Chapter 9, Problem 20P

(a)

To determine

The profit maximizing price, quantity, consumer surplus and producer surplus.

Explanation of Solution

From the given data, the inverse demand function can be calculated using Equation (1) as follows:

$\text{P}=15-0.33\text{Q}$ (1)

The marginal revenue can be calculated as follows:

$\begin{array}{c}\text{MR}=\frac{\partial \left(\text{P}\times \text{Q}\right)}{\partial \text{Q}}\\ =\frac{\partial \left(15-0.33\text{Q}\right)\text{Q}}{\partial \text{Q}}\\ =\frac{\partial \left(15\text{Q}-0.33{\text{Q}}^2\right)}{\partial \text{Q}}\\ =15-0.66\text{Q}\end{array}$

The marginal revenue function is $\text{MR}=15-0.66\text{Q}$.

Given the marginal cost is $1.

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

$\begin{array}{c}\text{MR}=\text{MC}\\ 15-0.66\text{Q}=1\\ 15-1=0.66\text{Q}\\ \text{Q}=\frac{14}{0.66}\\ =21.21\end{array}$

The profit is maximized when the quantity is 21.21.

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

$\begin{array}{c}\text{P}=15-0.33\left(21.21\right)\\ =15-6.993\\ \approx 8\end{array}$

Thus, the profit maximizing output is 21.21 and the profit maximizing price is $8.

Figure 1 given below shows the demand and market revenue.

The horizontal axis of Figure 1 measures the quantity of bottles and the vertical axis measures the price per bottle. The demand curve and the marginal revenue curve are downward sloping curves. The marginal cost is constant and is fixed at $1.

The consumer surplus is the sum area of A and B. The consumer surplus can be calculated as follows:

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

$\begin{array}{c}\text{Consumer surplus}=\frac{1}{2}\times \left(15-8\right)\times \text{21}\text{.21}\\ =\frac{1}{2}\times 7\times 21.21\\ =74.24\end{array}$

Thus, the consumer surplus is $74.24.

The producer surplus is the area of C+D calculated as follows:

$\text{Producer surplus}=\left(\text{Equilibrium price}-\text{Minimum willing price}\right)\times \text{Quantity}$

$\begin{array}{c}\text{Producer surplus}=\left(8-1\right)\times \text{21}\text{.21}\\ =7\times 21.21\\ =148.47\end{array}$

Thus, the producer surplus is $148.87.

The competitive equilibrium in the market is attained when the marginal cost is equal to the price which is equal to $1.

$\text{P}=1$

When the price is $1, the quantity demanded can be calculated by substituting the respective values in Equation (1).

$\begin{array}{c}\text{1}=15-0.33\left(\text{Q}\right)\\ 0.33\text{Q}=15-1\\ \text{Q}=\frac{14}{0.33}\\ \approx 42.42\end{array}$

The deadweight loss is the area given by DWL. It can be calculated as follows:

$\begin{array}{c}\text{DWL}=\frac{1}{2}\times \left(\begin{array}{l}\text{Profit maximizing price}\\ -\text{Competitive price}\end{array}\right)\times \left(\begin{array}{l}\text{Profit maximizing quantity}\\ -\text{Competitive quantity}\end{array}\right)\\ =\frac{1}{2}\left(8-1\right)\times \left(42.42-21.21\right)\\ =\frac{1}{2}\times 7\times 21.21\\ =74.24\end{array}$

Thus, the deadweight loss is $74.24.

Economics Concept Introduction

Consumer Surplus: The 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 he pays for it.

Producer Surplus: The 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.

(b)

To determine

The price that the townsfolk would be willing to pay.

Explanation of Solution

The townsfolk would be willing to pay an amount equal to the consumer surplus when the price is equal to the marginal cost. The consumer surplus is the area given by the areas C+D+E.

The consumer surplus can also be calculated as the sum of the producer surplus and the deadweight loss.

$\begin{array}{c}\text{Consumer surplus}=\text{Producer surplus}+\text{Deadweight loss}\\ =148.47+74.24\\ =222.71\end{array}$

The consumer surplus $222.71 is the price which the townsfolk would be willing to pay.

Economics Concept Introduction

Consumer Surplus: The 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 he pays for it.

(c)

To determine

The minimum payment the brewery would be willing to pay.

Explanation of Solution

The minimum payment that the brewery would be willing to accept is the amount that can compensate the producer surplus at the competitive price of $1. Thus, the minimum payment made by the brewery is $148.47.

Economics Concept Introduction

Producer Surplus: The 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.

(d)

To determine

The potential bargain.

Explanation of Solution

Clearly, there exists a bargain. The agreed payment would probably fall in the range $148.47 and $222.71. There would be no deadweight loss if a bargain is struck. The actual payment would depend on the strength of the bargaining of the two sides.

Economics Concept Introduction

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