In a particular region, there are two lakes rich in fish. The quantity of fish c

Question

In a particular region, there are two lakes rich in fish. The quantity of fish caught in each lake depends on the number of persons who fish in each according to

                        Q1 = 15 N1- 0.5 N12         and      Q2 = 8 N2 - 0.2 N22

                        where N1 and N2 denote the # of fishers at each lake. 

                        In all there are 28 fishers, i.e. N1 + N2 = 28.

(a) Suppose N1 = 16 and N2 = 12. At which lake is the average catch per fisher greater? In light of this fact, how would you expect the fishers to redeploy themselves? 

(b) How many fishers will settle at each lake?

(c) The Department of Fisheries seeks a division of fishers that will maximize the total catch at the two lakes. What division of the 28 fishers would you recommend to the Department?

(d)   What interpretation would you give to the Lagrangian multiplie,  ?

Explanation / Answer

a) Q1= 15*16-0.5 16^2= 112 112/16= avg of 7
Q2= 12*8 -.2 *12^2= 67.2 67.2/12= avg of 5.6
Fishers will move from 2 to 1.
b) Where average catch is the same
15-.5n1=8-.2n2
n1+n2=28
15-.5*(28-n2)=8-.2n2
15-14 +.5n2= 8-.2n2
.7n2= 7
n2=10
n1=18
c) TQ=Q1+Q2= 15n1 +8n2-0.5n1^2-0.2n2^2
TQ1= 15-n1
TQ2= 8-.4n2
n1+n2=28 (partial derivatives of both are 1)
So we have 15-n1=

8-.4n2=

15-n1= 8-.4n2

n1+n2=28

15-(28-n2)= 8-.4n2

1.4n2= 21

n2= 15 n1=13

lambda is the marginal fish caught; the additional fish caught by one more fisherman.

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