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1 month ago

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

Business

1 answer:

Mariulka [3.8K]1 month ago

5 0

Answer:

A) The equation for net benefits is given by 20 + 24Q - 4Q²

B) 40; 40

C) MNB(Q) = 24 - 8Q

D) 16; -16

E) Q = 3

F) 0

Explanation:

a) This outlines the formula for net benefits.

Essentially, net benefits arise when total benefits surpass total costs from Q units of the control variable.

From this definition, we derive the equation

N(Q)= B(Q) - C(Q)

Where

N(Q)= Net benefits from Q units of control variable

B(Q)= Total benefits from Q control variable units

C(Q)= Total cost from Q units of the control variable

We are informed that B(Q)= 100 + 36Q - $4Q^{2}$ and C(Q) = 80 +12Q

This implies:

N(Q)= 100 + 36Q - 4Q² - (80 + 12Q)

= 100 + 36Q - 4Q² - 80 - 12Q

= 20 + 24Q - 4Q²

b) To find the net benefits when Q = 1 and Q=5

Step 1) For Q=1, we apply the 20 + 24Q - 4Q² formula

= 20 + 24(1) - 4(1²)

= 20 + 24 - 4

= 40

Step 2) For Q=5, we use the 20 + 24Q - 4Q² formula

= 20 + 24(5) - 4(5²)

= 20 + 120 - 100

= 40

c) To determine the marginal net benefits equation

Marginal net benefits reflect the change in net benefits caused by a change of one unit in the control variable

The formula is formulated as follows:

MNB(Q) = MB (Q) - MC(Q)

Where

MNB(Q)= The marginal net benefits at the Q level of the control variable

MB(Q)= marginal benefits

MC(Q)=Marginal Costs

We are already informed: MB(Q) = 36 – 8Q and MC(Q) = 12.

This indicates

MNB(Q)=MB (Q)-MC(Q)

= 36-8Q-12

= 36-12-8Q

= 24-8Q

Thus, MNB(Q) = 24-8Q

d) To find marginal net benefits

Step 1) When Q=1 we apply the marginal net benefit formula MNB(Q)= 24-8Q

=MNB(Q)= 24-8(1)

= 16

Step 2) For Q=5 we utilize the marginal net benefit MNB(Q)= 24-8Q

=MNB(Q)= 24-8(5)

=24-40

=-16

e) To calculate maximum net benefits, which occur when marginal costs equal marginal benefits (indicating the control variable level).

MB(Q)=MC(C)

MB(Q)= Marginal benefits

MC(Q)= Marginal Costs

We know that MB(Q) =36-8Q and MC(Q)=12

Therefore, setting maximum benefit

= MB(Q)= MC(Q)

=36-8Q=12

8Q=36-12

8Q= 24

So, Q= 3.

This indicates that at Q equal to 3, net benefits achieve maximum levels

f) To compute marginal net benefits - the difference between marginal costs and benefits

MNB(Q)=MB(Q)-MC(Q)

MNB(Q)= Marginal net benefits

MB(Q)= Marginal benefits

MC(Q)= Marginal Costs

We are aware that MB(Q) =36-8Q and MC(Q)=12

MNB(Q)= 36-8Q-12

=36-12-8Q

=24-8Q

As established, maximum levels of net benefits occur at Q = 3, substituting Q with 3 in the equation

MNB(Q)= =24-8(3)

= 24-24

This indicates that at Q = 3, net costs balance out with net benefits, maximizing net benefits.

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