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

Without tax, the profit-maximizing production level is 1500 units per week at a price of 90 cents per unit, resulting in a total profit of 20000 cents (or $200). With a tax of 14 cents per unit, the new production level is 1150 units per week at a price of 98 cents per unit, with a total profit of 3100 cents (or $31).

Step-by-step solution

Calculating Marginal Revenue (MR)

The Marginal Revenue (MR) is the derivative of the revenue function. The revenue can be calculated as the product of price and quantity, i.e., \(R = P \cdot Q = (120 - 0.02Q) \cdot Q\). Taking the derivative of this function gets us the marginal revenue function, \(MR = 120 - 0.04Q\).

Calculating Marginal Cost (MC)

The Marginal Cost (MC) is the derivative of the total cost, \(C\), with respect to quantity produced. Therefore, the Marginal cost is the derivative of \(C = 60Q + 25000\), which results in \(MC = 60\).

Find the Profit-maximizing Quantity

To maximize profits, a firm will produce up until the point where MR = MC. Therefore, solving \(120 - 0.04Q = 60\) for \(Q\) gives us the profit maximizing quantity. From this equation, \(Q = 1500\).

Find the Profit-maximizing Price

The corresponding price can still be obtained by substituting \(Q = 1500\) into the price equation \(P = 120 - 0.02Q\). This yields \(P = 90\). (in cents per unit)

Calculating Total Profit

The total profit can be found by the formula: \(Profit = Revenue - Cost\), where \(Revenue = P \cdot Q\) and \(Cost = 60Q + 25000\). Substituting in \(Q = 1500\) and \(P = 90\), we get \(Profit = 90 \cdot 1500 - (60 \cdot 1500 + 25000) = 20000 \) (in cents).

Adjusting for Tax

If the government imposes a tax of 14 cents per unit, the marginal cost increases by the same amount, with MC now equal to 74. Using MR = MC, we find the new quantity \(Q = 1150\). The associated price is obtained by substituting \(Q = 1150\) in the price equation, yielding \(P = 98\).

Calculating Total Profit under Tax

The total profit under tax can be found by using the same formula as in Step 5. Substituting in \(Q = 1150\) and \(P = 98\) yields \n\(Profit = 98 \cdot 1150 - (74 \cdot 1150 + 25000) = 3100 \) (in cents).

Key concepts

Marginal Revenue

Marginal revenue (MR) is a crucial concept in microeconomics representing the additional income earned from selling one more unit of a good or service. It is calculated by taking the derivative of the total revenue function with respect to quantity. In simpler terms, it's the change in revenue that comes from selling one extra item.

For example, if a company sells pens, and by selling one more pen, their revenue increases by \(1.00, then the MR of that pen is \)1.00. Understanding MR is essential because it helps firms decide the number of products to produce and sell to maximize their profits. If MR exceeds the marginal cost of producing an item, the firm profits from selling additional units. However, when MR falls below the marginal cost, it's a signal that producing more will not increase profits.

Marginal Cost

Marginal cost (MC) is the cost of producing an additional unit of product. In the context of our exercise, MC is consistent and is derived from the total cost function. The firm's total cost function is given by the equation \(C = 60Q + 25000\), where \(C\) represents the total cost, \(Q\) is the number of units produced, and 25000 is a fixed cost that does not vary with production.

By taking the derivative with respect to \(Q\), we find that the MC is $60 per unit. This information is essential in profit maximization because a profit-maximizing firm will continue to produce additional units as long as the MR is greater than the MC. Once MR equals MC, the production of additional units would not contribute to increased profits but might start incurring losses instead.

Profit-Maximizing Quantity

Profit-maximizing quantity is the number of units a firm should produce and sell to achieve the highest possible profits. This is found at the point where marginal revenue equals marginal cost, often referred to as MR=MC rule. In our problem, the firm maximizes profit by setting \(MR = MC\) and solving for quantity, \(Q\).

In our exercise, this rule led to finding that the firm should produce 1,500 units per week to maximize profit. The MR=MC rule is a standard guideline because producing fewer units would mean losing potential profits, while producing more would lead to an increase in costs that are not matched by revenue, thus decreasing net profit.

Government Taxation Impact

Government taxation imposes an additional cost on the production of goods, affecting both marginal cost and potentially the price. In microeconomic analysis, a tax per unit can be considered as an increase in marginal cost because it adds to the cost of producing each additional unit. When the government levies a tax, as in our exercise where a tax of 14 cents per unit is imposed, the marginal cost rises by the amount of tax.

After the tax increase, firms will often produce less and charge a higher price to offset the increased costs, which also results in lower total profits. The exercise demonstrated that after the tax imposition, the profit-maximizing quantity and total profit decreased, illustrating the direct impact government policies can have on a firm's production decisions and economics at large.