Current star formation in S0 galaxies: NGC 4710

Elliptical (E) and lenticular (S0) galaxies lack the substantial interstellar medium (ISM) found in the star-forming spiral galaxies. However, significant numbers of E and S0 galaxies are known to contain detectable amounts of interstellar matter (e.g., Jura 1988). Thus, it is worth investigating whether these galaxies are currently able to form stars from their ISM, or whether they should be consigned to the dustbin of inert objects (Thronson and Bally 1987). The results strongly imply that current star formation is responsible for NGC 4710's far infrared and radio continuum properties. If this is indeed the case, then one expects this star formation to be fueled by molecular gas, which is presumably dominated by H2 and can be traced by the CO-12 J=1 to 0 line. Both Kenney and Young (1988) and Sage and Wrobel (1989) have detected such an emission line from NGC 4710, and infer the presence of more than 10(exp 8) solar mass of H2. The origin of the molecular gas in NGC 4710 remains a mystery. The galaxy is very deficient in HI (Kenney and Young, in preparation), suggesting that it originally was a spiral galaxy from which the outer, mainly atomic, gas was stripped by the ram pressure of the Virgo Cluster's intracluster medium, leaving only a central interstellar medium (ISM) rich in molecular gas. Alternatively, the CO may have originated via stellar mass loss with subsequent cooling, cooling flows, or capture from a gas-rich companion. Information on the morphology and kinematics of the CO can be compared with that of the galaxy's other gases and stars to distinguish among these various possible origins for the molecular gas. Major axis CO mapping with single dishes indicate an unresolved source. Thus, a millimeter array is currently being used to image NGC 4710 in CO to provide the needed morphological and kinematical data

Efficiency rents of pumped-storage plants and their uses for operation and investment decisions

We apply duality methods of linear and convex programming to the problems of operation and rental valuation of facilities for conversion and storage of cyclically priced goods, e.g. , energy. Both problems are approached by shadow-pricing the stock (which is a purely intermediate commodity); and if the given market price p for the final good is a continuous function of time, then the stock's shadow price function ? is shown to be unique (and continuous). Therefore, despite being perfect Allen-Hicks complements, the plant's capacities have definite and separate marginal values, which are expressed in terms of ? (and p). In particular, the unit reservoir rent equals the total positive variation of ? over the cycle. The optimal storage policy is also given in terms of ? and p). The marginal capacity values are used to determine the optimum investment. The framework can accommodate related storage problems (such as hydroelectric generation)

Characterizations of long-run producer optima and the short-runapproach to long-run market equilibrium: a general theory withapplications to peak-load pricing

This is a new formal framework for the theory of competitive equilibrium and its applications.Our "short-run approach" means the calculation of long-run producer optimaand general equilibria from the short-run solutions to the producer's profit maximizationprogramme and its dual. The marginal interpretation of the dual solution means that itcan be used to value the capital and other fixed inputs, whose levels are then adjustedaccordingly (where possible). But short-run profit can be a nondifferentiable function ofthe fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigidcapacity constraint. Nondifferentiability of the optimal value requires the introductionof nonsmooth calculus into equilibrium analysis, and subdifferential generalizations ofsmooth-calculus results of microeconomics are given, including the key Wong-Viner EnvelopeTheorem. This resolves long-standing discrepancies between "textbook theory"and industrial experience. The other tool employed to characterise long-run produceroptima is a primal-dual pair of programmes. Both marginalist and programming characterizationsof producer optima are given in a taxonomy of seventeen equivalent systemsof conditions. When the technology is described by production sets, the most usefulsystem for the short-run approach is that using the short-run profit programme andits dual. This programme pair is employed to set up a formal framework for long-rungeneral-equilibrium pricing of a range of commodities with joint costs of production.This gives a practical method that finds the short-run general equilibrium en route tothe long-run equilibrium, exploiting the operating policies and plant valuations that mustbe determined anyway. These critical short-run solutions have relatively simple formsthat can greatly ease the fixed-point problem of solving for equilibrium, as is shownon an electricity pricing example. Applicable criteria are given for the existence of theshort-run solutions and for the absence of a duality gap. The general analysis is speltout for technologies with conditionally fixed coefficients, a concept extending that of thefixed-coefficients production function to the case of multiple outputs. The short-run approachis applied to the peak-load pricing of electricity generated by thermal, hydro andpumped-storage plants. This gives, for the first time, a sound method of valuing thefixed assets-in this case, river flows and the sites suitable for reservoirs.general equilibrium, fixed-input valuation, nondifferentiable joint costs,Wong-Viner Envelope Theorem, public utility pricing

The Short-Run Approach to LRMC Pricing for Multiple Outputs with Nondifferentiable Costs

Using convex calculus, we extend the Wong-Viner Theorem to nondifferentiable costs by equating the capital inputs' rental prices to their profit-imputed marginal values. Thus extended, the short-run approach to LRMC pricing is applied to peak-load pricing with storage.Wong-Viner theorem, multiple outputs, peak-load pricing, energy storage

Efficiency Rents of Storage Plants in Peak-Load Pricing, II: Hydroelectricity - (Now published as Efficiency rents of hydroelectric storage plants in continuous-time peak-load pricing, in The Current State of Economic Science, by S B Dahiya (ed.), vol.1, pp.453-480 (Spellbound Publications, Rohtak, 1999).

Duality methods of linear and convex programming are applied to impute definite marginal values to the fixed inputs of a hydroelectric plant from the operating profit. Our earlier analysis of pumped storage (of energy and other cyclically priced goods) is thus extended to valuation of an external inflow to the reservoir. Given a continuous time-of-use price for electricity, the profit-imputed hydro values are uniquely determined - unlike the corresponding values imputed from fuel savings for a mixed hydro-thermal system. In particular the water inflow is assigned a unique, time-dependent shadow price. The short-run profit is then differentiable in all the fixed inputs, so that unique and separate marginal values can be imputed to the reservoir and the turbine capacities (despite their perfect complementarity). The two rents can be expressed in terms of the shadow price for water (which determines the optimal storage policy). In particular, the unit reservoir rent equals the total positive variation of the shadow price over the cycle. Evaluation of profit-imputed rents is shown to be useful not only to a profit-maximising industry but also to a public utility aiming to price its outputs at long-run marginal cost and to optimise its capital stock on the basis of purely short-run calculations. In addition we verify the production set properties that are needed to incorporate such a storage problem into a continuous-time model of general competitive equilibrium with the space of bounded functions of time as the commodity space.storage hydro, rental valuation, peak-load pricing, linear programming.

The Wong-Viner Envelope Theorem for subdifferentiable functions

The Wong-Viner Envelope Theorem on the equality of long-run and short-run marginalcosts (LRMC and SRMC) is reformulated for convex but generally nondifferentiable costfunctions. The marginal cost can be formalized as the multi-valued subdifferential a.k.a.the subgradient set but, in itself, this is insufficient to extend the result effectively, i.e., toidentify suitable SRMCs as LRMCs. This goal is achieved by equating the profit-imputedvalues of the fixed inputs to their prices. Thus reformulated, the theorem is proved froma lemma on the sections of the joint subdifferential of a bivariate convex function. Thenew technique is linked to the Partial Inversion Rule of convex calculus.Wong-Viner Envelope Theorem, nondifferentiable joint costs, profit-imputedvaluation of fixed inputs, general equilibrium, public utility pricing.

Continuity of the Equilibrium Price Density and its Uses in Peak-Load Pricing

With L8 as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Pólya theory of rearrangements; and it includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. For continuous-time peak-load pricing of, e.g., electricity, this allows the inclusion of storage and of cross-price dependent demands. In this context a continuously varying price has two uses. First, it excludes the demand jumps that arise from discontinuous switches from one price rate to another. Second, in the operation and valuation of hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks and, in the case of hydro also the river flows, have well-defined marginal values.Price density, continuous-time peak-load pricing, pumped storage.