Question

A firm faces the following linear demand for its product, \(Q D=30-P / 2\). The firm has a marginal cost of production given by \(M C=8\). Find the expression for the firm's marginal revenue. Plot the \(M C\) and \(M R\) curves on a graph. What is the amount of output that the firm should produce? At what price is the output sold?

Short answer

The firm should produce 13 units of output and sell at a price of $34 per unit.

Step-by-step solution

Determine Demand Function

The given linear demand function is \(QD = 30 - \frac{P}{2}\). We rearrange this to express \(P\) in terms of \(Q\):\[ P = 60 - 2Q \]

Find Total Revenue and Marginal Revenue

Total Revenue (TR) can be found by multiplying price (P) by quantity (Q):\[ TR = P \times Q = (60 - 2Q) Q = 60Q - 2Q^2 \]Marginal Revenue (MR) is the derivative of Total Revenue with respect to quantity (Q):\[ MR = \frac{d(TR)}{dQ} = \frac{d(60Q - 2Q^2)}{dQ} = 60 - 4Q \]

Set Marginal Revenue Equal to Marginal Cost

To find the profit-maximizing output, set Marginal Revenue (MR) equal to Marginal Cost (MC):\[ 60 - 4Q = 8 \]Solving for \(Q\), we have:\[ 60 - 8 = 4Q \]\[ 52 = 4Q \]\[ Q = 13 \]

Determine Price at Profit-Maximizing Quantity

Substitute \(Q = 13\) back into the price equation from Step 1 to determine the price at this level of output:\[ P = 60 - 2(13) = 60 - 26 = 34 \]

Sketch MR and MC Curves

Plot the Marginal Cost (MC) and Marginal Revenue (MR) curves. \(MC\) is constant at 8, thus a horizontal line \(MC = 8\). The MR curve is \(60 - 4Q\). Both curves intersect at \(Q = 13\), indicating the profit-maximizing output level.

Key concepts

Linear Demand

A linear demand function is a crucial concept in economics. It represents how the quantity demanded of a good changes with its price. The given equation in this problem is:

• \( Q_D = 30 - \frac{P}{2} \)

This simple algebraic expression shows the inverse relationship between price \( P \) and quantity demanded \( Q_D \). As price increases, the quantity demanded decreases and vice versa. This negative slope is typical for most goods. By rearranging the demand equation, we find the price as a function of quantity:

• \( P = 60 - 2Q \)

Here, you can see that for every additional unit of quantity, the price decreases by 2 units. This linear relationship is easy to graph and interpret, making it a common method for depicting demand in economic studies.

Profit Maximization

Profit maximization is a key objective for businesses. It involves finding the level of output where profits are at their highest. To find this point, we use the concepts of total revenue (TR) and marginal revenue (MR).
Total Revenue is calculated as:

• \( TR = P \times Q = (60 - 2Q)Q = 60Q - 2Q^2 \)

Marginal revenue, on the other hand, is the change in total revenue from selling one more unit. It is derived by differentiating the total revenue with respect to quantity:

• \( MR = \frac{d(TR)}{dQ} = 60 - 4Q \)

To maximize profits, set the marginal revenue equal to the marginal cost (MC), which in the problem is constant at 8:

• \(60 - 4Q = 8\)
• Solving this, we find \(Q = 13\).

This value of \(Q\) represents the most profitable level of output for the firm. It’s where the additional cost of producing one more unit equals the additional revenue generated from its sale.

Marginal Cost

Marginal cost (MC) is the cost of producing one more unit of a good. This is a vital component for making decisions about production. In the exercise, marginal cost is given as a constant:

• \( MC = 8 \)

A constant marginal cost implies that each additional unit costs the same to produce. This simplifies calculations since the MC curve on a graph is a horizontal line at \( MC = 8 \). Consistent costs make predicting expenses easier for the firm and are helpful in planning how much to produce.
To maximize profit, the firm sets its output where marginal cost equals marginal revenue. In this case, it happens at \(Q = 13\), meaning this is the quantity where they produce optimally. By understanding MC along with MR, firms can smartly decide their production levels to enhance profitability while keeping costs under control.