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 : &lt; + &lt; be dened by f ( x ) = ( if x = 0 x sin 1 x 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 nd the local maxima and minima (if any) in the set where each f is dened: (a) f ( x ) = x 3 + ax + b where x &lt; . (b) f ( x ) = ln ( x 2-9) where | x | &gt; 3. (c) f ( x ) = x 2 / 3 ( x-1) 4 where 0 x 1. (3 points) 1...
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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