Question

A firm faces the following average revenue (demand) curve: $$P=120-0.02 Q$$ where \(Q\) is weekly production and \(P\) is price, measured in cents per unit. The firm's cost function is given by \(C=60 Q+25,000 .\) Assume that the firm maximizes profits. a. What is the level of production, price, and total profit per week? b. If the government decides to levy a tax of 14 cents per unit on this product, what will be the new level of production, price, and profit?

Short answer

In absence of the tax, optimal production is 1500 units, price is 90 cents per unit, and profit is \$500 weekly. After introducing the tax, optimal production reduces to 1375 units, price increases to 97 cents, and profit decreases to \$332.25 weekly.

Step-by-step solution

Define the Firm’s Profit Equation

The profit \(\Pi\) for a firm is defined as total revenues \(R\) minus total costs \(C\). In this case, the revenue is given by the price \(P\) times quantity \(Q\) and cost is given as \(C = 60Q + 25000\). So, we have the profit function as: \(\Pi = P \cdot Q - C = (120 - 0.02Q) \cdot Q - (60Q + 25000)\)

Find the Quantity That Maximizes Profit

In order to maxmise the profit, we take the derivative of the profit function with respect to the quantity \(Q\) and set it equal to zero. This gives us: \(\frac{d \Pi}{d Q} = 120 - 0.04Q - 60 = 0\). Solving this equation gives us the optimal quantity \(Q^* = (120 - 60) / 0.04 = 1500 \) units.

Calculate the Price and Profit at the Optimal Quantity

Plug the optimal quantity \(Q^*\) into the price and cost functions to find the price \(P^* = 120 - 0.02 \cdot 1500 = 90\) cents per unit, and total cost \(C^* = 60 \cdot 1500 + 25000 = 115000\) cents. Now, we can determine the total profit, which is \( \Pi^* = P^* \cdot Q^* - C^* = 90 \cdot 1500 - 115000 = 50000\) cents, or \$500.

Introduce a Unit Tax and Re-calculate

Because of the 14 cents per unit tax levied by the government, the cost function changes to \(C = 60Q + 1400 + 25000\). Now, we need to recalculate the optimal quantity, price, and profit. Following the same steps as before, we find the new quantity \(Q^{**} = 1375\) units, price \(P^{**} = 97\) cents per unit, and profit \( \Pi^{**} = P^{**} \cdot Q^{**} - C^{**} = 33225\) cents, or \$332.25.

Key concepts

Average Revenue

Average revenue is the income earned by selling one unit of a product, and it is synonymous with the price in the context of this exercise. If you think of a chocolate bar, average revenue is the price a customer pays for one bar. In mathematical terms, it is represented by dividing total revenue by the quantity of goods sold, but with a linear demand function like the one provided, it's directly shown as the price function:

• In this exercise, the average revenue curve is defined by the equation \( P = 120 - 0.02Q \).
• This equation tells us how price (and hence, average revenue) changes with different levels of production, \( Q \).
• A higher \( Q \) leads to a lower average revenue because the firm's demand curve has a negative slope, implying that prices need to fall to sell more units.

Understanding how average revenue relates to quantity produced is essential for profit maximization. Firms aim to produce where their marginal costs and marginal revenues balance for the highest profitability.

Demand Curve

The demand curve visually represents the relationship between the price of a product and the quantity demanded. It shows how consumers respond to price changes. In our scenario, the firm's demand equation \( P = 120 - 0.02Q \) displays a linear demand curve:

• This negative slope indicates that as price decreases, the quantity demanded (produced) increases. Hence, it's always descending from left to right on a graph.
• The equation implies that if the firm wants to sell more, it should lower the price; otherwise, consumers will not purchase additional goods.

The demand curve is critical in understanding market dynamics, providing insights into how changes in prices affect production levels. Firms use it to determine optimal pricing and production strategies to maximize profit.

Cost Function

The cost function represents the total cost associated with producing a certain quantity of goods, including both fixed and variable costs. In our exercise, the firm's cost function is given by

• \( C = 60Q + 25000 \).
• The term \( 60Q \) represents variable costs, which depend directly on the level of production. As \( Q \) (quantity) increases, so does this part of the cost.
• The \( 25000 \) term is a fixed cost, incurred regardless of the production level, like rent or salaries.

Understanding the cost function helps a firm identify cost behavior and determine profitability. It's used to identify which production level minimizes costs, especially when deciding how much to produce or how pricing strategies need to change under different economic conditions.

Unit Tax

A unit tax is a fixed amount that a government taxes on each unit of goods sold, which directly raises production costs. In the exercise, a unit tax of 14 cents was introduced, altering the production landscape:

• This tax increases the cost function to \( C = 60Q + 1400 + 25000 \), adding an additional \( 1400 \) cents to account for the tax across the \( Q \) units.
• The firm calculates the new cost, price, and production levels to adapt to this increase.
• The introduction of a unit tax typically results in higher consumer prices and potentially reduced production levels, reflecting its economic burden.

Taxes like unit tax must be accounted for as they affect both the firm's profitability and the consumer's purchasing power. They are a practical illustration of how external financial policies can impact business decisions.