target automobile""s roofline to lowest point of coverage below the roofline after inflation) of at least 17.
The term xe2x80x9ccharacteristic leak-down timexe2x80x9d is intended to encompass the measurement of time required for the entire amount of inflation gas introduced within an already-inflated (to a peak initial pressure which xe2x80x9copensxe2x80x9d up the areas of weak sealing) and deflated airbag cushion upon subsequent re-inflation at a constant pressure at 10 psi. It is well known and well understood within the airbag art, and particularly concerning side curtain (low permeability) airbag cushions, that retention of inflation gas for long periods of time is of utmost importance during a collision that results in rollover and other subsequent problems. Side curtain airbags are designed to inflate as quickly as driver- and passenger-side bags, but facing prices , and let  denote the firm's supply function, i.e.,

Let  (the price of good ) and fix the other goods' prices at . Applying Theorem 1 to  yields  (the firm's optimal supply of good ). Applying Theorem 2 (whose assumptions are verified when  is restricted to a bounded interval) yields

i.e. the producer surplus  can be obtained by integrating under the firm's supply curve for good .

Applications to mechanism design and auction theory
Consider an agent whose utility function  over outcomes  depends on his type . Let  represent the "menu" of possible outcomes the agent could obtain in the mechanism by sending different messages. The agent's equilibrium utility  in the mechanism is then given by (1), and the