Quantity and number

Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity

Emissions trading without a quantity constraint.

This paper examines the differences between standard “cap-and-trade” emissions trading plans and “credit” plans in which individual agents create credits by reducing emissions below a firmspecific baseline. The two are equivalent if the baseline is a fixed quantity, but not if the baseline is specified as a baseline emissions ratio times current output. In the latter case there is no exogenous constraint on aggregate emissions. It may be called the case of “(ratio-based) credit trading”. Examples include the Clean Development Mechanism (CDM) of the Kyoto Protocol and the Canadian Pilot Emissions Reduction Trading plan (PERT). Unlike the case of cap-and-trade, the theoretical properties of ratio-based credit trading plans are not well known. In the absence of a binding quantity constraint, it is even difficult to understand how an ERC plan can generate a positive price. This paper studies the difference between ratiobased credit trading and conventional “cap-and-trade” plans in the context of a very simple model. It also considers how the two plans might interact if, for example, credits from a credit plan could be applied to commitments under a quantity-based cap-and-trade plan, and applies its findings to current plans for credit trading, including PERT and the clean development mechanism. The paper demonstrates that ratio-based credit trading is more like a tax instrument than a quantity instrument. It shows that there is no incentive to trade in a ratio-based market in which all firms receive baselines computed using their “business as ususal” emission ratios. Combining ratio-based credit trading with “cap-and-trade” allowance markets effectively relaxes the quantity constraint in the cap-and-trade plan and reduces the price of traded allowances. In the long run, there will be no effective constraint on emissions. The results have strong implications for current policy. In particular, they suggest that mixing quantity-based and ratio-based emission trading plans is inappropriate.

Exact and Superlative Price and Quantity Indicators

The traditional economic approach to index number theory is based on a ratio concept. The Konüs true cost of living index is a ratio of cost functions evaluated at the same utility level but with the prices of the current period in the cost function that appears in the numerator and the prices of the base period in the denominator cost function. The Allen quantity index is also a ratio of cost functions where the utility levels vary but the price vector is held constant in the numerator and denominator. There is a corresponding theory for differences of cost functions that was initiated by Hicks and the present paper develops this approach. Diewert defined superlative price and quantity indexes as observable indexes which were exact for a ratio of unit cost functions or for a ratio of linearly homogeneous utility functions. The present paper looks for counterparts to his results in the difference context, for both flexible homothetic and flexible nonhomothetic preferences. The Bennet indicators of price and quantity change turn out to be superlative for the nonhomothetic case. The underlying preferences are of the translation homothetic form discussed by Balk, Chambers, Dickenson, Färe and Grosskopf.Price and quantity aggregates, index number theory, equivalent and compensating variations, exact and superlative indexes, flexible functional forms,

Four-wave mixing wavelength conversion efficiency in semiconductor traveling-wave amplifiers measured to 65 nm of wavelength shift

The efficiency of broadband optical wavelength conversion by four-wave mixing in semiconductor traveling-wave amplifiers is measured for wavelength shifts up to 65 nm using a tandem amplifier geometry. A quantity we call the relative conversion efficiency function, which determines the strength of the four-wave mixing nonlinearity, was extracted from the data. Using this quantity, gain requirements for lossless four-wave mixing wavelength conversion are calculated and discussed. Signal to background noise ratio is also measured and discussed in this study

Approximate Integrated Likelihood via ABC methods

We propose a novel use of a recent new computational tool for Bayesian inference, namely the Approximate Bayesian Computation (ABC) methodology. ABC is a way to handle models for which the likelihood function may be intractable or even unavailable and/or too costly to evaluate; in particular, we consider the problem of eliminating the nuisance parameters from a complex statistical model in order to produce a likelihood function depending on the quantity of interest only. Given a proper prior for the entire vector parameter, we propose to approximate the integrated likelihood by the ratio of kernel estimators of the marginal posterior and prior for the quantity of interest. We present several examples.Comment: 28 pages, 8 figure