Chebyshev polynomial approximation to approximate partial differential equations

This paper suggests a simple method based on Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. The methodology simply consists in determining the value function by using a set of nodes and basis functions. We provide two examples. Pricing an European option and determining the best policy for chatting down a machinery. The suggested method is flexible, easy to program and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.

Revealed cardinal preference

I prove that as long as we allow the marginal utility for money (lambda) to vary between purchases (similarly to the budget) then the quasi-linear and the ordinal budget-constrained models rationalize the same data. However, we know that lambda is approximately constant. I provide a simple constructive proof for the necessary and sufficient condition for the constant lambda rationalization, which I argue should replace the Generalized Axiom of Revealed Preference in empirical studies of consumer behavior. 'Go Cardinals!' It is the minimal requirement of any scientifi c theory that it is consistent with the data it is trying to explain. In the case of (Hicksian) consumer theory it was revealed preference -introduced by Samuelson (1938,1948) - that provided an empirical test to satisfy this need. At that time most of economic reasoning was done in terms of a competitive general equilibrium, a concept abstract enough so that it can be built on the ordinal preferences over baskets of goods - even if the extremely specialized ones of Arrow and Debreu. However, starting in the sixties, economics has moved beyond the 'invisible hand' explanation of how -even competitive- markets operate. A seemingly unavoidable step of this 'revolution' was that ever since, most economic research has been carried out in a partial equilibrium context. Now, the partial equilibrium approach does not mean that the rest of the markets are ignored, rather that they are held constant. In other words, there is a special commodity -call it money - that reflects the trade-offs of moving purchasing power across markets. As a result, the basic building block of consumer behavior in partial equilibrium is no longer the consumer's preferences over goods, rather her valuation of them, in terms of money. This new paradigm necessitates a new theory of revealed preference.

Stability of Growth Models with Generalised Lag Structures

This paper considers the lag structures of dynamic models in economics, arguing that the standard approach is too simple to capture the complexity of actual lag structures arising, for example, from production and investment decisions. It is argued that recent (1990s) developments in the the theory of functional differential equations provide a means to analyse models with generalised lag structures. The stability and asymptotic stability of two growth models with generalised lag structures are analysed. The paper concludes with some speculative discussion of time-varying parameters.

Tractable valuations under uncertainty

I put forward a concise and intuitive formula for the calculation of the valuation for a good in the presence of the expectation that further, related, goods will soon become available. This valuation is tractable in the sense that it does not require the explicit resolution of the consumerís life-time problem.

Dynamic Relational Contracts under Complete Information

This paper considers a long-term relationship between two agents who both undertake a costly action or investment that together produces a joint benefit. Agents have an opportunity to expropriate some of the joint benefit for their own use. Two cases are considered: (i) where agents are risk neutral and are subject to limited liability constraints and (ii) where agents are risk averse, have quasi-linear preferences in consumption and actions but where limited liability constraints do not bind. The question asked is how to structure the investments and division of the surplus over time so as to avoid expropriation. In the risk-neutral case, there may be an initial phase in which one agent overinvests and the other underinvests. However, both actions and surplus converge monotonically to a stationary state in which there is no overinvestment and surplus is at its maximum subject to the constraints. In the risk-averse case, there is no overinvestment. For this case, we establish that dynamics may or may not be monotonic depending on whether or not it is possible to sustain a first-best allocation. If the first-best allocation is not sustainable, then there is a trade-off between risk sharing and surplus maximization. In general, surplus will not be at its constrained maximum even in the long run.