standard repeated moral hazard models, actions taken by CEO's have a persistent effect in time. Second, we derive the effect of effort on stock prices from primitives; i.e., effort affects directly the conditional distribution of profits, and not the distribution of prices. The stock market determines the price of the stock of the firm using information about past profits. A complete characterization of the Second Best contract assuming limited liability is given as a benchmark. Allowing for an arbitrary number of option grants to be awarded, sufficient conditions are given for the implementation of the Second Best contract by an Options Scheme. For a stylized scheme with a unique option grant, the characteristics of the solution are analyzed. We find that the optimal time of exercise balances the increase in quality of information of waiting one extra period with the cost of the poorer smoothing of incentives of doing so. The number of options in the grant, the constant wage, and especially the exercise price are used to best exploit the correlation between the changes in prices and in the likelihood ratios of the histories of profits generating them. As an example, whenever low prices are poorly correlated with the likelihood ratios, the optimal option scheme implies a positive exercise price, which allows for a better use of a higher correlation over the high stock price range than a simple restricted stock scheme. Our results suggest caution regarding regulations that influence the setting of exercise prices. Este artículo estudia el problema de incentivos que surge entre los dueños de una empresa y el ejecutivo que la dirige, fruto de la imposibilidad de observar directamente las acciones del directivo. El modelo difiere del modelo estándar en la literatura en dos puntos clave. En primer lugar, tiene en cuenta que las acciones que toma el directivo tienen un efecto persistente en el tiempo; esta persistencia no la consideran los modelos estándar de riesgo moral repetido. En segundo lugar, el efecto del esfuerzo del directivo en el precio de las acciones de la empresa se deriva de los primitivos del modelo: el esfuerzo determina la distribución de probabilidad de los beneficios de la empresa, y no directamente la distribución de precios. Los compradores en el mercado de valores determinan el precio de las acciones basándose en la información disponible sobre los beneficios pasados. El artículo presenta, como marco de referencia, una caracterización del contrato óptimo asumiendo responsabilidad limitada por parte del directivo. Para el caso en que se pueden emitir múltiples paquetes de opciones, se presentan condiciones suficientes para la implementación del contrato óptimo. Para un caso simplificado en el que la compensación se realiza con un solo paquete de opciones, se analizan las características del mismo. Los resultados del análisis indican que la fecha de ejercicio óptima se determina balanceando los beneficios y los costes de esperar un periodo más: por un lado, aumenta la calidad de información; por el otro, aumenta el coste de proveer incentivos, por tener que estar estos concentrados en un horizonte temporal menor. El número de opciones en el paquete, el salario, y especialmente el precio de ejercicio se usan para explotar la correlación entre los cambios en precios y los cocientes de probabilidad relativa correspondientes a las historias de beneficios que generan esos precios. Por ejemplo, cuando los precios bajos están débilmente correlacionados con los correspondientes cocientes, el paquete óptimo de opciones tiene un precio de ejercicio positivo, que permite explotar la correlación existente en el rango de precios alto mejor que un paquete que incluyera simplemente acciones (i.e, acciones de venta restringida). Estos resultados sugieren cautela a la hora de aprobar regulación que pueda distorsionar la elección de los precios de ejercicio de las opciones en los paquetes de compensación de directivos de empresa.Riesgo Moral, Contratos Óptimos, Persistencia, Compensación de Directivos, Opciones Moral Hazard, Optimal Contracts, Persistence, CEO Compensation, Stock

Unobservable Persistant Productivity and Long Term Contracts

We study the problem of a firm that faces asymmetric information about the productivity of its potential workers. In our framework, a worker’s productivity is either assigned by nature at birth, or determined by an unobservable initial action of the worker that has persistent effects over time. We provide a characterization of the optimal dynamic compensation scheme that attracts only high productivity workers: consumption –regardless of time period– is ranked according to likelihood ratios of output histories, and the inverse of the marginal utility of consumption satisfies the martingale property derived in Rogerson (1985). However, in the case of i.i.d. output and square root utility we show that, contrary to the features of the optimal contract for a repeated moral hazard problem, the level and the variance of consumption are negatively correlated, due to the influence of early luck into future compensation. Moreover, in this example long-term inequality is lower under persistent private informatio

Unobservable Persistant Productivity and Long Term Contracts

We study the problem of a firm that faces asymmetric information about the productivity of its potential workers. In our framework, a worker’s productivity is either assigned by nature at birth, or determined by an unobservable initial action of the worker that has persistent effects over time. We provide a characterization of the optimal dynamic compensation scheme that attracts only high productivity workers: consumption –regardless of time period– is ranked according to likelihood ratios of output histories, and the inverse of the marginal utility of consumption satisfies the martingale property derived in Rogerson (1985). However, in the case of i.i.d. output and square root utility we show that, contrary to the features of the optimal contract for a repeated moral hazard problem, the level and the variance of consumption are negatively correlated, due to the influence of early luck into future compensation. Moreover, in this example long-term inequality is lower under persistent private informationMechanism design, Moral hazard, Persistence

Wandering domains for composition of entire functions

C. Bishop has constructed an example of an entire function f in Eremenko-Lyubich class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in Eremenko-Lyubich class such that the Fatou set of f compose with g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. Singh and results of W. Bergweiler and A. Hinkkanen related to this problem.Comment: 21 pages, 3 figure

Mediation: Incomplete information bargaining with filtered communication

We analyze a continuous-time bilateral double auction in the presence of two-sided incomplete information and a smallest money unit. A distinguishing feature of our model is that intermediate concessions are not observable by the adversary: they are only communicated to a passive auctioneer. An alternative interpretation is that of mediated bargaining. We show that an equilibrium using only the extreme agreements always exists and display the necessary and sucient condition for the existence of (perfect Bayesian) equilibra which yield intermediate agreements. For the symmetric case with uniform type distribution we numerically calculate the equilibria. We find that the equilibrium which does not use compromise agreements is the least ecient, however, the rest of the equilibria yield the lower social welfare the higher number of compromise agreements are used.

Absorbing sets and Baker domains for holomorphic maps

We consider holomorphic maps $f: U \to U$ for a hyperbolic domain $U$ in the complex plane, such that the iterates of $f$ converge to a boundary point $\zeta$ of $U$. By a previous result of the authors, for such maps there exist nice absorbing domains $W \subset U$. In this paper we show that $W$ can be chosen to be simply connected, if $f$ has parabolic I type in the sense of the Baker--Pommerenke--Cowen classification of its lift by a universal covering (and $\zeta$ is not an isolated boundary point of $U$). Moreover, we provide counterexamples for other types of the map $f$ and give an exact characterization of parabolic I type in terms of the dynamical behaviour of $f$

On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.Comment: 34 pages, 10 figure