Chapter 8, Problem 22P

(a)

To determine

The supply curve.

Explanation of Solution

The long-run total cost function using extra ordinary operation managers is given by Equation (1) as follows:

${\text{LTC}}_{\text{E}}=200+{\text{Q}}^2$ (1)

It is evident that the total variable cost is Q², in the given function.

The long-run average total cost can be calculated using Equation (2) as follows:

$\text{Long-run average variable cost}=\frac{\text{Long run variable cost}}{\text{Quantity of output }}$ (2)

$\begin{array}{c}\text{Long-run average variable cost}=\frac{{\text{Q}}^2}{\text{Q}}\\ =\text{Q}\end{array}$

The long-run average variable cost function is Q, which gets minimized when Q is zero.

The marginal cost function is given by Equation (3) as follows:

${\text{LMC}}_{\text{E}}=2\text{Q}$ (3)

The supply curve of the firm is the same as the marginal cost function. Thus, the supply curve of the firm is ${\text{LMC}}_{\text{E}}=2\text{Q}$.

Economics Concept Introduction

Marginal cost: Marginal cost is defined as an additional cost that is incurred due to the production of an extra unit of output.

Average variable cost: Average variable cost is defined as the total variable cost divided by the quantity of output.

Supply: Supply is defined as the quantity of a commodity offered for sale at different market prices.

(b)

To determine

The supply curve of the firm.

Explanation of Solution

The long-run total cost function using average operation managers is given by Equation (1) as follows.

${\text{LTC}}_{\text{A}}=200+2{\text{Q}}^2$ (4)

It is evident that the total variable cost is 2Q², in the given function.

The long-run average total cost can be calculated using Equation (2) as follows:

$\begin{array}{c}\text{Long-run average variable cost}=\frac{{\text{2Q}}^2}{\text{Q}}\\ =2\text{Q}\end{array}$

The long-run average variable cost function is 2Q, which gets minimized when Q is zero.

The marginal cost function is given by Equation (5) as follows:

${\text{LMC}}_{\text{A}}=4\text{Q}$ (5)

The supply curve of the firm is the same as the marginal cost function. Thus, the supply curve of the firm is ${\text{LMC}}_{\text{A}}=4\text{Q}$.

Economics Concept Introduction

Marginal cost: Marginal cost is defined as an additional cost that is incurred due to the production of an extra unit of output.

Average variable cost: Average variable cost is defined as the total variable cost divided by the quantity of output.

Supply: Supply is defined as the quantity of a commodity offered for sale at different market prices.

(c)

To determine

The long-run equilibrium price.

Explanation of Solution

To calculate the long-run equilibrium, we need additional information about the demand function of the firm. The decision of the firm to produce will depend on the strength of the demand functions to enhance production of firms.

(d)

To determine

The demand in the long run.

Explanation of Solution

It is given that there are 50 firms with extra ordinary managers who have an individual supply curve $\text{P}=2\text{Q}$. The long-run average total cost of exceptional managers attains the minimum value when the firm produces 14 units of output.

The total units of output produced can be calculated as follows:

$\begin{array}{c}\text{Total output}=\text{Output of individual firm}\times \text{Number of firms}\\ =14\times 50\\ =700\end{array}$

Thus, the total output produced is 700 units.

The market demand for ice creams is given as follows:

$\begin{array}{c}{\text{Quantity}}_{\text{Demanded}}=8,000-100\text{P}\\ \text{700}=8,000-100\text{P}\end{array}$

The value of price can be calculated as follows:

$\begin{array}{c}100\text{P}=8,000-\text{700}\\ \text{P}=\frac{7,300}{100}\\ =73\end{array}$

The equilibrium price is $73.

The price exceeds the minimum long-run average total cost of average managers that is $40. This implies that the total demand cannot be filled solely by extraordinary managers. Thus, it is profitable for the average managers to supply the ice cream. The entry of the new firms will reduce the price of ice creams in the market.

Economics Concept Introduction

Total cost: Total cost is defined as the sum of variable cost and fixed cost.

Average total cost: Average total cost is defined as the total cost divided by the quantity of output.

(e)

To determine

The long run equilibrium price.

Explanation of Solution

The equilibrium price of ice creams in the long run is the minimum cost of the long-run average total cost of average operation managers, which is $40.

Economics Concept Introduction

Total cost: Total cost is defined as the sum of variable cost and fixed cost.

Average total cost: Average total cost is defined as the total cost divided by the quantity of output.

(f)

To determine

The number of firms in the industry.

Explanation of Solution

When the price is $40, the quantity demanded can be calculated as follows:

$\begin{array}{c}{\text{Quantity}}_{\text{Demanded}}=8,000-100\text{P}\\ =8,000-100\left(40\right)\\ =8,000-4,000\\ =4,000\end{array}$

Thus, the quantity demanded is 4,000 gallons of ice cream.

The quantity supplied by the extraordinary managers can be calculated by equating the marginal cost and the marginal cost.

$\begin{array}{c}\text{Marginal revenue}=\text{Marginal cost}\\ 40=2\text{Q}\\ \text{Q}=\frac{40}{2}\\ =20\end{array}$

Thus, the equilibrium quantity is 20.

There are 50 firms in the market. Thus, the total quantity supplied can be calculated as follows:

$\begin{array}{c}\text{Total output}=\text{Output of individual firm}\times \text{Number of firms}\\ =20\times 50\\ =1,000\end{array}$

The total quantity supplied by extra ordinary managers is one million gallon tubes of ice cream. The average managers will produce 10,000 gallon tubs of ice cream.

The number of firms can be calculated as follows:

$\begin{array}{c}\text{The number of firms}=\frac{\text{Total quantity supplied}}{\text{Quantity supplied by individual unit}}\\ =\frac{3,000,000}{10,000}\\ =300\end{array}$

Thus, the number of firms in the industry is 300.

Economics Concept Introduction

Average total cost: Average total cost is defined as the total cost divided by the quantity of output.

(g)

To determine

The price determined by the average managers.

Explanation of Solution

A firm operated by average managers will earn only zero profit. This is due to the reason that the firms operate at the minimum point of the long run average total cost curve at $40.

The profit can be calculated as the difference between the total revenue and the total cost as follows:

$\begin{array}{c}\text{Profit}=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(200+2{\left(\text{Quantity}\right)}^2\right)\\ =\left(40\times 10\right)-\left(200+2{\left(10\right)}^2\right)\\ =400-400\\ =0\end{array}$

Thus, the long-run profit is zero.

Economics Concept Introduction

Total revenue: Total revenue is defined as the total income earned from the sale of output produced.

Total cost: Total cost is defined as the sum of variable cost and fixed cost.

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

(h)

To determine

The price determined by the extra ordinary managers.

Explanation of Solution

The profit can be calculated as the difference between the total revenue and the total cost as follows:

$\begin{array}{c}\text{Profit}=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{Quantity}\right)-\left(200+{\left(\text{Quantity}\right)}^2\right)\\ =\left(40\times 20\right)-\left(200+{\left(20\right)}^2\right)\\ =800-600\\ =200\end{array}$

Thus, the extra ordinary manager will earn a profit, which is $200,000.

Economics Concept Introduction

Total revenue: Total revenue is defined as the total income earned from the sale of output produced.

Total cost: Total cost is defined as the sum of variable cost and fixed cost.

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