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# ps4 - may be used to prove that a solution exists to this...

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Atsushi Inoue ECG 765: Mathematics for Economists Fall 2009 Problem Set ] 4 Due in class on Tuesday, September 22 1. A ﬁshery earns a proﬁt of π ( x ) from catching and selling x units of ﬁsh. The ﬁrm owns a pool which currently has y 1 ﬁsh in it. If x [0 ,y 1 ] ﬁsh are caught this period, the remaining i = y 1 - x will grow to f ( i ) ﬁsh by the beginning of the next period, where f : < + → < + is the growth function for the ﬁsh population. The ﬁshery wishes to set the volume of its catch in each of the next two periods so as to maximize the sum of its proﬁts over this horizon. That is, it solves: max x 1 ,x 2 π ( x 1 ) + π ( x 2 ) subject to 0 x 1 y 1 0 x 2 f ( y 1 - x 1 ) Show that if π and f are continuous on < + then the Weiestrass theorem
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Unformatted text preview: may be used to prove that a solution exists to this problem. (3 points) 2. Do exercise 13.21 of Simon and Blume (1994, p.295). (2 points) 3. Let f : < + → < be deﬁned by f ( x ) = ( if x = 0 x sin ± 1 x ² if x 6 = 0 Prove that f is continuous at 0. (2 points) 4. In each of the following cases, determine the intervals in which the function f is increasing or decreasing and the ﬁnd the local maxima and minima (if any) in the set where each f is deﬁned: (a) f ( x ) = x 3 + ax + b where x ∈ < . (b) f ( x ) = ln ( x 2-9) where | x | > 3. (c) f ( x ) = x 2 / 3 ( x-1) 4 where 0 ≤ x ≤ 1. (3 points) 1...
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