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Comparing monopoly bundling with separate sales is relatively straightforward in an environment with a large number of goods. We show that results similar to those for the asymptotic case can be obtained in the more realistic case with a given finite number of goods, provided that the distributions of valuations are symmetric and log‐concave.

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Constrained efficient provision of an excludable public good is studied in a model where preferences are private information. The provision level is asymptotically deterministic, making it possible to approximate the optimal mechanism with a mechanism that provides a fixed quantity of the good and charges fixed user fees for access. In general, the fixed fees involve third degree price discrimination, but, if names are uninformative about preferences, the analysis provides a justification for average cost pricing.

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We investigated how the luminance of an immediate surround to a visual target affects the measurement of contrast sensitivity. We examined four young, healthy, subjects under typical lighting conditions for each test but varied the surround luminance systematically by controlling a backlit surround. The effect of varying the surround on contrast sensitivity measurements made with a Pelli-Robson chart was minimal. The only statistically significant difference was between measurements made with a 900 cd.m-2 surround compared to those made with 5.6, 9, or 30 cd.m-2 surrounds when tested at a viewing distance of 4 m. The higher luminance surround produced higher levels of contrast sensitivity. The effect of the surround luminance was greater when measuring contrast sensitivity to sinewave gratings using a method of constant stimuli. Optimum performance was found using a surround luminance of 9 cd.m-2 (20% of the mean target luminance). Statistically, a wide range of surround luminances (3-30 cd.m-2, or 7-67% mean target luminance) gave similar levels of visual performance, but higher luminance surrounds led to reduced contrast sensitivity. Possible reasons for the difference in the effect of the surround depending upon the test are discussed. Ophthalmic practitioners using contrast sensitivity measurements with letter charts probably need not make special efforts to illuminate the test surround appropriately. However, practitioners using sinewave gratings, especially computer generated sinewave gratings, to measure contrast sensitivity, may have their measurements affected by inappropriately lit task surrounds.

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