Question

Suppose a firm's total revenues depend on the amount produced ( \(q\) ) according to the function \\[ T R=70 q-q^{2} \\] Total costs also depend on \(q:\) \\[ T C=q^{2}+30 q+100 \\] a. What level of output should the firm produce in order to maximize profits \((T R-T C) ?\) What will profits be? b. Show that the second-order conditions for a maximum are satisfied at the output level found in part (a). c. Does the solution calculated here obey the "marginal revenue equals marginal cost" rule? Explain.

Short answer

Answer: The output level that maximizes the firm's profit is 10 units, and the corresponding profit amount is $100.

Step-by-step solution

Find Profit Function

Since the profit is the difference between the total revenue and the total cost, we can write the profit function as: \\[ P(q) = TR - TC = (70q - q^2) - (q^2 + 30q + 100) \\] Now, simplify the equation: \\[ P(q) = 70q - q^2 - q^2 - 30q - 100 = -2q^2 + 40q - 100 \\]

Take the First Derivative of the Profit Function

Now, we need to find the critical points at which the profit function has a maximum or minimum value. To do this, we'll take the first derivative of the profit function and set it equal to zero: \\[ \frac{dP}{dq} = -4q + 40 \\] Setting this equal to zero, we get: \\[ -4q + 40 = 0 \\]

Solve for q

Now, we solve for the production level q: \\[ q = \frac{40}{4} = 10 \\]

Take the Second Derivative of the Profit Function

To check if this value of q corresponds to a maximum, we will take the second derivative of the profit function, and plug in the value of q: \\[ \frac{d^2P}{dq^2} = -4 \\] Since the second derivative is negative, the profit function is concave down, and thus q=10 identifies a maximum.

Determine if Solution Obeys "Marginal Revenue Equals Marginal Cost" Rule

Finally, we need to check if the marginal revenue (MR) equals the marginal cost (MC) at q=10. We find the marginal revenue by taking the first derivative of the total revenue function: \\[ \frac{dTR}{dq} = 70 - 2q \\] We find the marginal cost by taking the first derivative of the total cost function: \\[ \frac{dTC}{dq} = 2q + 30 \\] Now, we plug in the value of q=10 into both equations: \\[ MR = 70 - 2(10) = 50 \\ MC = 2(10) + 30 = 50 \\] Since MR equals MC at q=10, the solution obeys the "marginal revenue equals marginal cost" rule. In conclusion, the firm should produce 10 units in order to maximize profits, and at this level, the profit will be P(10) = -2(10)^2 + 40(10) - 100 = $100.

Key concepts

Marginal Revenue

Marginal revenue (MR) is crucial for understanding how much a firm can expect to earn from producing one more unit of a good or service. It represents the change in total revenue from selling an additional unit. Mathematically, it's found by taking the first derivative of the total revenue (TR) function with respect to quantity (q).

In the given exercise, the total revenue function is \( TR = 70q - q^2 \). Calculating the derivative gives us the marginal revenue function: \( \frac{dTR}{dq} = 70 - 2q \). This indicates that for every unit increase in production, the revenue would increase by amounts decreasingly less due to the term \( -2q \).

When firms make decisions about production levels, they aim to equate MR to marginal cost (MC) to maximize profit. Let's further investigate how marginal revenue matches up with marginal cost at the profit-maximizing level of output in the next sections.

Marginal Cost

Marginal cost (MC) is the cost of producing an additional unit of output. This is a cornerstone concept in microeconomics, as it helps determine the most efficient level of production for a firm. To find the MC, one would take the first derivative of the total cost (TC) function with respect to the quantity. For our example, the TC function given is \( TC = q^2 + 30q + 100 \), leading to the marginal cost function: \( \frac{dTC}{dq} = 2q + 30 \).

This equation tells us that as output increases, the cost to produce each additional unit also increases. In other words, MC rises with an increase in the number of goods produced, which reflects the law of increasing costs. Given the marginal costs and revenue, a firm can identify the level of production that maximizes its profits, which is when \( MR = MC \). We'll analyze this equilibrium further, relating it to the optimal output in the given exercise.

Concavity of Profit Function

The concavity of the profit function is useful in determining whether a critical point identified by setting the first derivative to zero is a maximum or a minimum. If the second derivative of the profit function is negative, it indicates that the function is concave down, which typically points to a maximum profit point.

In our exercise, after calculating \( \frac{d^2P}{dq^2} \), we found it to be -4, a negative value, implying that the profit function \( P(q) \) is concave down. Visualizing this, imagine a downward-opening parabola; the highest point on this curve corresponds to the maximum profit figure that the firm can achieve. By confirming the concavity of the profit function, we provide mathematical evidence that the production level \( q = 10 \) units will indeed give the firm its maximum profit.

First Derivative Test

The first derivative test is applied to the profit function to locate the maximum profit point. It involves taking the derivative of the profit function with respect to quantity and setting it equal to zero, thus finding the critical points. For the function \( P(q) \), we determined \( \frac{dP}{dq} = -4q + 40 \).

After setting the derivative equal to zero, we solved \( q \) to be 10. This tells us that at \( q = 10 \) units, there is either a maximum or a minimum profit. To confirm whether it's a maximum, we applied the second derivative test, as discussed earlier. Since the second derivative was negative, it validated that \( q = 10 \) is a maximum profit point. It's crucial to use the first derivative test effectively to identify the production level that maximizes a firm's profit, which is what microeconomic theory advises for profit maximization.