hw10-07-1-key - ECONOMICS 331 Mathematical Economics Kevin...

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Unformatted text preview: ECONOMICS 331 Mathematical Economics Kevin Wainwright Homework Assignment 10 ANSWER KEY 1. A new cellphone company &Yap.com&is setting up in Burnaby and it has to plan its capacity. The peak period demand is given by p 1 = 200 & : 25 q 1 and the o/-peak is given by p 2 = 190 & : 25 q 2 : Let K be the cell capacity which costs 15 per unit and is only paid once and is used in both periods. There are no other costs. The quantity of cell usage in either market ( q 1 ;q 2 ) cannot excede K . (a) write down the Kuhn-Tucker conditions. Z = 200 q 1 & : 25 q 2 1 + 190 q 2 & : 25 q 2 2 & 15 K + & 1 ( K & q 1 ) + & 2 ( K & q 2 ) Z 1 = 200 & : 5 q 1 & & 1 q 1 Z 2 = 190 & : 5 q 2 & & 2 q 2 Z K = & 15 + & 1 + & 2 K Z & 1 = K & q 1 & 1 Z & 2 = K & q 2 & 2 (b) nd the optimal outputs and capacity for this problem.How much of the capacity price is paid for by each market? In this case, both constraints are binding Z 1 = 200 & : 5 q 1 = & 1 Z 2 = 190 & : 5 q 2 = & 2 Z K = & 15 + & 1 + & 2 = 0 Z & 1 = K & q 1 = 0 Z & 2 = K & q 2 = 0 By substitution Z 1 = 200 & : 5 K = & 1 Z 2 = 190 & : 5 K = 15 & & 1 K = q 1 = q 2 = 375 & 1 = 12 : 5 ;& 2 = 2 : 5 (c) Suppose price of capacity is now 5 per unit of capacity. Redo part (b) Z 1 = 200 & : 5 q 1 = 5 Z 2 = 190 & : 5 q 2 = 0 & 1 = 5 K = q 1 K =...
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hw10-07-1-key - ECONOMICS 331 Mathematical Economics Kevin...

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