Question

Suppose the total benefit derived from a continuous decision $\mathit{Q}$,is role="math" localid="1657557240837" $\mathit{B}\mathbf{\left(}\mathit{Q}\mathbf{\right)}\mathbf{=}\mathbf{20}\mathit{Q}\mathbf{-}\mathbf{2}{\mathit{Q}}^{\mathbf{2}}$and the corresponding total cost is $\mathit{C}\mathbf{\left(}\mathit{Q}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{+}\mathbf{2}{\mathit{Q}}^{\mathbf{2}}$, so that $\mathit{M}\mathit{B}\left(Q\right)\mathbf{=}\mathbf{20}\mathbf{-}\mathbf{4}\mathit{Q}$ and $\mathit{M}\mathit{C}\left(Q\right)\mathbf{=}\mathbf{4}\mathit{Q}$.

1. What is the total benefit when9$\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{10}\mathbf{?}$

2. What is the marginal benefit when$\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{10}\mathbf{?}$

3. What level of Q maximizes total benefit?

4. What is the total cost when $\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{10}\mathbf{?}$

5. What is marginal cost when $\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{?}\mathit{Q}\mathbf{=}\mathbf{10}\mathbf{?}$

6. What level of Q maximizes net benefit?

Short answer

1. The total benefit if Q=2 is $32 and if Q=10 is $0.

2. The marginal benefit if Q=2 is $12 and if Q=10 is $(-20).

3. When Q is at 5, the total benefits are on its maximum.

4. The total cost when Q=2 is $12 and when Q=10 is $204.

5. The marginal cost when Q=2 is $8 and when Q=10 is $40.

6. The net benefits will be on maximum when the level of the Q is $2.5.

Step-by-step solution

(a) Total benefit when Q=2 and Q=10.

To find the total benefit, use the equation:$B\left(Q\right)=20Q-2Q^2$

If Q=2, substitute to the equation, then calculate:

$$\begin{gathered} B\left(Q\right)=20Q-2Q^2 \\ =20.2-2.2^2 \\ =40-8 \\ =32. \end{gathered}$$

The total benefit if Q=2 is $32.

If Q=10, substitute to the equation, then calculate:

$$\begin{gathered} B\left(Q\right)=20Q-2Q^2 \\ =20.10-2.{10}^2 \\ =200-200 \\ =0 \end{gathered}$$

The total benefit if Q=10 is $0.

(b) Marginal benefit when Q=2 and Q=10.

To find the marginal benefit, use the equation:$MB\left(Q\right)=20-4Q$

If Q=2, then calculate:

$$\begin{gathered} MB\left(Q\right)=20-4Q \\ =20-4.2 \\ =20-8 \\ =12 \end{gathered}$$

If Q=2 the marginal benefit is $12.

If Q=10, then calculate:

$$\begin{gathered} MB\left(Q\right)=20-4Q \\ =20-4.10 \\ =20-40 \\ =-20 \end{gathered}$$

If Q=10 the marginal benefit is $(-20).

(c) Q maximizes total benefit at,

To find the maximum of the total benefits of Q level, use the equation for total benefits,

$B\left(Q\right)=20-2Q^2$

Then, take the derivative of B(Q) and set to zero.

$$\begin{gathered} \frac{dB}{dQ}=20-4Q=0 \\ 4Q=20 \\ Q=\frac{20}{4} \\ Q=5 \end{gathered}$$

Therefore, when Q is 5, the total benefits are on its maximum.

(d) The total cost when Q=2 and Q=10.

To find the total cost, use the equation:$C\left(Q\right)=4+2Q^2$

If Q=2, then:

$$\begin{gathered} C\left(Q\right)=4+2Q^2 \\ =4+2.2^2 \\ =4+2.4 \\ =4+8 \\ =12. \end{gathered}$$

If Q=2, the total cost is $12.

If Q=10, then:

$$\begin{gathered} C\left(Q\right)=4+2Q^2 \\ =4+2.{10}^2 \\ =4+2.100 \\ =4+200 \\ =204. \end{gathered}$$

If Q=10, the total cost is $204.

(e) The marginal cost when Q=2 and Q=10

To find the marginal cost, use the equation:$MC\left(Q\right)=4Q$

If Q=2, then :

$$\begin{gathered} MC\left(Q\right)=4Q \\ =4.2 \\ =8 \end{gathered}$$

If Q=2, the marginal cost is $8.

If Q=10, then:

$$\begin{gathered} MC\left(Q\right)=4Q \\ =4.10 \\ =40 \end{gathered}$$

If Q=10, the marginal cost is 40.

(f) Q maximizes net benefit at,

To find the level of Q that maximizes the net benefits, use the equation:

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

With the given: $MB\left(Q\right)=20-4Q$and $MC\left(Q\right)=4Q$, then substitute in the equation,

$$\begin{gathered} MB\left(Q\right)=MC\left(Q\right) \\ 20-4Q=4Q \\ 20=4Q+4Q \\ 20=8Q \\ Q=\frac{20}{8} \\ =2.5 \end{gathered}$$

The net benefits will be on its maximum when the level Q is $2.5.