Chapter 13, Problem 11PA

Suppose that each firm in a competitive industry has the following costs:

Total cost: TC = 50 + ½ q²

Marginal cost: MC = q

where q is an individual firm’s quantity produced.

The market demand curve for this product is

Demand:    Q^D = 120 − P

where P is the price and Q is the total quantity of the good. Currently, there are 9 firms in the market.

a. What is each firm’s fixed cost? What is its variable cost? Give the equation for average total cost.

b. Graph average-total-cost curve and the marginal-cost curve for q from 5 to 15. At what quantity is average-total-cost curve at its minimum? What is marginal cost and average total cost at that quantity?

c Give the equation for each firm’s supply curve.

d. Give the equation for the market supply curve for the short run in which the number of firms is fixed.

e. What is the equilibrium price and quantity for this market in the short run?

f. In this equilibrium, how much does each firm produce? Calculate each firm’s profit or loss. Is there incentive for firms to enter or exit?

g. In the long run with free entry and exit, what is the equilibrium price and quantity in this market?

h. In this long-run equilibrium, how much does each firm produce? How many firms are in the market?

To determine

Calculate average total cost.

Explanation of Solution

The total cost equation, marginal cost equation and demand equation are given below:

$\text{Total cost}=50+\frac{1}{2}{\text{q}}^{\text{2}}$ (1)

$\text{Marginal cost}=\text{q}$ (2)

$\text{Quantity demand}=120-\text{P}$ (3)

The fixed cost and variable cost of each firm are determined from Equation (1). Here, the fixed cost is a part of the total cost and will not change in response to a change in quantity. The variable cost is a part of the total cost and it changes in response to a change in quantity. So, the fixed cost is $50 and the variable cost is $\frac{1}{2}{\text{q}}^{\text{2}}$.

Average total cost equation is represented below:

$\text{Average total cost}=\frac{\text{Total cost}}{\text{Quantity}}$ (4)

Or

$\text{Average total cost }=\frac{\text{50}}{\text{q}}+\frac{1}{2}\text{q}$.

Economics Concept Introduction

Concept introduction:

Average total cost: The average total cost is the total cost per unit of the output produced by a firm.

To determine

Draw average total cost curve.

Explanation of Solution

Figure 1 represents the average total cost curve and marginal cost curve.

From the above figure, the x-axis shows the quantity of the output and the y-axis shows the price level.

From the graph, the average total cost is at its minimum, when they produce 10 units of output. The average total cost and marginal cost are at the quantity of 10 units of output.

Economics Concept Introduction

Concept introduction:

Average total cost: The average total cost is the total cost per unit of the output produced by a firm.

To determine

Supply curve.

Explanation of Solution

In the perfect competition, the supply curve is even, as the marginal cost curve is beyond the intersecting point of the average total cost curve.

Supply curve for each firm is shown below:

$\text{Quantity supply}=\text{Price}<\$10$ (5)

$\text{Quantity supply}=\text{Price}\ge \$10$ (6)

Economics Concept Introduction

Concept introduction:

Marginal cost curve: The marginal cost is the additional cost incurred on the additional production and its curve is U-shaped, which represents the combination of price level and quantity of output.

To determine

Quantity supply.

Explanation of Solution

In the short-run, there are currently 9 firms. So, the market supply curve is determined by using the following formula:

$\text{Quantity supply}=\text{P}$ (7)

Hence, the short-run market supply curve is shown below:

$\text{Quantity supply}=9\text{P}$

Economics Concept Introduction

Concept introduction:

Supply: Supply refers to the total value of the goods and services that are available for the purchase at a particular price in a given period of time.

To determine

Equilibrium price and quantity.

Explanation of Solution

The equilibrium quantity and price are determined by using the following formula:

$\text{Quantity demand}=\text{Quantity supply}$ (8)

Substitute the respective values in Equation (8) to calculate the equilibrium quantity and price.

$\begin{array}{c}\text{120}-\text{P}=\text{9P}\\ 120=10\text{P}\\ \text{P}=\frac{120}{10}\\ =12\end{array}$

Thus, the equilibrium price is $12.

Substitute the price of $12 in Equation (3) to calculate the equilibrium quantity.

$\begin{array}{c}\text{Quantity demand }=\text{120}-\text{P}\\ =\text{120}-12\\ =108\end{array}$

Thus, the equilibrium quantity is 108 units.

Economics Concept Introduction

Concept introduction:

Equilibrium price: It is the price at which the quantity demanded of a good or service is equal to the quantity supplied.

Equilibrium: It is the market price and quantity determined by equating the supply to the demand. At this equilibrium point, the supply will be equal to the demand and there will be no excess demand or supply in the economy. Thus, the economy will be at equilibrium.

To determine

Calculate profit.

Explanation of Solution

In the short-run equilibrium, each firm produces 12 units $\left(\frac{108}{9}\right)$.

Profit can be calculated by using the following formula:

$\text{Profit}=\text{Total revenue}-\text{Total cost}$ (9)

Substitute the respective values in Equation (9) to calculate the profit.

$\begin{array}{c}\text{Profit}=\left(12\times 12\right)-\text{50}+\frac{1}{2}{\left(12\right)}^2\\ =144-122\\ =22\end{array}$

Thus, the profit is $22.

Since the firms have positive value in making profit, there will be a benefit for the firm entering the market.

Economics Concept Introduction

Concept introduction:

Profit: Profit refers to the excess revenue after subtracting the total cost from the total revenue.

To determine

Equilibrium quantity.

Explanation of Solution

In the long-run, the firm earns zero economic profit, so the price is equal to the minimum average total cost. Since the average total cost is $10, the equilibrium price is also be $10.

Equilibrium quantity can be determined by using Equation (3), when the equilibrium price is $10.

$\begin{array}{c}\text{Quantity demand }=\text{120}-\text{P}\\ =\text{120}-10\\ =110\end{array}$

Thus, the long-run equilibrium quantity is 110.

Economics Concept Introduction

Concept introduction:

Equilibrium: It is the market price and quantity determined by equating the supply to the demand. At this equilibrium point, the supply will be equal to the demand and there will be no excess demand or supply in the economy. Thus, the economy will be at equilibrium.

To determine

Number of firms in the long run.

Explanation of Solution

In the long-run, the firm produces 10 units of output. This is because in the long-run, the production price of the firm is equal to the minimum average total cost. The average total cost is 10 units. Also, there are 11 firms $\left(\frac{110}{10}=11\right)$  entering the market in the long-run.

Economics Concept Introduction

Long run: Thelong run refers to the time, which changes the production variable to adjust to the market situation.