Peter Duck

SLADI: A Semi-Lagrangian Alternating-Direction Implicit Method for the Numerical Solution of Advection–Diffusion Problems with Application to Electricity Storage Valuations

In this paper, an efficient and novel methodology for numerically solving advection–diffusion problems is presented: a semi-Lagrangian approach for hyperbolic problems of advection is combined with an alternating-direction implicit method for parabolic problems involving diffusion. This is used to value a four-dimensional “storage option” (linked to storing electricity) involving three space variables and time. Efficiency is obtained by solving (only) tridiagonal systems of equations at every time step by incorporating the alternating-direction methodology. Extensive numerical experimentation indicates that the method is stable and accurate; three variants of the scheme are assessed and excellent numerical convergence can be observed. Further, a methodology for determining and results for optimal storage operation are presented.

Optimal costless extraction rate changes from a non-renewable resource

This paper considers the role of costless decisions relating to the extraction of a non-renewable resource in the presence of uncertainty. We begin by deriving a size scale of the extractable resource, above which the solution to the valuation and optimal control strategy can be described by analytic solutions; we produce solutions for a general form of operating cost function. Below this critical resource size level the valuation and optimal control strategy must be solved by numerical means; we present a robust numerical algorithm that can solve such a class of problem. We also allow for the embedding of an irreversible investment decision (abandonment) into the optimisation. Finally, we conduct experimentation for each of these two approaches (analytical and numerical), and show how they are consistent with one another when used appropriately. The extensions of this papers techniques to renewable resources are explored.

Joint economic and physical constraints on nuclear power: how much uranium would be needed to decarbonize electricity supply?

A new uranium supply and usage analysis takes into account growth in physical demand for electricity, both before and after decarbonization. It also models the economic willingness of investors to build the required reactors as a profit-maximizing decision. The analysis computes the optimum path of investment and disinvestment in one or more uranium-based fuel cycles as dynamic profit maximization under physical constraints, and the resulting economic decisions are compared with physical decarbonization goals. Investor decisions prove to be insensitive to the electricity price (and hence to higher carbon tax) above a low threshold. Above that price threshold, their decisions on reactor investment are dominated by the total supply of uranium and the speed at which reactors can be built. In order to pay for the very large generating capacity needed, profit-maximizing investors seem to need of the order of 36u2009Mu2009tons of uranium reserves to use (some seven times the worlds presently identified reserves). Even given this supply, they will limit their maximum investment in capacity if the build rate of reactors is too slow. If the world does choose to decarbonize, it will be necessary to manage uranium prospecting, and reactor building, purchase and operation as an interdependent system, and there will be a steep task of uranium exploration over the next decade or two.

Patent now or later? Corporate financing decisions, agency costs and social benefits

We analyze the incentives of firms to delay patenting a product they intend to commercialize to maximize the period they can exploit the market under patent protection. We model the patenting and market-launching decisions and consider partial financing of these costs with debt. Agency conflicts between equityholders and debtholders arise concerning the optimal patenting and market-launch timing and represent a classical moral hazard problem. We show that delaying patenting increases the value of the firm significantly in the absence of preemption risk. In the presence of preemption risk, the firm that aims to maximize the market exploitation period under patent protection accelerates the market-launch of the product. The use of debt financing reduces the incentives to delay patenting, but generates significant agency costs in terms of loss of firm value, debt capacity and increases in the fair credit spreads. When considered in terms of social effects, the impact of the agency conflicts is overall positive, as it accelerates patenting and market-launching the product and delays default.