Three Essays on Dynamic Games with Asymmetric Agents

Abstract

[eng] The thesis is divided into three chapters (excluding the concluding chapter and this introduction). Each of them falls within a separate field in Economics. The first is on business cycle theory, a field within Macroeconomics. The second is on life insurance models, which is generally regarded as a field within the actuarial sciences. The third is on the well-known fish war model, a topic within Environmental and Resource Economics. In Chapter 2, we construct a Macroeconomic differential game where the agents are the fiscal and monetary authorities of an economy, i.e. the government and the central bank. The authorities use the government de cit and the nominal interest rate as policy instruments to steer the economy, the state of which is captured by aggregate output (or real gross domestic product) and price in ation. The objective of the agents is to maintain those variables at some given desired levels, which we think of as the economy's general equilibrium. As is standard in the literature, we assume that the use of policy instruments brings a cost, which must be balanced against the cost of remaining in disequilibrium. We investigate the development of the economy under the assumption that the government and the central bank discount future utilities at constant but different rates, which implies that a coalition formed by the two has inconsistent time preferences. We compute both noncooperative and cooperative solutions. The main finding in Chapter 2 is that in the noncooperative solution, the authorities will enact policies that are mutually contradictory. By this we mean that the government and the central bank try to steer aggregate output in opposite directions, and therefore \cancel out" the actions of one another. This implies that there is room for Pareto improvement. Importantly, we show that this situation is a result of the asymmetry in the discount rates, and that in the cooperative solution the mutually contradictory policies are removed. Based on this, we make the claim that policy coordination between the government and the central bank is especially useful when they operate under different discount rates. Since, in general, there's no reason to assume that governments and central banks operate under the same discount rates, our findings have practical implications for the conduct of fiscal and monetary policy. For example, we suggest that in countries with a democratic political system, the government may have a higher discount rate because it is controlled by elected politicians, while the central bank is controlled by unelected technocrats. In Chapter 3, we construct a continuous time life insurance model in the tradition of Richard (1975) and Pliska and Ye (2007). We model two agents that share a common household budget, but have separate expenditures on consumption and life insurance. At each point in time, each agent withdraws money from the budget at some rate to spend on his consumption expenditures and insurance-premium. At some randomly determined point in time one agent dies, and the survivor receives the life insurance payment and continues to consume until he too dies. The unusual feature of our model is, as in Chapter 2, that the agents have different discount rates of future events, meaning that the household as a whole has inconsistent time preferences. Thus, we combine two different features which up to now have been developed separately within the literature on life insurance models: on the one hand, households with multiple agents, and one the other hand, households with inconsistent time preferences. As is standard in life insurance models, we only compute the cooperative solution. That is, it is assumed that no agent will leave the household under any conditions. The model of Chapter 3 contains quite a large number of parameters for describing the properties of the agents and the environment in which they act. Hence, there is plenty of opportunities for making comparative dynamics. The most interesting result, perhaps, is that if one agent becomes less patient (i.e. his discount rate increases), the household reacts by shifting spending to his insurance premium from that of the other insurance premiums. We provide an economic interpretation of this and several other results. In Chapter 4, we investigate the fish war model due to Levhari and Mirman (1980), under the assumption that the agents have asymmetric quasi-hyperbolic discount functions, which implies that the agents have inconsistent time preferences. Hence, this chapter differs from Chapter 3 and Chapter 4 in that the individual agents have nonconstant discount rates. We look at three types of fish war models. In the first, the agents have logarithmic utility functions and the growth function is nonlinear, which is the standard case. What is nonstandard are the discount functions. The behaviors under noncooperation and cooperation are compared. We find that the stability of the coalition in the cooperative solution is not sensitive to the inconsistency in time preferences introduced by quasi-hyperbolic discounting. In the second model, we build a nonstandard fish war game with power utility functions, quasi-hyperbolic discount functions and a linear stochastic autocorrelated growth function. In contrast to the previous model, we find that stability breaks down quickly when the inconsistency in time preference increases. Then, as our third model, we extend the second model by allowing asymmetric utility functions and a nonlinear growth function. Here, we make some observations on how cooperative solution is affected by the presence of risk and by asymmetry in risk aversions. Chapter 4 may be regarded as the "experimental" chapter of the thesis: we develop a new set of dynamic programming algorithms and apply them to solve specific problems.

[spa] La tesis trata sobre el tema de los juegos dinámicos con agentes asimétricos. En cada uno de los tres capítulos principales, resolvemos juegos dinámicos en los que los agentes pueden diferir en cuanto a sus funciones de utilidad y sus funciones de descuento. Se consideran tanto juegos de tiempo discreto como continuo, y los juegos se resuelven tanto en cooperación como en no cooperación. La herramienta matemática que usamos para derivar soluciones es la programación dinámica. La tesis también hace un uso extensivo de los resultados numéricos para investigar las propiedades de las soluciones, en particular en los casos en que no se pueden derivar soluciones analíticas.

En el primero de los capítulos principales, resolvemos un juego dinámico de tiempo continuo de política monetaria y fiscal, un tema dentro de la macroeconomía. Los agentes del juego son el gobierno y el banco central de un país, y su objetivo es estabilizar las fluctuaciones del ciclo económico de la economía. El elemento principal del juego es la presencia de asimetría en las preferencias temporales. Suponemos que el gobierno tiene una tasa de descuento más alta que la del banco central, lo que implica que está más preocupado por el futuro cercano. Mostramos que esta asimetría conduce a una situación en la que la cooperación entre los agentes es más beneficiosa permitiéndoles alcanzar sus objetivos. El juego se resuelve utilizando un algoritmo de programación dinámica de la literatura sobre juegos dinámicos con agentes asimétricos.

En el segundo capítulo, resolvemos un modelo de seguro de vida en tiempo continuo, un tema de las ciencias actuariales. Los dos agentes son los miembros de un hogar, y cada agente compra su propio seguro de vida, de modo que cuando un agente fallece, el otro recibe el pago del seguro. Nuestro modelo permite que los agentes difieran en términos de sus tasas de descuento y tasas de mortalidad, y se deriva una solución cooperativa. Mostramos que, en igualdad de condiciones, el agente menos paciente tiene un mayor gasto en su prima de seguro. El juego se resuelve mediante un algoritmo de programación dinámica que demostramos en un apéndice del capítulo.

En el tercer capítulo, resolvemos varios modelos diferentes vinculados con el conocido como great fish war model, sobre la explotación conjunta de recursos pesqueros (o de recursos naturales renovables en general), que pertenecen al campo de la economía del medioambiente y de los recursos naturales. Este capítulo, a diferencia de los dos anteriores, trata de juegos de tiempo discreto. Primero desarrollamos un nuevo conjunto de algoritmos de programación dinámica, que se aplica tanto a juegos cooperativos como no cooperativos, con un horizonte de planificación finito e infinito con funciones arbitrarias de utilidad y descuento. Luego usamos esta teoría para resolver dos juegos de explotación de un recurso natural (pesquero) en particular. Los juegos difieren en términos de qué funciones de utilidad y funciones de crecimiento usamos, pero ambos asumen que los agentes tienen funciones de descuento casi hiperbólicas. Luego investigamos un tercer modelo de guerra de peces, en el que se permite que los agentes tengan funciones de utilidad asimétricas. Este tercer modelo se resuelve solo para la solución cooperativa, usando inducción hacia atrás completamente numérica. Para cada uno de los tres modelos, usamos soluciones numéricas para investigar el comportamiento de los agentes. Las investigaciones importantes incluyen la comparación del comportamiento bajo cooperación con el de no cooperación, los efectos de la preferencia temporal, la aversión al riesgo y el tema de la estabilidad de las soluciones cooperativas.