2. Suppose that firms in an industry have the following cost function: C = 100 + 0.25q2, and the industry faces an inverse demand curve of P = 90 – 2Q.
a. I f the industry is competitive, find the long-run equilibrium price, quantity, and profit of a typical firm. How many firms are in the industry?
b. Now suppose one firm (with the same cost function) has monopolized the industry. What is the monopoly price, quantity and level of profit?
c. What is the change in consumer surplus of the monopoly equilibrium relative to the competitive equilibrium?
d. What is the deadweight loss?
e. Graphically illustrate the equilibria under the 2 market structures.
p=230-q and TC=10=0.25q2
a. To find the long-run equilibrium price and quantity of a typical firm in a competitive industry, we need to set the industry's demand and supply equal to each other and solve for the equilibrium quantity and price.
The inverse demand curve for the industry is given by P = 90 - 2Q, where P represents the price of the product and Q represents the total quantity demanded in the industry.
The supply curve of a typical firm in a competitive industry is derived from its cost function. The cost function in this case is given by C = 100 + 0.25q^2, where C represents total cost and q represents the quantity produced by the firm.
In the long-run equilibrium, the industry's total quantity supplied equals its total quantity demanded, so we set the demand and supply curves equal to each other:
90 - 2Q = 100 + 0.25Q^2
Now, we can solve this quadratic equation to find the equilibrium quantity:
0.25Q^2 + 2Q - 10 = 0
Using a quadratic equation solver or completing the square method, we find that the equilibrium quantity is Q = 8 units.
To find the equilibrium price, substitute the equilibrium quantity into the inverse demand curve:
P = 90 - 2Q
P = 90 - 2(8)
P = 90 - 16
P = 74
Therefore, in the long-run competitive equilibrium, the price is $74 per unit and the quantity is 8 units.
To determine the profit of a typical firm in the industry, we can subtract the cost function from the equilibrium price multiplied by the equilibrium quantity:
Profit = P * Q - C
Profit = (74 * 8) - (100 + 0.25(8^2))
Profit = 592 - 100 - 16
Profit = $476
The number of firms in the industry can be determined by dividing the total industry quantity (Q) by the quantity produced by a typical firm. In this case, Q = 8 units and a typical firm produces 8 units, so there is 1 firm in the industry.
b. If one firm monopolizes the industry, it has the ability to set its price and quantity to maximize profit. To find the monopoly price and quantity, we need to calculate the profit-maximizing level where marginal revenue (MR) equals marginal cost (MC). In a monopoly, MR is less than the price:
MR = 90 - 4Q
MC = dC/dq = 0.5q
Set MR equal to MC and solve for the quantity:
90 - 4Q = 0.5Q
90 = 4.5Q
Q = 20 units
Substitute the quantity back into the inverse demand curve to find the monopoly price:
P = 90 - 2Q
P = 90 - 2(20)
P = 90 - 40
P = $50
Therefore, the monopoly price is $50 per unit and the quantity is 20 units.
To calculate the level of profit, subtract the cost function from the price multiplied by the quantity:
Profit = P * Q - C
Profit = (50 * 20) - (100 + 0.25(20^2))
Profit = 1000 - 100 - 100
Profit = $800
c. Consumer surplus is a measure of the difference between what consumers are willing to pay for a good and what they actually pay. To calculate the change in consumer surplus between the monopoly and competitive equilibrium, we need to find the consumer surplus in both situations.
In the competitive equilibrium, consumer surplus can be calculated by finding the area under the demand curve and above the competitive price. The area represents the value consumers receive above what they paid:
Consumer Surplus (competitive) = (1/2)(P - Pc)(Q) = (1/2)(90 - 74)(8) = $64
In the monopoly equilibrium, consumer surplus is given by the area under the demand curve and above the monopoly price:
Consumer Surplus (monopoly) = (1/2)(P - Pm)(Q) = (1/2)(90 - 50)(20) = $400
The change in consumer surplus is the difference between the monopoly and competitive consumer surplus:
Change in Consumer Surplus = Consumer Surplus (monopoly) - Consumer Surplus (competitive) = $400 - $64 = $336
Therefore, the change in consumer surplus relative to the competitive equilibrium is an increase of $336.
d. Deadweight loss represents the loss in economic efficiency caused by a market imperfection, such as a monopolistic market structure. It is the reduction in total surplus (consumer surplus + producer surplus) resulting from a deviation from the perfectly competitive equilibrium.
To calculate the deadweight loss, we need to find the difference between the total surplus in the competitive equilibrium and the monopoly equilibrium:
Total Surplus (competitive) = Consumer Surplus (competitive) + Producer Surplus (competitive) = $64 + $476 = $540
Total Surplus (monopoly) = Consumer Surplus (monopoly) + Producer Surplus (monopoly) = $400 + $800 = $1200
Deadweight Loss = Total Surplus (competitive) - Total Surplus (monopoly) = $540 - $1200 = -$660
The deadweight loss in this case is negative, indicating a net gain in economic efficiency resulting from the monopoly.
e. To graphically illustrate the equilibria under the two market structures, we can plot the demand curve, the competitive supply curve, and the monopoly supply curve on a graph.
On the vertical axis, we have price (P), and on the horizontal axis, we have quantity (Q). The demand curve is downward-sloping, reflecting the inverse relationship between price and quantity demanded. The competitive and monopoly supply curves are upward-sloping, representing the relationship between price and quantity supplied.
For the competitive equilibrium, we plot the demand curve (P = 90 - 2Q) and the competitive supply curve (C = 100 + 0.25q^2) on the same graph. The intersection of these two curves represents the competitive equilibrium price and quantity.
For the monopoly equilibrium, we plot the demand curve (P = 90 - 2Q) and the monopoly supply curve (MC = 0.5q) on the same graph. The intersection of these two curves represents the monopoly equilibrium price and quantity.
The competitive equilibrium will have a lower price and a higher quantity compared to the monopoly equilibrium. The graph will show the deadweight loss as the shaded area between the competitive supply curve and the monopoly supply curve, representing the loss in economic efficiency caused by the monopoly.
Please note that it's difficult to provide a visual representation here without an actual graph, but I hope this explanation helps you understand how to graphically illustrate the equilibria under the two market structures.