Диссертация (1138062), страница 31
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ЛеммыВ этом приложении приведены некоторые вспомогательные леммы,упрощающие доказательства утверждений.Лемма 1. Формула двойной эластичности:ℰ x [ℰ x [ f ]] = 1 + ℰ x [ f ′ ] − ℰ x [ f ].Доказательство. Непосредственными вычислениями получаем:(︃ ′ )︃′dxf x fℰ x [ℰ x [ f ]] =[ℰ x [ f ]] ·=f ′xdxℰx[ f ]fx)︃(︃ ′′′′2′′f x ff xf ′xf x f+ − 2= ′ +1−=fff′fff= 1 + ℰ x [ f ′ ] − ℰ x [ f ].(A.1)Следствие 1. Применяя Лемму 1 к rlv, получаем следующее:ℰ x [ru ] = 1 −ru′ + ru .Доказательство. Посколькуru = −ℰ x [u′ ]иru′ = −ℰ x [u′′ ],чается непосредственным применением (A.1) toрезультат полуf ≡ u′ .Следствие 2. Применяя Следтствие 1 , имеем следующую эквивалентность:2 − ru′T11 − ru⇔ru′ T 0.(A.2)Доказательство. Из следствия 1, имеем:ℰ x [ru ] = 1 − ru′ + ru = (2 − ru′ ) − (1 − ru ) T 0⇔ru′ T 0.184Эквивалентность знаков эластичности и производнойruвытекает непосредственно из определения эластичности.
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