Suppose our target is to max Pi n gi log Pi, i=1 n subject to 0...

 

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Suppose our target is to max Pi n Σgi log Pi, i=1 n subject to 0 ≤ P; ≤ 1 for all i and p₁ = 1, Pi i=1 - = qi, i = 1,...,n, is = qi, i - where q; qi 9i is known, 0 ≤ qį ≤ 1 for all i, and Σ19 = 1. Show that pi the maximizer. Hint: argue that (i) it is without loss of generality to assume qi € (0, 1); (ii) under (i), the optimal p; is an interior so the restriction 0 < pi ≤ 1 does not count. Suppose our target is to max Pi n Σgi log Pi, i=1 n subject to 0 ≤ P; ≤ 1 for all i and p₁ = 1, Pi i=1 - = qi, i = 1,...,n, is = qi, i - where q; qi 9i is known, 0 ≤ qį ≤ 1 for all i, and Σ19 = 1. Show that pi the maximizer. Hint: argue that (i) it is without loss of generality to assume qi € (0, 1); (ii) under (i), the optimal p; is an interior so the restriction 0 < pi ≤ 1 does not count.

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Solution To show that pi gi i 1 n is the maximizer of the given objective function we can follow the ... View the full answer