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Answer:
A)= 20+24Q-4Q^{2} (This is the equation for the net benefits)
B) 40; 40
C)MNB(Q)= 24-8Q
D) 16; -16
E) Q=3
F)0
Explanation:
a) To write out the equation for the net benefits.
First, net benefits represents the difference obtained when the total benefits exceed the total costs derived form Q units of the control variable
Based on the definition, the equation is as follows
N(Q)= B(Q) - C(Q)
representing
N(Q)= The Net benefits that are derived from Q level of control variable
B(Q)= The total benefits derived from Q units of control variable
C(Q)= The total cost form Q units of control variable
As we have been given the following:B(Q)= 100 + 36Q - and C(Q) = 80 +12Q
This means:
N(Q)= 100+36Q-4Q^{2}- (80+12Q)
= 100+36Q-4Q^{2}-80-12Q
=100-80+36q-12Q-4Q^{2}
=20+24Q-4Q^{2}
= 20+24Q-4Q^{2} (This is the equation for the net benefits)
b) Determine he net benefits when Q = 1 and Q=5
Step 1) when Q=1 we use the formula 20+24Q-4Q^{2}
= 20+24Q-4Q^{2}
= 20(24x1)-(4x (1^{2))
=20+24-4
=40
Step 2) when Q=5 we use the formula 20+24Q-4Q^{2}
= 20+24Q-4Q^{2}
= 20(24x5)-(4x (5^{2))
=20+120-4x25
=140-100
=40
c)To write out the equation for the marginal net benefits
Marginal net benefits represent the change experienced in net benefits with change in one unit of the control variable
The formula therefore is as follows:
MNB(Q)=MB (Q)-MC(Q)
Representing
MNB (Q)= The marginal net benefits at Q level of control variable
MB(Q)= marginal benefits
MC(Q)=Marginal Costs
We are already given: MB(Q) = 36 – 8Q and MC(Q) = 12.)
This means
MNB(Q)=MB (Q)-MC(Q)
= 36-8Q-12
36-12-8Q
=24-8Q
MNB(Q)= 24-8Q
d) Find the marginal net benefits
Step 1) when Q=1 we use the formula for marginal net benefit MNB(Q)= 24-8Q
=MNB(Q)= 24-8(1)
= 16
Step 2) when Q=5 we use the formula for marginal net benefit MNB(Q)= 24-8Q
=MNB(Q)= 24-8(5)
=24-40
=-16
e) Calculate the maximum net benefits which represent the maximum level where the marginal costs= marginal benefits (it is a control variable level).
MB(Q)=MC(C)
MB(Q)= Marginal benefits
MC(Q)= Marginal Costs
We already know that MB(Q) =36-8Q and MC(Q)=12
Therefore, maximum benefit
= MB(Q)= MC(Q)
=36-8Q=12
8Q=36-12
8Q= 24
Q= 3.
This means that when Q is 3, then the net benefits will be at its maximum level
f) Compute marginal net benefits which is the difference between the marginal costs and benefits
MNB(Q)=MB(Q)-MC(Q)
MNB(Q)= Marginal net benefits
MB(Q)= Marginal benefits
MC(Q)= Marginal Costs
we already know that MB(Q) =36-8Q and MC(Q)=12
MNB(Q)= 36-8Q-12
=36-12-8Q
=24-8Q
Since the maximum level where net benefits are highest is 3 as calculated above, we then replace Q with 3 in the equation
MNB(Q)= =24-8(3)
= 24-24
=0
This means at Q level of 3, the net costs will be equal to the net benefits and the net benefits are at the maximum level.