Chapter 1, Problem 9CACQ

To determine

(a)

To find:

The total benefit when Qis 2 and when Qis 10.

Answer to Problem 9CACQ

When Qis 2, total benefit is 32 and when Qis 10, the total benefit is 0.

Explanation of Solution

Use the given equation,

Total Benefit is given by, $\text{B}\left(Q\right)=\text{20}Q-\text{2}{Q}^{\text{2}}$

When Quantity, Qis 2, total benefit will be,

$\begin{array}{c}\text{B}\left(Q\right)=\text{20}Q-\text{2}{Q}^{\text{2}}\\ =20\times 2-2\left(2\times 2\right)\\ =40-8\\ =32\end{array}$

Thus, when Quantity is 2, total benefit is 32.

When quantity ( Q ) is 10, total benefit is as follows:

$\begin{array}{c}\text{B}\left(Q\right)=20Q-2Q^2\\ =20\times 10-2\left(10\times 10\right)\\ =200-200\\ =0\end{array}$

Thus, when Qis 10, the total benefit is 0.

Economics Concept Introduction

Total benefit:

Total benefit refers to the total satisfaction a consumer can get after consumption. It can also be the maximum amount that a consumer will pay to get a commodity or service.

To determine

(b)

To compute:

The marginal benefit when Qis 2 and Qis 10.

Answer to Problem 9CACQ

When Qis 2, marginal benefit is 12 and when Qis 10, the marginal benefit is -20.

Explanation of Solution

Use the given equation, marginal benefit is as follows:

$\text{MB}\left(Q\right)=20-4Q$

When Qis 2, marginal benefit is as follows:

$\begin{array}{c}\text{MB}\left(Q\right)=20-4Q\\ =20-\left(4\times 2\right)\\ =20-8\\ =12\end{array}$

Thus, if Qis 2, marginal benefit is 12 and when Qis 10, marginal benefit is as follows:

$\begin{array}{c}\text{MB}\left(Q\right)=20-4Q\\ =20-\left(4\times 10\right)\\ =20-40\\ =-20\end{array}$

Thus, if Qis 10, the marginal benefit is -20.

Economics Concept Introduction

Marginal benefit:

Marginal benefit is the addition to total benefit when an additional unit of good is consumed.

To determine

(c)

To find:

The level of Q that maximizes total benefit

Answer to Problem 9CACQ

When Qis 5, total benefits are maximum.

Explanation of Solution

Total benefit is given by the equation,

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

Taking the derivative of B( Q ) and setting it equal to zero gives,

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

Thus, when Qis 5, total benefits are maximum. To verify that, second derivative should be negative.

$\frac{d^{2}B}{d{Q}^2}=-4\le 0$

Therefore, at Qis 5, total benefits are maximum.

Economics Concept Introduction

Total benefit:

Total benefit refers to the total satisfaction a consumer can get after consumption. It can also be the maximum amount that a consumer will pay to get a commodity or service.

To determine

(d)

To compute:

The total cost when Qis2 and when Q is 10.

Answer to Problem 9CACQ

When Qis 2, the total cost is $12 and when Q = 10, total cost is $204.

Explanation of Solution

Total cost is the $\text{C}\left(Q\right)=4+2Q^{\text{2}}$

When Q is 2,

$\begin{array}{c}\text{C}\left(Q\right)=4+2Q^2\\ =4+2\left(2^2\right)\\ =4+8\\ =12\end{array}$

Thus, when Qis 2, the total cost is 12.

When Q = 10, total cost will be,

$\begin{array}{c}\text{C}\left(Q\right)=4+2Q^2\\ =4+2\left({10}^2\right)\\ =4+200\\ =204\end{array}$

Thus, when Qis 10, total cost is 204.

Economics Concept Introduction

Total cost:

Total cost is the total amount incurred in production of goods or services.

To determine

(e)

To find:

The marginal cost when Qis 2 and when Q is 10.

Answer to Problem 9CACQ

When Qis 2, the marginal cost is 8 and when Qis 10, the marginal cost is 40.

Explanation of Solution

Marginal Cost

$\text{MC}=\text{}4Q$

When Qis 2, marginal cost is as follows:

$\begin{array}{c}\text{MC}=\text{}4Q\\ =4\times 2\\ =8\end{array}$

When Qis 10, marginal cost is as follows:

$\begin{array}{c}\text{MC}=\text{}4Q\\ =4\times 10\\ =40\end{array}$

Thus, if Qis 10, the marginal cost is 40.

Economics Concept Introduction

Marginal cost:

Marginal cost is the addition to total cost when an additional unit is produced.

To determine

(f)

To find:

The level of Q which minimizes cost.

Answer to Problem 9CACQ

When Q = 0, the cost is minimum.

Explanation of Solution

Total cost is,

$C\left(Q\right)=4+2Q^2$

Taking the derivative and setting it to zero,

$\begin{array}{c}C\left(Q\right)=4+2Q^2\\ \frac{dC}{dQ}=4Q=0\\ Q=0\end{array}$

To verify that it is minimum, check that the second derivative is positive,

$\begin{array}{l}C\left(Q\right)=4+2Q^2\\ \frac{dC}{dQ}=4Q=0\\ \frac{d^{2}C}{d{Q}^2}=4\ge 0\end{array}$

Therefore, at Qis 0 total cost is minimum.

Economics Concept Introduction

Total cost:

Total cost is the total amount incurred in production of goods or services.

To determine

(g)

To find:

The level of Q which maximizes net benefits.

Answer to Problem 9CACQ

Net benefits will be maximum when Qis 2.5.

Explanation of Solution

Since,

$\text{MB}\left(Q\right)=20-4Q$

And,

$MC\left(Q\right)=4Q$

For the maximum net benefits, it follows that,

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

Thus, the net benefits will be maximum when Qis 2.5

Economics Concept Introduction

Net benefits:

The net benefits are the maximum at that level of control variable ( Q ) where the marginal benefits are equal to the marginal costs. The following condition equation is used for maximum net benefits:

$\text{MB}\left(Q\right)=\text{MC}\left(Q\right)$

Where,

MB( Q ) ismarginal benefits, and

MC( Q ) is marginal costs.