Question

Suppose a profit-maximizing monopolist is producing 800 units of output and is charging a price of \(\$ 40\) per unit. a. If the elasticity of demand for the product is -2 find the marginal cost of the last unit produced. b. What is the firm's percentage markup of price over marginal cost? c. Suppose that the average cost of the last unit produced is \(\$ 15\) and the firm's fixed cost is \(\$ 2000\). Find the firm's profit.

Short answer

a. The marginal cost of the last unit produced is \$20. b. The firm's percentage markup of price over marginal cost is 100%. c. The firm's profit is \$17,000.

Step-by-step solution

Find the Marginal Cost

Using the formula for price elasticity of demand \(P(1 + 1/\epsilon) = MC\), where \(P = $40\) is the price per unit and \(\epsilon = -2\) is the price elasticity of demand. Note that the elasticity is negative because of the law of demand which states that as price increases, quantity demanded decreases. Substitute these values into the formula to find the Marginal Cost (MC).

Calculate the Percentage Markup

The percentage markup is calculated as \((P - MC) / MC * 100%\), where \(P\) is the price and \(MC\) is the marginal cost found in the previous step. This formula comes from the definition of percentage markup which is the difference between the price and the marginal cost, divided by the marginal cost, all multiplied by 100.

Compute the Firm's Profit

The monopoly firm's profit is given by \((P - AC)Q - FC\), where \(P = $40\) is the price per unit, \(AC = $15\) is the average cost per unit, \(Q = 800\) is the quantity of output, and \(FC = $2000\) is the firm's fixed cost. Substitute these values into the formula to find the profit.

Key concepts

Price Elasticity of Demand

Price Elasticity of Demand (PED) measures how much the quantity demanded of a good responds to changes in its price. When the demand is elastic, like in the case provided with an elasticity of -2, it means that for every 1% increase in price, the quantity demanded decreases by 2%. This elasticity is crucial for monopolists to understand because it influences their pricing strategies.

In a monopolistic market, understanding the elasticity of demand can help determine how sensitive consumers are to price changes. A more elastic demand suggests that consumers will significantly change their purchase quantity in response to price changes, which can be risky if prices are set too high.

For a monopolist, the price elasticity of demand is directly connected to their ability to set prices above marginal cost and still maintain sales. Due to the demand elasticity of -2, the monopolist in this problem optimally sets prices by considering this elasticity when calculating the marginal cost. Substituting into the elasticity formula, they aim to maximize profit while ensuring that the price remains attractive to buyers.

Marginal Cost

Marginal Cost (MC) represents the cost of producing one additional unit of output. For a monopoly, calculating marginal cost is essential for determining how much to produce and what price to charge.

In the context of the exercise, marginal cost links directly to the monopolist's pricing decision through the elasticity of demand. With the formula: \[P(1 + 1/\epsilon) = MC\]where \(P\) is the price per unit, and \(\epsilon\) is the price elasticity of demand, we find the MC by plugging in the values: \(P = 40\) and \(\epsilon = -2\). This provides a calculated marginal cost reflecting the last unit's production expense.

Accurate calculation of MC allows monopolies to set prices strategically, maximizing their profits without unnecessarily inflating production costs. This understanding of their cost structure forms the bedrock of profit maximization strategies, enabling these firms to adjust output and pricing to market conditions.

Profit Maximization

Profit Maximization is the primary goal of any monopolist, aiming to achieve the greatest price possible without losing customers. In the exercise example, profit maximization involves calculating the difference between total revenue and total costs (both variable and fixed costs).

The formula used is:\[(P - AC)Q - FC\]where \(P = \\(40\) is the price per unit, \(AC = \\)15\) is the average cost per unit, \(Q = 800\) is the quantity produced, and \(FC = \$2000\) is fixed cost.

• Total Revenue: It's the product of price and quantity, providing the gross income from sales.
• Total Cost: It comprises fixed costs (such as rent) and variable costs (affected by production level).
• Profit: The difference between total revenue and total cost, providing real earnings.

By analyzing these elements, a monopolist ensures they operate efficiently and maintain a competitive advantage, even in a market with no close substitutes. Balancing production and costs while setting optimal prices forms the cornerstone of monopolistic profit strategies.