Optimal monopoly investment and capacity utilization under random demand

Unique value-maximizing programs of irreversible capacity investment and capacity utilization are described and shown to exist under general conditions for monopolist exhibiting capital adjustment costs and serving random consumer demand for a nondurable good over an infinite horizon. Stationary properties of these programs are then fully characterized under the assumption of serially independent demand disturbances. Optimal monopoly behavior in this case includes acquisition of a constant and positive level of capacity, the maintenance of a positive expected value of excess capacity in each period, and an asymmetrical response of price to unanticipated fluctuations in consumer demand. Under a general form of Markovian demand, the effect of uncertainty on irreversible capacity investment is also described in terms of the discounted flow of expected revenue accruing to the marginal unit of existing capacity and the option value of deferring the acquisition of additional capital. The option value of deferring such acquisition, created by the irreversibility of capacity investment, is characterized directly in terms of the value function of the firm, and is then shown to be zero in a stationary equilibrium with serially independent demand disturbances. The response of investment to increase demand uncertainty depends, as a result, directly on the properties of the marginal revenue product of capital. A non-negative response of optimal capacity to increased uncertainty in market demand is demonstrated for a general class of aggregate consumer preferences.Industrial capacity

Supply Function Equilibria with Capacity Constraints and Pivotal Suppliers

The concept of a supply function equilibrium (SFE) has been widely used to model generators’ bidding behavior and market power issues in wholesale electricity markets. Observers of electricity markets have noted how generation capacity constraints may contribute to market power of generation firms. If a generation firm’s rivals are capacity constrained then the firm may be pivotal; that is, the firm could substantially raise the market price by unilaterally withholding output. However the SFE literature has not properly analyzed the impact of capacity constraints and pivotal firms on equilibrium predictions. We characterize the set of symmetric supply function equilibria for uniform price auctions when firms are capacity constrained and show that this set is increasing as capacity per firm rises. We provide conditions under which asymmetric equilibria exist and characterize these equilibria. In addition, we compare results for uniform price auctions to those for discriminatory auctions, and we compare our SFE predictions to equilibrium predictions of models in which bidders are constrained to bid on discrete units of output.supply function equilibrium, pivotal firm, wholesale electricity market

Intermittency and the Value of Renewable Energy

A key problem with renewable energy is intermittency. This paper develops a method to quantify the social costs of large-scale renewable energy generation. The method is based on a theoretical model of electricity system operations that allows for endogenous choices of generation capacity investment, reserve operations, and demand-side management. We estimate the model using generator characteristics, solar output, electricity demand, and weather forecasts for an electric utility in southeastern Arizona. The estimated welfare loss associated with a 20% solar photovoltaic mandate is 11% higher than the average cost difference between solar generation and natural gas generation. Unforecastable intermittency yields welfare loss equal to 3% of the average cost of solar. Eliminating a mandate provision requiring a minimum percentage of distributed solar generation increases welfare. With a $21/ton social cost of CO2 this mandate is welfare neutral if solar capacity costs decrease by 65%.

Price caps, oligopoly, and entry

We extend the analysis of price caps in oligopoly markets to allow for sunk entry costs and endogenous entry. In the case of deterministic demand and constant marginal cost, reducing a price cap yields increased total output, consumer welfare, and total welfare; results consistent with those for oligopoly markets with a fixed number of firms. With deterministic demand and increasing marginal cost these comparative static results may be fully reversed, and a welfare-improving cap may not exist. Recent results in the literature show that for a fixed number of firms, if demand is stochastic and marginal cost is constant then lowering a price cap may either increase or decrease output and welfare (locally); however, a welfare improving price cap does exist. In contrast to these recent results, we show that a welfare-improving cap may not exist if entry is endogenous. However, within this stochastic demand environment we show that certain restrictions on the curvature of demand are sufficient to ensure the existence of a welfare-improving cap when entry is endogenous

Price caps, oligopoly, and entry

We extend the analysis of price caps in oligopoly markets to allow for sunk entry costs and endogenous entry. In the case of deterministic demand and constant marginal cost, reducing a price cap yields increased total output, consumer welfare, and total welfare; results consistent with those for oligopoly markets with a fixed number of firms. With deterministic demand and increasing marginal cost these comparative static results may be fully reversed, and a welfare-improving cap may not exist. Recent results in the literature show that for a fixed number of firms, if demand is stochastic and marginal cost is constant then lowering a price cap may either increase or decrease output and welfare (locally); however, a welfare improving price cap does exist. In contrast to these recent results, we show that a welfare-improving cap may not exist if entry is endogenous. However, within this stochastic demand environment we show that certain restrictions on the curvature of demand are sufficient to ensure the existence of a welfare-improving cap when entry is endogenous

Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold

The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly-connected ``links'' and multiply-connected ``blobs.'' Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block is a graph that cannot be separated into disconnected subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and 3-blocks are special cases of $k$-blocks with $k=1$, 2, and 3, respectively. We study bond percolation clusters at the percolation threshold on 2-dimensional square lattices and 3-dimensional cubic lattices and, using Monte-Carlo simulations, determine the distribution of the sizes of the 3-blocks into which the blobs are decomposed. We find that the 3-blocks have fractal dimension $d_3=1.2\pm 0.1$ in 2D and $1.15\pm 0.1$ in 3D. These fractal dimensions are significantly smaller than the fractal dimensions of the blobs, making possible more efficient calculation of percolation properties. Additionally, the closeness of the estimated values for $d_3$ in 2D and 3D is consistent with the possibility that $d_3$ is dimension independent. Generalizing the concept of the backbone, we introduce the concept of a ``$k$-bone'', which is the set of all points in a percolation system connected to $k$ disjoint terminal points (or sets of disjoint terminal points) by $k$ disjoint paths. We argue that the fractal dimension of a $k$-bone is equal to the fractal dimension of $k$-blocks, allowing us to discuss the relation between the fractal dimension of $k$-blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when printe

Stochastic Renormalization Group in Percolation: I. Fluctuations and Crossover

A generalization of the Renormalization Group, which describes order-parameter fluctuations in finite systems, is developed in the specific context of percolation. This ``Stochastic Renormalization Group'' (SRG) expresses statistical self-similarity through a non-stationary branching process. The SRG provides a theoretical basis for analytical or numerical approximations, both at and away from criticality, whenever the correlation length is much larger than the lattice spacing (regardless of the system size). For example, the SRG predicts order-parameter distributions and finite-size scaling functions for the complete crossover between phases. For percolation, the simplest SRG describes structural quantities conditional on spanning, such as the total cluster mass or the minimum chemical distance between two boundaries. In these cases, the Central Limit Theorem (for independent random variables) holds at the stable, off-critical fixed points, while a ``Fractal Central Limit Theorem'' (describing long-range correlations) holds at the unstable, critical fixed point. This first part of a series of articles explains these basic concepts and a general theory of crossover. Subsequent parts will focus on limit theorems and comparisons of small-cell SRG approximations with simulation results.Comment: 33 pages, 6 figures, to appear in Physica A; v2: some typos corrected and Eqs. (26)-(27) cast in a simpler (but equivalent) for

Regular Expression Matching and Operational Semantics

Many programming languages and tools, ranging from grep to the Java String library, contain regular expression matchers. Rather than first translating a regular expression into a deterministic finite automaton, such implementations typically match the regular expression on the fly. Thus they can be seen as virtual machines interpreting the regular expression much as if it were a program with some non-deterministic constructs such as the Kleene star. We formalize this implementation technique for regular expression matching using operational semantics. Specifically, we derive a series of abstract machines, moving from the abstract definition of matching to increasingly realistic machines. First a continuation is added to the operational semantics to describe what remains to be matched after the current expression. Next, we represent the expression as a data structure using pointers, which enables redundant searches to be eliminated via testing for pointer equality. From there, we arrive both at Thompson's lockstep construction and a machine that performs some operations in parallel, suitable for implementation on a large number of cores, such as a GPU. We formalize the parallel machine using process algebra and report some preliminary experiments with an implementation on a graphics processor using CUDA.Comment: In Proceedings SOS 2011, arXiv:1108.279