and that capacity be chosen such that𝐷𝑌𝐿=𝑐0−𝑛∑𝑖=1𝜆𝑖≤0𝑌≥0𝑌(𝑐0−𝑛∑𝑖=1𝜆𝑖)= 0(5.109)where𝐿is the Lagrangean𝐿(𝑦, 𝑌, 𝜆) =𝑛∑𝑖=1∫𝑦𝑖0(𝑝𝑖(𝜏)−𝑐𝑖)𝑑𝜏−𝑐0𝑌−𝑛∑𝑖=1𝜆𝑖(𝑦𝑖−𝑌)In off-peak periods (𝑦𝑖< 𝑌), complementary slackness requires that𝜆𝑖= 0 and there-fore from (5.108)𝑝𝑖(𝑦𝑖) =𝑐𝑖assuming𝑦𝑖>0. In peak periods (𝑦𝑖=𝑌)𝑝𝑖(𝑦𝑖) =𝑐𝑖+𝜆𝑖We conclude that it is optimal to price at marginal cost in off-peak periods and chargea premium during peak periods. Furthermore, (5.109) implies that the total premiumis equal to the marginal capacity cost𝑛∑𝑖=1𝜆𝑖=𝑐0Furthermore, note that𝑛∑𝑖=1𝜆𝑖
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