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Suppose that the total benefit and total cost from a continuous activity are, respectively, given by the following equations:

B(Q) = 100 + 36Q – 4Q2 and C(Q) =80 + 12Q.

(Note: MB(Q) = 36 – 8Q and MC(Q) = 12.)

Instructions: Use a negative sign (-) where appropriate.

a. Write out the equation for the net benefits.

N(Q) = + Q + Q2

b. What are the net benefits when Q = 1? Q = 5?

Net benefits when Q = 1:
Net benefits when Q = 5:

c. Write out the equation for the marginal net benefits.

MNB(Q) = + Q

d. What are the marginal net benefits when Q = 1? Q = 5?

Marginal net benefits when Q = 1:
Marginal net benefits when Q = 5:

e. What level of Q maximizes net benefits?
f. At the value of Q that maximizes net benefits, what is the value of marginal net benefits?

Respuesta :

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 - [tex]4Q^{2}[/tex] 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.

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