Chapter 9, Problem 17P

(a)

To determine

The competitive price and quantity.

Explanation of Solution

The competitive price in the given situation is equal to the marginal cost. Thus, the competitive price is $0.10 per package. The inverse demand function can be calculated using Equation (1) as follows:

$\text{P}=1-0.01\text{Q}$ (1)

The quantity can be calculated by substituting the value of price in Equation (1) as follows:

$\begin{array}{c}\text{0}\text{.1}=1-0.01\text{Q}\\ 0.01\text{Q}=1-\text{0}\text{.1}\\ \text{Q}=\frac{0.90}{0.01}\\ =90\end{array}$

Thus, the competitive quantity is 90.

(b)

To determine

The consumer surplus and producer surplus.

Explanation of Solution

Figure 1 shows the price and quantity of packages.

The horizontal axis of Figure 1 measures the quantity of packages and the vertical axis measures the price.

The consumer surplus is the area of the triangle ABD. The consumer surplus is given by Equation (2) as follows:

$\text{Consumer surplus}=\frac{1}{2}\times \left(\text{Maximum willing price}-\text{Equilibrium price}\right)\times \text{Quantity}$ (2)

$\begin{array}{c}{\text{Consumer surplus}}_{\text{Initial}}=\frac{1}{2}\times \left(1-0.1\right)\times \text{90}\\ =\frac{1}{2}\times 0.9\times 90\\ =40.5\end{array}$

Thus, the consumer surplus is $40.50.

In the given situation, the producer surplus is zero.

Economics Concept Introduction

Consumer Surplus: Consumer surplus is defined as the difference between the maximum amount a person is willing to pay for consuming a commodity and the actual price the person pays for it.

Producer Surplus: Producer surplus is defined as the difference between the actual market price for which a commodity is sold and the minimum cost at which a producer is willing to sell a commodity. This minimum accepted price is usually the cost of production of the commodity.

(c)

To determine

The monopoly price and quantity.

Explanation of Solution

The marginal revenue can be calculated as follows in Equation.

$\begin{array}{c}\text{MR}=\frac{\partial \left(\text{P}\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(1-0.01\text{Q}\right)\text{Q}}{\partial \text{Q}}\\ =\frac{\partial \left(1\text{Q}-0.01{\text{Q}}^2\right)}{\partial \text{Q}}\\ =1-0.02\text{Q}\end{array}$

The marginal revenue function is $\text{MR}=1-0.02\text{Q}$.

Given that the marginal cost is $0.10 and the marginal revenue is $\text{MR}=1-0.02\text{Q}$.

The profit of a firm is maximized when marginal revenue is equal to the marginal cost.

$\begin{array}{c}\text{MR}=\text{MC}\\ 1-0.02\text{Q}=0.10\\ 1-0.10=0.02\text{Q}\\ \text{Q}=\frac{0.9}{0.02}\\ =45\end{array}$

The profit is maximized when the quantity of output produced is 45.

The price charged by the monopolist to maximize the profit can be calculated by substituting the value of profit maximizing output in Equation (1).

$\begin{array}{c}\text{P}=1-0.01\left(45\right)\\ =1-0.45\\ =0.55\end{array}$

Thus, the profit maximizing output is 45 and the profit maximizing price is $0.55.

Economics Concept Introduction

Profit: Profit is defined as the excess revenue earned over the total cost of production.

(d)

To determine

The consumer surplus and producer surplus.

Explanation of Solution

The new consumer surplus can be calculated as follows:

$\text{Consumer surplus}=\frac{1}{2}\times \left(\text{Maximum willing price}-\text{Equilibrium price}\right)\times \text{Quantity}$

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times \left(1-0.55\right)\times \text{45}\\ =\frac{1}{2}\times 0.45\times 45\\ =10.125\end{array}$

Thus, the consumer surplus is $10.125.

The producer surplus can be calculated as follows.

$\text{Producer surplus}=\left(\text{Equilibrium price}-\text{Minimum willing price}\right)\times \text{Quantity}$$\begin{array}{c}\text{Producer surplus}=\left(0.55-0.1\right)\times \text{45}\\ =0.45\times 45\\ =20.25\end{array}$

Thus, the producer surplus is $20.25.

Economics Concept Introduction

Consumer Surplus: Consumer surplus is defined as the difference between the maximum amount a person is willing to pay for consuming a commodity and the actual price the person pays for it.

Producer Surplus: Producer surplus is defined as the difference between the actual market price for which a commodity is sold and the minimum cost at which the producer is willing to sell the commodity. This minimum accepted price is usually the cost of production of the commodity.

(e)

To determine

The deadweight loss.

Explanation of Solution

The change in consumer surplus can be calculated as follows.

$\begin{array}{c}\text{Change in consumer surplus}={\text{Consumer surplus}}_{\text{New}}-{\text{Consumer surplus}}_{\text{Initial}}\\ =10.125-40.50\\ =-30.375\end{array}$

Consumer surplus is decreased by $30.375.

The change in producer surplus can be calculated as follows:

$\begin{array}{c}\text{Change in producer surplus}={\text{Producer surplus}}_{\text{New}}-{\text{Producer surplus}}_{\text{Initial}}\\ =20.25-0\\ =20.25\end{array}$

Producer surplus is increased by $20.25.

The dead weight loss can be calculated as the difference between the loss in consumer surplus and the gain in producer surplus.

$\begin{array}{c}\text{The dead weight loss}=\text{Change in producer surplus}-\text{Change in consumer surplus}\\ =20.25-\left(30.375\right)\\ =-10.125\end{array}$

Thus, the dead weight loss is $10.125.

Economics Concept Introduction

Consumer Surplus: Consumer surplus is defined as the difference between the maximum amount a person is willing to pay for consuming a commodity and the actual price the person pays for it.

Producer Surplus: Producer surplus is defined as the difference between the actual market price for which a commodity is sold and the minimum cost at which the producer is willing to sell the commodity. This minimum accepted price is usually the cost of production of the commodity.

Deadweight loss: Deadweight loss is defined as the loss of the total consumer surplus and producer surplus due to overproduction or underproduction.