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Essays in Mathematical Economics

  • Author(s): Vasquez, Markus Antonio
  • Advisor(s): Anderson, Robert M
  • et al.
Abstract

We apply mathematical techniques in the context of economic decision making. First, we are interested in understanding the behaviors and beliefs of agents playing economic games in which the underlying action spaces are possibly non-compact and the agents’ payoff functions are possibly discontinuous. Under these circumstances, there is no guarantee of the existence of a Nash equilibrium in randomized strategies. In fact, there are games for which no Nash equilibrium exists. To restore equilibrium we allow each agent access to randomized strategies that are not necessarily countably additive. This has the unfortunate side effect of introducing uncertainty into the players’ payoff functions due to the failure of Fubini’s theorem for finitely additive measures. We introduce two ways of resolving this ambiguity and show that for one we are able to recover a general equilibrium existence result.

Next, we turn to the problem that expected utility theory typically assumes that agents use concave utility functions. This is problematic since this implies that agents are risk averse and, consequently, will not gamble. We speculate that non-concavity may be the result of agents’ utility functions arising from solving the the knapsack problem, a combinatorial optimization problem. We introduce a class of utility of wealth functions, called knapsack utility functions, which are appropriate for agents who must choose an optimal collection of indivisible goods from a countably infinite collection. We find that these functions are pure jump processes. Moreover, we find that localized regions of convexity–and thus a demand for gambling–is the norm, but that the incentive to gamble is much more pronounced at low wealth levels. We consider an intertemporal version of the problem in which the agent faces a credit constraint. We find that the agent’s utility of wealth function closely resembles a knapsack utility function when the agent’s saving rate is low.

Finally, we turn our attention to the Black-Scholes model of security price movements. Our goal is to understand the beliefs and incentives of individual agents required for the Black-Scholes model to be self-predicting. We consider a model in which each agent believes that the Black-Scholes model is correct. Each agent observes a private stream of information, which she uses to update her beliefs about future movements of the security price. Each agent is then faced with an optimization problem whose solution tells us her optimal portfolio for any given price (i.e. her demand function). Imposing market clearing conditions then determines a price at each point in time. That is, the agents prior beliefs about the security price process along with their private information streams generate a price process. We may then ask under which conditions the distribution of this process matches the agents’ prior belief. We find that that condition is fairly restrictive and imposes significant constraints on the drift of the price process when agents are homogenous and use utility functions with constant absolute risk aversion or constant relative risk aversion.

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