Question

Suppose that the total benefit and the total cost from a continuous activity are respectively, given by the following equations: $\mathit{B}\left(Q\right)\mathbf{=}\mathbf{100}\mathbf{+}\mathbf{36}\mathit{Q}\mathbf{-}\mathbf{4}{\mathit{Q}}^{\mathbf{2}}$and$\mathit{C}\left(Q\right)\mathbf{=}\mathbf{80}\mathbf{+}\mathbf{12}\mathit{Q}$ [Note: $\mathit{M}\mathit{B}\left(Q\right)\mathbf{=}\mathbf{36}\mathbf{-}\mathbf{8}\mathit{Q}$and $\mathit{M}\mathit{C}\left(Q\right)\mathbf{=}\mathbf{12}$]

(a)Write out the equation for the net benefits.

(b) What are the net benefits when $\mathit{Q}\mathbf{=}\mathbf{1}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{5}\mathbf{?}$

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

(d) What are the marginal net benefits when $\mathit{Q}\mathbf{=}\mathbf{1}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{5}\mathbf{?}$

(e) What level of $\mathit{Q}$maximizes net benefits?

(f) At the value of $\mathit{Q}$that maximizes net benefits, what is the value of marginal net benefits?

Short answer

1. The equation for the net benefits is $N\left(\mathrm{Q}\right)=20+24Q-4Q^2$

2. If $Q=1$, the net benefit is $40$and if $Q=5$, the net benefit is 40.

3. The equation for the marginal Net benefit is $MNB\left(Q\right)=24-8Q$

4. The marginal net benefit is $16$if $Q=1$and $(-16)$if $Q=5$.

5. The net benefit will be on its maximum when $Q$level is $3$.

6. If $Q$level is $3$, the net benefit in its maximum and on that level, marginal net benefits are zero.

Step-by-step solution

(a) The equation for net benefit.

To find the equation for the net benefits, use the following equation:

$N\left(\mathrm{Q}\right)=B\left(\mathrm{Q}\right)-C\left(\mathrm{Q}\right)$where $N\left(Q\right)$is the net benefits from $Q$units of controlled variables, $B\left(Q\right)$is the total benefit from $Q$units of controlled variables.

With the given values:

role="math" localid="1657541096053" $B\left(Q\right)=100+36Q-4Q^2$;

$C\left(Q\right)=80+12Q$;

By substitute all the given numbers, we gt

$N\left(Q\right)=100+36Q-4Q^2-\left(80+12Q\right)$

$=100+36Q-4Q^2-80-12Q$

$=20+24Q-4Q^2$

Therefore, the equation for net benefit is$N\left(Q\right)=20+24Q-4Q^2$.

(b) The net benefits when Q=1 and Q=5.

To find the net benefits if $Q=1$and $Q=5$, we use the following equation for the net benefits:

$N\left(Q\right)=20+24Q-4Q^2$

Then we substitute $Q=1$,

$N\left(Q\right)=20+24Q-4Q^2$

$=20+24\left(1\right)-\left(4.1^2\right)$

$=20+24-4$

$=40$

Therefore if localid="1657542606974" $\mathit{Q}\mathbf{=}\mathbf{1}$, the net benefit is localid="1657542625312" $\mathbf{40}$.

If $Q=5$, we substitute,

$N\left(Q\right)=20+24Q-4Q^2$

$=20+24\left(5\right)-4\left(5^2\right)$

$=20+120-4\left(25\right)$

$=140-100$

$=40$

Therefore if $\mathit{Q}\mathbf{=}\mathbf{5}$, the net benefit is$\mathbf{40}$.

(c) The equation for marginal net benefit.

Marginal net benefits are the one unit change in a controlled variable from the net benefits. To find the equation for net benefits, use the following formula:$MNB\left(Q\right)=MB\left(Q\right)-MC\left(Q\right)$

With the given

$$\begin{gathered} MB\left(Q\right)=36-8Q; \\ MC\left(Q\right)=12, \end{gathered}$$

We substitute all the given numbers,

$$\begin{gathered} MNB\left(Q\right)=36-8Q-12 \\ =36-8Q-12 \\ =24-8Q \end{gathered}$$

Therefore, the equation for marginal net benefits is$MNB\left(Q\right)=24-8Q$

(d) The marginal benefits when Q=1 and Q=5.

To find the marginal net benefit if Q=1 and Q=5, we use the following equation:

$MNB\left(Q\right)=24-8Q$

Then substitute $Q=1$,

$$\begin{gathered} MNB\left(Q\right)=24-8Q \\ MNB\left(Q\right)=24-8\left(1\right) \\ MNB\left(Q\right)=16. \end{gathered}$$

Thus, the marginal net benefit is role="math" localid="1657544946727" $\mathbf{16}$if role="math" localid="1657544963451" $\mathit{Q}\mathbf{=}\mathbf{1}$.

If $Q=5$,

$$\begin{gathered} MNB\left(Q\right)=24-8Q \\ MNB\left(Q\right)=24-8\left(5\right) \\ MNB\left(Q\right)=24-40 \\ MNB\left(Q\right)=-16. \end{gathered}$$

Thus, the marginal net benefits are (-16) if Q=5 .

(e) Q maximizes the net benefit at,

To find the level of Q where it maximizes its net benefits, use the following formula:

$MB\left(Q\right)=MC\left(Q\right)$

With the given:

$$\begin{gathered} MB\left(Q\right)=36-8Q; \\ MC\left(Q\right)=12; \end{gathered}$$

Substituting all the numbers, we get

$$\begin{gathered} MB\left(Q\right)=MC\left(Q\right) \\ 36-8Q=12 \\ 8Q=24 \\ Q=\frac{24}{8}=3 \\ Q=3 \end{gathered}$$

So, the net benefits will be on its maximum when $\mathit{Q}$level is $3$.

(f) vale of marginal net value.

To find the value of managerial benefits of the value of Q that maximizes the net benefits, use the following equation: $MNB\left(Q\right)=MB\left(Q\right)-MC\left(Q\right)$

With the given:

$$\begin{gathered} MB\left(Q\right)=36-8Q; \\ MC\left(Q\right)=12; \end{gathered}$$

By substitute all the given numbers, we get

$$\begin{gathered} MNB\left(Q\right)=MB\left(Q\right)-MC\left(Q\right) \\ MNB\left(Q\right)=36-8Q-12 \\ MNB\left(Q\right)=24-8Q \\ MNB\left(Q\right)=24-8\left(3\right) \\ MNB\left(Q\right)=24-24 \\ MNB\left(Q\right)=0 \end{gathered}$$

So, if $\mathit{Q}$level is $\mathbf{3}$, the net benefits in its maximum and on that level, managerial net benefit is $\mathbf{0}$.